Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy Topological Systems

Apostolos Syropoulos1113578 Valeria de Paiva1113579

1113578Greek Molecular Computing Group

Xanthi Greece

asyropoulosgmailcom1113579School of Computer ScienceUniversity of Birmingham UK

valeriadepaivagmailcom

8th Panhellenic Logic Symposium

July 7 2011

Ioannina Greece

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Outline

1 Vagueness

General Ideas

Many-valued Logics and Vagueness

Fuzzy Sets and Relations

2 Dialectica Spaces

3 Fuzzy Topological Spaces

4 Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569

CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894

Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Outline

1 Vagueness

General Ideas

Many-valued Logics and Vagueness

Fuzzy Sets and Relations

2 Dialectica Spaces

3 Fuzzy Topological Spaces

4 Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569

CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894

Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569

CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894

Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569

CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894

Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569

CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894

Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569

CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894

Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569

CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894

Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569

CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894

Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569

CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894

Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569

CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894

Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569

CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894

Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569

CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894

Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

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de Paiva

Vagueness

General Ideas

Many-valued Logics and

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Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

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Systems

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de Paiva

Vagueness

General Ideas

Many-valued Logics and

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Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

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de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

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de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

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de Paiva

Vagueness

General Ideas

Many-valued Logics and

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Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

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de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569

CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894

Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extent that it

has borderline cases

What is a borderline case

A case in which it seems possible either to apply or not to

apply a vague term

The adjective ldquotallrdquo is a typical example of a vague term

Examples

Serena is 175m tall is she tall

Is Betelgeuse a huge star

If you cut one head off of a two headed man have you

decapitated him

Eubulides of Miletus The Sorites Paradox

The old puzzle about bald people

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569

CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894

Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569

CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894

Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569

CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894

Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

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Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

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Fuzzy

Topological

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Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

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Systems

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de Paiva

Vagueness

General Ideas

Many-valued Logics and

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Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

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Systems

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Vagueness

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Many-valued Logics and

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Fuzzy

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Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

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Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

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Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

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Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

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Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

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Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

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Systems

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Vagueness

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Many-valued Logics and

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Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

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de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

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de Paiva

Vagueness

General Ideas

Many-valued Logics and

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Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

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de Paiva

Vagueness

General Ideas

Many-valued Logics and

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Dialectica Spaces

Fuzzy

Topological

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Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

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de Paiva

Vagueness

General Ideas

Many-valued Logics and

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Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

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de Paiva

Vagueness

General Ideas

Many-valued Logics and

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Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

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Vagueness

General Ideas

Many-valued Logics and

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Fuzzy

Topological

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Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

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de Paiva

Vagueness

General Ideas

Many-valued Logics and

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Dialectica Spaces

Fuzzy

Topological

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Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

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Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

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de Paiva

Vagueness

General Ideas

Many-valued Logics and

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Dialectica Spaces

Fuzzy

Topological

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Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

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de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

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Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

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Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569

CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894

Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569

CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894

Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation is vague when

the relation of the representing system to the represented

system is not one-one but one-many

Example A small-scale map is usually vaguer than a large-scale

map

Charles Sanders Peirce A proposition is vague when there are

possible states of things concerning which it is intrinsically

uncertain whether had they been contemplated by the

speaker he would have regarded them as excluded or

allowed by the proposition By intrinsically uncertain we

mean not uncertain in consequence of any ignorance of the

interpreter but because the speakerrsquos habits of language were

indeterminate

Max Black demonstrated this definition using the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569

CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894

Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569

CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894

Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569

CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894

Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569

CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894

Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569

CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894

Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569

CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894

Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambibuity and generality are entirely different

notions

A term or phrase is ambiguous if it has at least two specific

meanings that make sense in context

Example He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a general

conceptmdasheg lsquoThis is a manrsquomdashwill be verified by a number of

facts such as lsquoThisrsquo being Brown or Jones or Robinsonrdquo

Example Again consider the word chair

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569

CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894

Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569

CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894

Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569

CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894

Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569

CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894

Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569

CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894

Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

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de Paiva

Vagueness

General Ideas

Many-valued Logics and

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Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

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de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

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de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

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de Paiva

Vagueness

General Ideas

Many-valued Logics and

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Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

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Many-valued Logics and

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Fuzzy

Topological

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Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

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Systems

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Vagueness

General Ideas

Many-valued Logics and

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Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

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Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

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Systems

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Vagueness

General Ideas

Many-valued Logics and

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Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

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Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

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Systems

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de Paiva

Vagueness

General Ideas

Many-valued Logics and

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Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

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Systems

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Vagueness

General Ideas

Many-valued Logics and

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Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569

CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894

Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894

Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the

problem of future contingencies (ie what is the truth value of

a proposition about a future event)

Can someone tell us what is the truth value of the proposition

ldquoThe Higgs boson existsrdquo

Jan Łukasiewicz defined a three-valued logic to deal with future

contingencies

Emil Leon Post introduced the formulation of 119899 truth values

where 119899 ge 1113569CC Chang proposed MV-algebras that is an algebraic model

of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy set theory and

its accompanying fuzzy logic

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894

Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894

Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

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de Paiva

Vagueness

General Ideas

Many-valued Logics and

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Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

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de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

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de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

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de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

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de Paiva

Vagueness

General Ideas

Many-valued Logics and

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Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

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Many-valued Logics and

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Fuzzy

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Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

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Systems

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de Paiva

Vagueness

General Ideas

Many-valued Logics and

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Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

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Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

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de Paiva

Vagueness

General Ideas

Many-valued Logics and

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Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

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Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

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Systems

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de Paiva

Vagueness

General Ideas

Many-valued Logics and

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Dialectica Spaces

Fuzzy

Topological

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Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

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Systems

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Vagueness

General Ideas

Many-valued Logics and

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Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894

Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894

Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

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de Paiva

Vagueness

General Ideas

Many-valued Logics and

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Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

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Systems

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de Paiva

Vagueness

General Ideas

Many-valued Logics and

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Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

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de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

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de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

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de Paiva

Vagueness

General Ideas

Many-valued Logics and

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Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

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de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Fuzzy (Sub)sets

Fuzzy set theory is based on the idea that elements may

belong to a set to a certain degree

Given a universe 119883 a fuzzy subset 119860 of 119883 is characterized by

a function 119860 ∶ 119883 rarr 1113690 1113690 = [1113567 1113568] where 119860(119909) = 119894 means that 119909belongs to 119860 with degree equal to 119894Example One can easily build a fuzzy subset consisting of all

the tall students of a class

Assume that 119880 and 119883 are nonempty sets A binary fuzzy

relation 119877 in 119880 and 119883 is a fuzzy subset of 119880 times 119883 The value

of 119877(119906 119909) is interpreted as the degree of membership of the

ordered pair (119906 119909) in 119877

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematical theory

that unifies all branches of mathematics

Categories have been used to model and study logical systems

Dialectica spaces which were introduced by de Paiva form a

category which is a model of linear logic

Dialectica categories have been used to model Petri nets the

Lambek Calculus state in programming and to define fuzzy

petri nets

Recent development fuzzy topological systems that is the

fuzzy counterpart of Vickersrsquos topological systems

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

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Fuzzy

Topological

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Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

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de Paiva

Vagueness

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Many-valued Logics and

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Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

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Systems

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Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

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Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

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Systems

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de Paiva

Vagueness

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Fuzzy

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Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

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de Paiva

Vagueness

General Ideas

Many-valued Logics and

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Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

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de Paiva

Vagueness

General Ideas

Many-valued Logics and

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Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

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Vagueness

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Many-valued Logics and

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Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

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de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

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Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

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Systems

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de Paiva

Vagueness

General Ideas

Many-valued Logics and

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Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

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Systems

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de Paiva

Vagueness

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Many-valued Logics and

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Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

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Vagueness

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Many-valued Logics and

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Dialectica Spaces

Fuzzy

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Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

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de Paiva

Vagueness

General Ideas

Many-valued Logics and

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Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

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Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

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de Paiva

Vagueness

General Ideas

Many-valued Logics and

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Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Lineales

A quintuple (119871 le ∘ 1113568⊸) is a lineale if

(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication such that

(119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887 thenthis element is denoted 119886 ⊸ 119887 and is called the

pseudo-complement of 119886 with respect to 119887

The quintuple (1113690 le and 1113568rArr) where 1113690 is the unit interval

119886 and 119887 = 111361211136081113613119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 111361211136001113623119886 119887) is a lineale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are sets and 120572is a map 119880 times 119883 rarr 1113690

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is a pair

of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883 such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 11988411136081113603119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times 11136081113603119884

120573- 1113690

120572

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the following arrows

(119880 119883 120572)(119891119892)⟶ (119881 119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860 otimes 119861 = (119880 times 119881 119883119881 times 119884119880 120572 times 120573) where 120572 times 120573 is the

relation that using the lineale structure of 119868 takes theminimum of the membership degrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573) whereagain the relation 120572 rarr 120573 is given by the implication in

the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr 1113690 is the fuzzy relation that is

defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 1113567)120573(119907 119910) if 119911 = (119910 1113568)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is defined similar

to 120574

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Topological Systems

A triple (119880 ⊧ 119883) where 119883 is a frame and 119880 is a set is a

topological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114003119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114005119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 120572 119883) where 119883 is a frame 119880 is a set and 120572 ∶ 119880 times 119883 rarr 1113690a binary fuzzy relation is a fuzzy topological system if

when 119878 is a finite subset of 119883 then

120572(1199061114003119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114005119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1113568 and 120572(119906 perp) = 1113567 for all 119906 isin 119880

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzy topological

systems and the arrows between them form the category

119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883) is a fuzzy topological system

(119880 120580 119883) where

120580(119906 119909) = 11141081113568 when 119906 ⊧ 1199091113567 otherwise

The category of topological systems is a full subcategory of

1202271202581202501202611113716(119826119838119853)

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 120572 119883) is a fuzzy topological

space Then the extent of an open 119909 is a function whose

graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by the extents of

the members of 119860 correspond to a fuzzy topology on 119883

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bit stream

and elements of 119883 are assertions about bit streams

Example If 119906 generates the bit stream 010101010101hellip and

ldquostarts 01010rdquo isin 119883 then 119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program that produces bit

streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of some

interaction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo to some

degree

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

Fuzzy

Topological

Systems

Syropoulos

de Paiva

Vagueness

General Ideas

Many-valued Logics and

Vagueness

Fuzzy Sets and Relations

Dialectica Spaces

Fuzzy

Topological

Spaces

Finale

Finale

We have briefly presented

The notion of vagueness

Dialectica categories

Fuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • Fuzzy Sets and Relations
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale