CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

Categorical Propositions

PHIL UA-70: LogicApril 7–9, 2015

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CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

OUTLINE

CATEGORICAL PROPOSITIONS

THE COMPONENTS OF A CATEGORICAL PROPOSITION

QUALITY, QUANTITY, AND DISTRIBUTION

THE SQUARE OF OPPOSITION

THE TRADITIONAL SQUARE

THE MODERN SQUARE

CATEGORICAL SYLLOGISMS

MAJOR TERMS, MINOR TERMS, MIDDLE TERMS

MOOD AND FIGURE

VENN DIAGRAMS

REPRESENTING STANDARD FORM CATEGORICAL

PROPOSITIONS WITH VENN DIAGRAMS

TESTING THE VALIDITY OF CATEGORICAL SYLLOGISMS

WITH VENN DIAGRAMS

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

A BRIEF HISTORICAL INTERLUDE

▶ Before looking at the system of logic known asQuantificational Logic, or ‘QL’, we’ll look at the ancient andmedieval system of logic which studied categoricalsyllogisms.

▶ Two goals:

▶ Gain an informal understanding of what’s going on in QL

▶ In particular, how that theory understands ‘all’ and ‘some’

▶ come to appreciate just how much of an improvement QLrepresents.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

A BRIEF HISTORICAL INTERLUDE

▶ Before looking at the system of logic known asQuantificational Logic, or ‘QL’, we’ll look at the ancient andmedieval system of logic which studied categoricalsyllogisms.

▶ Two goals:

▶ Gain an informal understanding of what’s going on in QL

▶ In particular, how that theory understands ‘all’ and ‘some’

▶ come to appreciate just how much of an improvement QLrepresents.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

A BRIEF HISTORICAL INTERLUDE

▶ Before looking at the system of logic known asQuantificational Logic, or ‘QL’, we’ll look at the ancient andmedieval system of logic which studied categoricalsyllogisms.

▶ Two goals:▶ Gain an informal understanding of what’s going on in QL

▶ In particular, how that theory understands ‘all’ and ‘some’▶ come to appreciate just how much of an improvement QL

represents.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

A BRIEF HISTORICAL INTERLUDE

▶ Before looking at the system of logic known asQuantificational Logic, or ‘QL’, we’ll look at the ancient andmedieval system of logic which studied categoricalsyllogisms.

▶ Two goals:▶ Gain an informal understanding of what’s going on in QL

▶ In particular, how that theory understands ‘all’ and ‘some’

▶ come to appreciate just how much of an improvement QLrepresents.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

A BRIEF HISTORICAL INTERLUDE

▶ Before looking at the system of logic known asQuantificational Logic, or ‘QL’, we’ll look at the ancient andmedieval system of logic which studied categoricalsyllogisms.

▶ Two goals:▶ Gain an informal understanding of what’s going on in QL

▶ In particular, how that theory understands ‘all’ and ‘some’▶ come to appreciate just how much of an improvement QL

represents.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

OUTLINE

CATEGORICAL PROPOSITIONS

THE COMPONENTS OF A CATEGORICAL PROPOSITION

QUALITY, QUANTITY, AND DISTRIBUTION

THE SQUARE OF OPPOSITION

THE TRADITIONAL SQUARE

THE MODERN SQUARE

CATEGORICAL SYLLOGISMS

MAJOR TERMS, MINOR TERMS, MIDDLE TERMS

MOOD AND FIGURE

VENN DIAGRAMS

REPRESENTING STANDARD FORM CATEGORICAL

PROPOSITIONS WITH VENN DIAGRAMS

TESTING THE VALIDITY OF CATEGORICAL SYLLOGISMS

WITH VENN DIAGRAMS

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

CATEGORICAL PROPOSITIONS

▶ examples of categorical propositions:

1. Everyone loves a Georgia peach.2. No animal eats its young.3. Somebody loves Kesha.4. Some hats are not fashionable.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

CATEGORICAL PROPOSITIONS

▶ examples of categorical propositions:1. Everyone loves a Georgia peach.

2. No animal eats its young.3. Somebody loves Kesha.4. Some hats are not fashionable.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

CATEGORICAL PROPOSITIONS

▶ examples of categorical propositions:1. Everyone loves a Georgia peach.2. No animal eats its young.

3. Somebody loves Kesha.4. Some hats are not fashionable.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

CATEGORICAL PROPOSITIONS

▶ examples of categorical propositions:1. Everyone loves a Georgia peach.2. No animal eats its young.3. Somebody loves Kesha.

4. Some hats are not fashionable.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

CATEGORICAL PROPOSITIONS

▶ examples of categorical propositions:1. Everyone loves a Georgia peach.2. No animal eats its young.3. Somebody loves Kesha.4. Some hats are not fashionable.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

CATEGORICAL PROPOSITIONS

▶ They say something about how the members of one classor category are related to the members of another class orcategory.

1. Everything in the class of people is contained in the class ofthings that love a Georgia peach.

2. Nothing in the class of animals is contained in the class ofthings that eat their young.

3. Something in the class of people is contained in the class ofthings that love Kesha.

4. Something in the class of hats is not contained in the classof things that are fashionable clothing.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

CATEGORICAL PROPOSITIONS

▶ They say something about how the members of one classor category are related to the members of another class orcategory.

1. Everything in the class of people is contained in the class ofthings that love a Georgia peach.

2. Nothing in the class of animals is contained in the class ofthings that eat their young.

3. Something in the class of people is contained in the class ofthings that love Kesha.

4. Something in the class of hats is not contained in the classof things that are fashionable clothing.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

CATEGORICAL PROPOSITIONS

▶ They say something about how the members of one classor category are related to the members of another class orcategory.

1. Everything in the class of people is contained in the class ofthings that love a Georgia peach.

2. Nothing in the class of animals is contained in the class ofthings that eat their young.

3. Something in the class of people is contained in the class ofthings that love Kesha.

4. Something in the class of hats is not contained in the classof things that are fashionable clothing.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

CATEGORICAL PROPOSITIONS

▶ They say something about how the members of one classor category are related to the members of another class orcategory.

1. Everything in the class of people is contained in the class ofthings that love a Georgia peach.

2. Nothing in the class of animals is contained in the class ofthings that eat their young.

3. Something in the class of people is contained in the class ofthings that love Kesha.

4. Something in the class of hats is not contained in the classof things that are fashionable clothing.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

CATEGORICAL PROPOSITIONS

▶ They say something about how the members of one classor category are related to the members of another class orcategory.

1. Everything in the class of people is contained in the class ofthings that love a Georgia peach.

2. Nothing in the class of animals is contained in the class ofthings that eat their young.

3. Something in the class of people is contained in the class ofthings that love Kesha.

4. Something in the class of hats is not contained in the classof things that are fashionable clothing.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

STANDARD FORM CATEGORICAL PROPOSITIONS

▶ Four standard form categorical propositions:

A) All S are P

All people are Georgia peach-lovers.

E) No S are P

No animals are young-eaters.

I) Some S are P

Some people are Kesha-lovers.

O) Some S are not P

Some hats are not fashionable clothing.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

STANDARD FORM CATEGORICAL PROPOSITIONS

▶ Four standard form categorical propositions:A) All S are P

All people are Georgia peach-lovers.E) No S are P

No animals are young-eaters.

I) Some S are P

Some people are Kesha-lovers.

O) Some S are not P

Some hats are not fashionable clothing.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

STANDARD FORM CATEGORICAL PROPOSITIONS

▶ Four standard form categorical propositions:A) All S are P

All people are Georgia peach-lovers.

E) No S are P

No animals are young-eaters.I) Some S are P

Some people are Kesha-lovers.

O) Some S are not P

Some hats are not fashionable clothing.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

STANDARD FORM CATEGORICAL PROPOSITIONS

▶ Four standard form categorical propositions:A) All S are P

All people are Georgia peach-lovers.

E) No S are P

No animals are young-eaters.

I) Some S are P

Some people are Kesha-lovers.O) Some S are not P

Some hats are not fashionable clothing.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

STANDARD FORM CATEGORICAL PROPOSITIONS

▶ Four standard form categorical propositions:A) All S are P

All people are Georgia peach-lovers.

E) No S are P

No animals are young-eaters.

I) Some S are P

Some people are Kesha-lovers.

O) Some S are not P

Some hats are not fashionable clothing.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

STANDARD FORM CATEGORICAL PROPOSITIONS

▶ Four standard form categorical propositions:A) All S are P All people are Georgia peach-lovers.E) No S are P

No animals are young-eaters.

I) Some S are P

Some people are Kesha-lovers.

O) Some S are not P

Some hats are not fashionable clothing.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

STANDARD FORM CATEGORICAL PROPOSITIONS

▶ Four standard form categorical propositions:A) All S are P All people are Georgia peach-lovers.E) No S are P No animals are young-eaters.I) Some S are P

Some people are Kesha-lovers.

O) Some S are not P

Some hats are not fashionable clothing.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

STANDARD FORM CATEGORICAL PROPOSITIONS

▶ Four standard form categorical propositions:A) All S are P All people are Georgia peach-lovers.E) No S are P No animals are young-eaters.I) Some S are P Some people are Kesha-lovers.

O) Some S are not P

Some hats are not fashionable clothing.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

STANDARD FORM CATEGORICAL PROPOSITIONS

▶ Four standard form categorical propositions:A) All S are P All people are Georgia peach-lovers.E) No S are P No animals are young-eaters.I) Some S are P Some people are Kesha-lovers.

O) Some S are not P Some hats are not fashionable clothing.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

OUTLINE

CATEGORICAL PROPOSITIONS

THE COMPONENTS OF A CATEGORICAL PROPOSITION

QUALITY, QUANTITY, AND DISTRIBUTION

THE SQUARE OF OPPOSITION

THE TRADITIONAL SQUARE

THE MODERN SQUARE

CATEGORICAL SYLLOGISMS

MAJOR TERMS, MINOR TERMS, MIDDLE TERMS

MOOD AND FIGURE

VENN DIAGRAMS

REPRESENTING STANDARD FORM CATEGORICAL

PROPOSITIONS WITH VENN DIAGRAMS

TESTING THE VALIDITY OF CATEGORICAL SYLLOGISMS

WITH VENN DIAGRAMS

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE COMPONENTS OF A CATEGORICAL PROPOSITION

▶ A categorical proposition has four components:

1. A quantifier—either ‘All’, ‘No’, or ‘Some’2. A subject term—the thing denoted in A,E, I, and O above

with an ‘S’3. A copula—either ‘are’ or ‘are not’4. A predicate term—the thing denoted in A,E, I, and O above

with a ‘P’

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE COMPONENTS OF A CATEGORICAL PROPOSITION

▶ A categorical proposition has four components:1. A quantifier—either ‘All’, ‘No’, or ‘Some’

2. A subject term—the thing denoted in A,E, I, and O abovewith an ‘S’

3. A copula—either ‘are’ or ‘are not’4. A predicate term—the thing denoted in A,E, I, and O above

with a ‘P’

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE COMPONENTS OF A CATEGORICAL PROPOSITION

▶ A categorical proposition has four components:1. A quantifier—either ‘All’, ‘No’, or ‘Some’2. A subject term—the thing denoted in A,E, I, and O above

with an ‘S’

3. A copula—either ‘are’ or ‘are not’4. A predicate term—the thing denoted in A,E, I, and O above

with a ‘P’

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE COMPONENTS OF A CATEGORICAL PROPOSITION

▶ A categorical proposition has four components:1. A quantifier—either ‘All’, ‘No’, or ‘Some’2. A subject term—the thing denoted in A,E, I, and O above

with an ‘S’3. A copula—either ‘are’ or ‘are not’

4. A predicate term—the thing denoted in A,E, I, and O abovewith a ‘P’

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE COMPONENTS OF A CATEGORICAL PROPOSITION

▶ A categorical proposition has four components:1. A quantifier—either ‘All’, ‘No’, or ‘Some’2. A subject term—the thing denoted in A,E, I, and O above

with an ‘S’3. A copula—either ‘are’ or ‘are not’4. A predicate term—the thing denoted in A,E, I, and O above

with a ‘P’

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE COMPONENTS OF A CATEGORICAL PROPOSITION

All︸︷︷︸quantifier

people︸ ︷︷ ︸subject term

are︸︷︷︸copula

Georgia peach lovers︸ ︷︷ ︸predicate term

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE COMPONENTS OF A CATEGORICAL PROPOSITION

No︸︷︷︸quantifier

animals︸ ︷︷ ︸subject term

are︸︷︷︸copula

young-eaters︸ ︷︷ ︸predicate term

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE COMPONENTS OF A CATEGORICAL PROPOSITION

Some︸ ︷︷ ︸quantifier

people︸ ︷︷ ︸subject term

are︸︷︷︸copula

Kesha-lovers︸ ︷︷ ︸predicate term

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE COMPONENTS OF A CATEGORICAL PROPOSITION

Some︸ ︷︷ ︸quantifier

hats︸︷︷︸subject term

are not︸ ︷︷ ︸copula

fashionable clothing︸ ︷︷ ︸predicate term

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE COMPONENTS OF A CATEGORICAL PROPOSITION

▶ ‘subject term’ and ‘predicate term’ as they are being usedhere need not line up with the grammatical subject and thegrammatical predicate.

▶ ‘Some S are P’ does not imply that some S are not P.

▶ ‘Everyone loves Kesha’ and ‘Someone loves Kesha’ areconsistent.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE COMPONENTS OF A CATEGORICAL PROPOSITION

▶ ‘subject term’ and ‘predicate term’ as they are being usedhere need not line up with the grammatical subject and thegrammatical predicate.

▶ ‘Some S are P’ does not imply that some S are not P.

▶ ‘Everyone loves Kesha’ and ‘Someone loves Kesha’ areconsistent.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE COMPONENTS OF A CATEGORICAL PROPOSITION

▶ ‘subject term’ and ‘predicate term’ as they are being usedhere need not line up with the grammatical subject and thegrammatical predicate.

▶ ‘Some S are P’ does not imply that some S are not P.▶ ‘Everyone loves Kesha’ and ‘Someone loves Kesha’ are

consistent.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

OUTLINE

CATEGORICAL PROPOSITIONS

THE COMPONENTS OF A CATEGORICAL PROPOSITION

QUALITY, QUANTITY, AND DISTRIBUTION

THE SQUARE OF OPPOSITION

THE TRADITIONAL SQUARE

THE MODERN SQUARE

CATEGORICAL SYLLOGISMS

MAJOR TERMS, MINOR TERMS, MIDDLE TERMS

MOOD AND FIGURE

VENN DIAGRAMS

REPRESENTING STANDARD FORM CATEGORICAL

PROPOSITIONS WITH VENN DIAGRAMS

TESTING THE VALIDITY OF CATEGORICAL SYLLOGISMS

WITH VENN DIAGRAMS

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

QUALITY, QUANTITY, AND DISTRIBUTION

▶ three properties of categorical propositions:

▶ their quality (either affirmative or negative)▶ their quantity (either universal or particular)▶ whether or not they distribute their subject or predicate

terms

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

QUALITY, QUANTITY, AND DISTRIBUTION

▶ three properties of categorical propositions:▶ their quality (either affirmative or negative)

▶ their quantity (either universal or particular)▶ whether or not they distribute their subject or predicate

terms

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

QUALITY, QUANTITY, AND DISTRIBUTION

▶ three properties of categorical propositions:▶ their quality (either affirmative or negative)▶ their quantity (either universal or particular)

▶ whether or not they distribute their subject or predicateterms

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

QUALITY, QUANTITY, AND DISTRIBUTION

▶ three properties of categorical propositions:▶ their quality (either affirmative or negative)▶ their quantity (either universal or particular)▶ whether or not they distribute their subject or predicate

terms

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

QUALITY

▶ In both of the claims1) All people are Georgia peach lovers.3) Some people are Kesha-lovers.

membership in the predicate class is affirmed.

▶ In both of the claims

2) No animals are young-eaters.4) Some hats are not fashionable clothing.

membership in the predicate class is denied

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

QUALITY

▶ In both of the claims1) All people are Georgia peach lovers.3) Some people are Kesha-lovers.

membership in the predicate class is affirmed.

▶ In both of the claims

2) No animals are young-eaters.4) Some hats are not fashionable clothing.

membership in the predicate class is denied

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

QUALITY

▶ In both of the claims1) All people are Georgia peach lovers.3) Some people are Kesha-lovers.

membership in the predicate class is affirmed.▶ In both of the claims

2) No animals are young-eaters.4) Some hats are not fashionable clothing.

membership in the predicate class is denied

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

QUALITY

▶ In both of the claims1) All people are Georgia peach lovers.3) Some people are Kesha-lovers.

membership in the predicate class is affirmed.▶ In both of the claims

2) No animals are young-eaters.4) Some hats are not fashionable clothing.

membership in the predicate class is denied

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

QUALITY

▶ If membership in the predicate class if affirmed, then thecategorical proposition has affirmative quality.

▶ If membership in the predicate class is denied, then thecategorical proposition has negative quality.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

QUALITY

▶ If membership in the predicate class if affirmed, then thecategorical proposition has affirmative quality.

▶ If membership in the predicate class is denied, then thecategorical proposition has negative quality.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

QUALITY

▶ A and I standard form categorical propositions haveaffirmative quality.

A) All S are P.I) Some S are P.

▶ E and O standard form categorical propositions havenegative quality.

E) No S are P.O) Some S is not P

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

QUALITY

▶ A and I standard form categorical propositions haveaffirmative quality.

A) All S are P.

I) Some S are P.

▶ E and O standard form categorical propositions havenegative quality.

E) No S are P.O) Some S is not P

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

QUALITY

▶ A and I standard form categorical propositions haveaffirmative quality.

A) All S are P.I) Some S are P.

▶ E and O standard form categorical propositions havenegative quality.

E) No S are P.O) Some S is not P

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

QUALITY

▶ A and I standard form categorical propositions haveaffirmative quality.

A) All S are P.I) Some S are P.

▶ E and O standard form categorical propositions havenegative quality.

E) No S are P.O) Some S is not P

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

QUALITY

▶ A and I standard form categorical propositions haveaffirmative quality.

A) All S are P.I) Some S are P.

▶ E and O standard form categorical propositions havenegative quality.

E) No S are P.

O) Some S is not P

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

QUALITY

▶ A and I standard form categorical propositions haveaffirmative quality.

A) All S are P.I) Some S are P.

▶ E and O standard form categorical propositions havenegative quality.

E) No S are P.O) Some S is not P

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

QUANTITY

▶ In both of the claims1) All people are Georgia peach lovers.2) No animals are young-eaters.

we are told something about all of the things in the subjectclass.

▶ In both of the claims

3) Some people are Kesha-lovers.4) Some hats are not fashionable clothing.

We are told something only about some of the things in thesubject class.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

QUANTITY

▶ In both of the claims1) All people are Georgia peach lovers.2) No animals are young-eaters.

we are told something about all of the things in the subjectclass.

▶ In both of the claims

3) Some people are Kesha-lovers.4) Some hats are not fashionable clothing.

We are told something only about some of the things in thesubject class.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

QUANTITY

▶ In both of the claims1) All people are Georgia peach lovers.2) No animals are young-eaters.

we are told something about all of the things in the subjectclass.

▶ In both of the claims3) Some people are Kesha-lovers.4) Some hats are not fashionable clothing.

We are told something only about some of the things in thesubject class.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

QUANTITY

▶ In both of the claims1) All people are Georgia peach lovers.2) No animals are young-eaters.

we are told something about all of the things in the subjectclass.

▶ In both of the claims3) Some people are Kesha-lovers.4) Some hats are not fashionable clothing.

We are told something only about some of the things in thesubject class.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

QUANTITY

▶ If a standard form categorical proposition says somethingabout all of the members of the subject class, then it hasuniversal quantity.

▶ If a standard form categorical proposition says somethingabout only some of the members of the subject class, then ithas particular quantity.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

QUANTITY

▶ If a standard form categorical proposition says somethingabout all of the members of the subject class, then it hasuniversal quantity.

▶ If a standard form categorical proposition says somethingabout only some of the members of the subject class, then ithas particular quantity.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

QUANTITY

▶ A and E have universal quantity.

A) All S are P.E) No S are P.

▶ I and O have particular quantity.

I) Some S are PO) Some S are not P

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

QUANTITY

▶ A and E have universal quantity.A) All S are P.

E) No S are P.

▶ I and O have particular quantity.

I) Some S are PO) Some S are not P

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

QUANTITY

▶ A and E have universal quantity.A) All S are P.E) No S are P.

▶ I and O have particular quantity.

I) Some S are PO) Some S are not P

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

QUANTITY

▶ A and E have universal quantity.A) All S are P.E) No S are P.

▶ I and O have particular quantity.

I) Some S are PO) Some S are not P

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

QUANTITY

▶ A and E have universal quantity.A) All S are P.E) No S are P.

▶ I and O have particular quantity.I) Some S are P

O) Some S are not P

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

QUANTITY

▶ A and E have universal quantity.A) All S are P.E) No S are P.

▶ I and O have particular quantity.I) Some S are P

O) Some S are not P

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

QUALITY & QUANTITY

Affirmative Negativen

Universal A Eff g

Particular I Ormo

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

DISTRIBUTION

▶ Quality and Quantity are properties of categoricalpropositions.

▶ Distribution is a property of the subject and predicate termsappearing in categorical propositions.

▶ A class term is distributed in a categorical proposition ifand only if the proposition tells you something about everymember of that class.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

DISTRIBUTION

▶ Quality and Quantity are properties of categoricalpropositions.

▶ Distribution is a property of the subject and predicate termsappearing in categorical propositions.

▶ A class term is distributed in a categorical proposition ifand only if the proposition tells you something about everymember of that class.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

DISTRIBUTION

▶ Quality and Quantity are properties of categoricalpropositions.

▶ Distribution is a property of the subject and predicate termsappearing in categorical propositions.

▶ A class term is distributed in a categorical proposition ifand only if the proposition tells you something about everymember of that class.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

DISTRIBUTION—A

A) All S are P.

▶ S is distributed.▶ P is not distributed.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

DISTRIBUTION—A

A) All S are P.▶ S is distributed.

▶ P is not distributed.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

DISTRIBUTION—A

A) All S are P.▶ S is distributed.▶ P is not distributed.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

DISTRIBUTION—E

E) No S are P.

▶ S is distributed.▶ P is distributed.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

DISTRIBUTION—E

E) No S are P.▶ S is distributed.

▶ P is distributed.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

DISTRIBUTION—E

E) No S are P.▶ S is distributed.▶ P is distributed.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

DISTRIBUTION—I

I) Some S are P

▶ S is not distributed.▶ P is not distributed.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

DISTRIBUTION—I

I) Some S are P▶ S is not distributed.

▶ P is not distributed.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

DISTRIBUTION—I

I) Some S are P▶ S is not distributed.▶ P is not distributed.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

DISTRIBUTION—O

O) Some S are not P

▶ S is not distributed.▶ P is distributed.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

DISTRIBUTION—O

O) Some S are not P▶ S is not distributed.

▶ P is distributed.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

DISTRIBUTION—O

O) Some S are not P▶ S is not distributed.▶ P is distributed.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

DISTRIBUTION

Distributed?S P

A: All S are P ✓ ×E: No S are P ✓ ✓I: Some S are P × ×O: Some S are not P × ✓

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

OUTLINE

CATEGORICAL PROPOSITIONS

THE COMPONENTS OF A CATEGORICAL PROPOSITION

QUALITY, QUANTITY, AND DISTRIBUTION

THE SQUARE OF OPPOSITION

THE TRADITIONAL SQUARE

THE MODERN SQUARE

CATEGORICAL SYLLOGISMS

MAJOR TERMS, MINOR TERMS, MIDDLE TERMS

MOOD AND FIGURE

VENN DIAGRAMS

REPRESENTING STANDARD FORM CATEGORICAL

PROPOSITIONS WITH VENN DIAGRAMS

TESTING THE VALIDITY OF CATEGORICAL SYLLOGISMS

WITH VENN DIAGRAMS

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

OUTLINE

CATEGORICAL PROPOSITIONS

THE COMPONENTS OF A CATEGORICAL PROPOSITION

QUALITY, QUANTITY, AND DISTRIBUTION

THE SQUARE OF OPPOSITION

THE TRADITIONAL SQUARE

THE MODERN SQUARE

CATEGORICAL SYLLOGISMS

MAJOR TERMS, MINOR TERMS, MIDDLE TERMS

MOOD AND FIGURE

VENN DIAGRAMS

REPRESENTING STANDARD FORM CATEGORICAL

PROPOSITIONS WITH VENN DIAGRAMS

TESTING THE VALIDITY OF CATEGORICAL SYLLOGISMS

WITH VENN DIAGRAMS

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE LOGICAL RELATIONS BETWEEN A, E, I, AND O

The traditional view:▶ A and E cannot both be true.

▶ That is, A and E are contraries.

▶ A and O necessarily have different truth-values (if A istrue, then O is false, and if A is false, then O is true).

▶ That is, A and O are contradictories.

▶ I and E necessarily have different truth-values (if I is true,then E is false, and if I is false, then E is true).

▶ That is, I and E are contradictories.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE LOGICAL RELATIONS BETWEEN A, E, I, AND O

The traditional view:▶ A and E cannot both be true.

▶ That is, A and E are contraries.

▶ A and O necessarily have different truth-values (if A istrue, then O is false, and if A is false, then O is true).

▶ That is, A and O are contradictories.

▶ I and E necessarily have different truth-values (if I is true,then E is false, and if I is false, then E is true).

▶ That is, I and E are contradictories.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE LOGICAL RELATIONS BETWEEN A, E, I, AND O

The traditional view:▶ A and E cannot both be true.

▶ That is, A and E are contraries.

▶ A and O necessarily have different truth-values (if A istrue, then O is false, and if A is false, then O is true).

▶ That is, A and O are contradictories.

▶ I and E necessarily have different truth-values (if I is true,then E is false, and if I is false, then E is true).

▶ That is, I and E are contradictories.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE LOGICAL RELATIONS BETWEEN A, E, I, AND O

The traditional view:▶ A and E cannot both be true.

▶ That is, A and E are contraries.

▶ A and O necessarily have different truth-values (if A istrue, then O is false, and if A is false, then O is true).

▶ That is, A and O are contradictories.

▶ I and E necessarily have different truth-values (if I is true,then E is false, and if I is false, then E is true).

▶ That is, I and E are contradictories.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE LOGICAL RELATIONS BETWEEN A, E, I, AND O

The traditional view:▶ A and E cannot both be true.

▶ That is, A and E are contraries.

▶ A and O necessarily have different truth-values (if A istrue, then O is false, and if A is false, then O is true).

▶ That is, A and O are contradictories.

▶ I and E necessarily have different truth-values (if I is true,then E is false, and if I is false, then E is true).

▶ That is, I and E are contradictories.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE LOGICAL RELATIONS BETWEEN A, E, I, AND O

The traditional view:▶ A and E cannot both be true.

▶ That is, A and E are contraries.

▶ A and O necessarily have different truth-values (if A istrue, then O is false, and if A is false, then O is true).

▶ That is, A and O are contradictories.

▶ I and E necessarily have different truth-values (if I is true,then E is false, and if I is false, then E is true).

▶ That is, I and E are contradictories.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE TRADITIONAL SQUARE OF OPPOSITION

AAll S are P

ENo S are P

ISome S are P

OSome S are not P

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE TRADITIONAL SQUARE OF OPPOSITION

AAll S are P

ENo S are P

ISome S are P

OSome S are not P

contraries

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE TRADITIONAL SQUARE OF OPPOSITION

AAll S are P

ENo S are P

ISome S are P

OSome S are not P

contradictories

contraries

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE TRADITIONAL SQUARE OF OPPOSITION

AAll S are P

ENo S are P

ISome S are P

OSome S are not P

contradictories

contradictories

contraries

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE TRADITIONAL SQUARE OF OPPOSITION

AAll S are P

ENo S are P

ISome S are P

OSome S are not P

contradictories

contradictories

contraries

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE TRADITIONAL SQUARE OF OPPOSITION

AAll S are P

ENo S are P

ISome S are P

OSome S are not P

contradictories

contradictories

contraries

subcontraries

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE TRADITIONAL SQUARE OF OPPOSITION

AAll S are P

ENo S are P

ISome S are P

OSome S are not P

contradictories

contradictories

contraries

entails

subcontraries

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE TRADITIONAL SQUARE OF OPPOSITION

AAll S are P

ENo S are P

ISome S are P

OSome S are not P

contradictories

contradictories

contraries

entails

entails

subcontraries

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE TRADITIONAL SQUARE OF OPPOSITION

AAll S are P

ENo S are P

ISome S are P

OSome S are not P

contradictories

contradictories

contraries

entails

entails

subcontraries

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE TRADITIONAL SQUARE OF OPPOSITION

According to the Traditional Square of Opposition, thefollowing inferences are all formally deductively valid:

1. All libertarian presidents are still alive.2. So, some libertarian presidents are still alive.

1. No libertarian presidents are still alive .2. So, some libertarian presidents are not still alive.

1. It is false that some libertarian presidents are still alive.2. So, some libertarian presidents are not still alive.

1. It is false that some libertarian presidents are not still alive.2. So, some libertarian presidents are still alive.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE TRADITIONAL SQUARE OF OPPOSITION

According to the Traditional Square of Opposition, thefollowing inferences are all formally deductively valid:

1. All libertarian presidents are still alive.2. So, some libertarian presidents are still alive.

1. No libertarian presidents are still alive .2. So, some libertarian presidents are not still alive.

1. It is false that some libertarian presidents are still alive.2. So, some libertarian presidents are not still alive.

1. It is false that some libertarian presidents are not still alive.2. So, some libertarian presidents are still alive.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE TRADITIONAL SQUARE OF OPPOSITION

According to the Traditional Square of Opposition, thefollowing inferences are all formally deductively valid:

1. All libertarian presidents are still alive.2. So, some libertarian presidents are still alive.

1. No libertarian presidents are still alive .2. So, some libertarian presidents are not still alive.

1. It is false that some libertarian presidents are still alive.2. So, some libertarian presidents are not still alive.

1. It is false that some libertarian presidents are not still alive.2. So, some libertarian presidents are still alive.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE TRADITIONAL SQUARE OF OPPOSITION

According to the Traditional Square of Opposition, thefollowing inferences are all formally deductively valid:

1. All S are P. (A)2. So, some S are P. (I)

1. No libertarian presidents are still alive .2. So, some libertarian presidents are not still alive.

1. It is false that some libertarian presidents are still alive.2. So, some libertarian presidents are not still alive.

1. It is false that some libertarian presidents are not still alive.2. So, some libertarian presidents are still alive.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE TRADITIONAL SQUARE OF OPPOSITION

According to the Traditional Square of Opposition, thefollowing inferences are all formally deductively valid:

1. All libertarian presidents are still alive.2. So, some libertarian presidents are still alive.

1. No libertarian presidents are still alive .2. So, some libertarian presidents are not still alive.

1. It is false that some libertarian presidents are still alive.2. So, some libertarian presidents are not still alive.

1. It is false that some libertarian presidents are not still alive.2. So, some libertarian presidents are still alive.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE TRADITIONAL SQUARE OF OPPOSITION

According to the Traditional Square of Opposition, thefollowing inferences are all formally deductively valid:

1. All libertarian presidents are still alive.2. So, some libertarian presidents are still alive.

1. No libertarian presidents are still alive .2. So, some libertarian presidents are not still alive.

1. It is false that some libertarian presidents are still alive.2. So, some libertarian presidents are not still alive.

1. It is false that some libertarian presidents are not still alive.2. So, some libertarian presidents are still alive.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE TRADITIONAL SQUARE OF OPPOSITION

According to the Traditional Square of Opposition, thefollowing inferences are all formally deductively valid:

1. All libertarian presidents are still alive.2. So, some libertarian presidents are still alive.

1. No libertarian presidents are still alive .2. So, some libertarian presidents are not still alive.

1. It is false that some libertarian presidents are still alive.2. So, some libertarian presidents are not still alive.

1. It is false that some libertarian presidents are not still alive.2. So, some libertarian presidents are still alive.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE TRADITIONAL SQUARE OF OPPOSITION

According to the Traditional Square of Opposition, thefollowing inferences are all formally deductively valid:

1. All libertarian presidents are still alive.2. So, some libertarian presidents are still alive.

1. No S are P. (E)2. So, some S are not P. (O)

1. It is false that some libertarian presidents are still alive.2. So, some libertarian presidents are not still alive.

1. It is false that some libertarian presidents are not still alive.2. So, some libertarian presidents are still alive.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE TRADITIONAL SQUARE OF OPPOSITION

According to the Traditional Square of Opposition, thefollowing inferences are all formally deductively valid:

1. All libertarian presidents are still alive.2. So, some libertarian presidents are still alive.

1. No libertarian presidents are still alive .2. So, some libertarian presidents are not still alive.

1. It is false that some libertarian presidents are still alive.2. So, some libertarian presidents are not still alive.

1. It is false that some libertarian presidents are not still alive.2. So, some libertarian presidents are still alive.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE TRADITIONAL SQUARE OF OPPOSITION

According to the Traditional Square of Opposition, thefollowing inferences are all formally deductively valid:

1. All libertarian presidents are still alive.2. So, some libertarian presidents are still alive.

1. No libertarian presidents are still alive .2. So, some libertarian presidents are not still alive.

1. It is false that some libertarian presidents are still alive.2. So, some libertarian presidents are not still alive.

1. It is false that some libertarian presidents are not still alive.2. So, some libertarian presidents are still alive.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE TRADITIONAL SQUARE OF OPPOSITION

According to the Traditional Square of Opposition, thefollowing inferences are all formally deductively valid:

1. All libertarian presidents are still alive.2. So, some libertarian presidents are still alive.

1. No libertarian presidents are still alive .2. So, some libertarian presidents are not still alive.

1. It is false that some libertarian presidents are still alive.2. So, some libertarian presidents are not still alive.

1. It is false that some libertarian presidents are not still alive.2. So, some libertarian presidents are still alive.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE TRADITIONAL SQUARE OF OPPOSITION

According to the Traditional Square of Opposition, thefollowing inferences are all formally deductively valid:

1. All libertarian presidents are still alive.2. So, some libertarian presidents are still alive.

1. No libertarian presidents are still alive .2. So, some libertarian presidents are not still alive.

1. It is false that some S are P. (∼I)2. So, some S are not P. (O)

1. It is false that some libertarian presidents are not still alive.2. So, some libertarian presidents are still alive.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE TRADITIONAL SQUARE OF OPPOSITION

According to the Traditional Square of Opposition, thefollowing inferences are all formally deductively valid:

1. All libertarian presidents are still alive.2. So, some libertarian presidents are still alive.

1. No libertarian presidents are still alive .2. So, some libertarian presidents are not still alive.

1. It is false that some libertarian presidents are still alive.2. So, some libertarian presidents are not still alive.

1. It is false that some libertarian presidents are not still alive.2. So, some libertarian presidents are still alive.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE TRADITIONAL SQUARE OF OPPOSITION

According to the Traditional Square of Opposition, thefollowing inferences are all formally deductively valid:

1. All libertarian presidents are still alive.2. So, some libertarian presidents are still alive.

1. No libertarian presidents are still alive .2. So, some libertarian presidents are not still alive.

1. It is false that some libertarian presidents are still alive.2. So, some libertarian presidents are not still alive.

1. It is false that some libertarian presidents are not still alive.2. So, some libertarian presidents are still alive.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE TRADITIONAL SQUARE OF OPPOSITION

According to the Traditional Square of Opposition, thefollowing inferences are all formally deductively valid:

1. All libertarian presidents are still alive.2. So, some libertarian presidents are still alive.

1. No libertarian presidents are still alive .2. So, some libertarian presidents are not still alive.

1. It is false that some libertarian presidents are still alive.2. So, some libertarian presidents are not still alive.

1. It is false that some libertarian presidents are not still alive.2. So, some libertarian presidents are still alive.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE TRADITIONAL SQUARE OF OPPOSITION

According to the Traditional Square of Opposition, thefollowing inferences are all formally deductively valid:

1. All libertarian presidents are still alive.2. So, some libertarian presidents are still alive.

1. No libertarian presidents are still alive .2. So, some libertarian presidents are not still alive.

1. It is false that some libertarian presidents are still alive.2. So, some libertarian presidents are not still alive.

1. It is false that some S are not P. (∼O)2. So, some S are P. (I)

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE TRADITIONAL SQUARE OF OPPOSITION

According to the Traditional Square of Opposition, thefollowing inferences are all formally deductively valid:

1. All libertarian presidents are still alive.2. So, some libertarian presidents are still alive.

1. No libertarian presidents are still alive .2. So, some libertarian presidents are not still alive.

1. It is false that some libertarian presidents are still alive.2. So, some libertarian presidents are not still alive.

1. It is false that some libertarian presidents are not still alive.2. So, some libertarian presidents are still alive.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE TRADITIONAL SQUARE OF OPPOSITION

▶ Even though there are no libertarian presidents,

All libertarian presidents are still alive

and

No libertarian presidents are still alive

cannot both be true (they are contraries).

▶ Likewise,

Some libertarian presidents are still alive

and

Some libertarian presidents are not still alive

cannot both be false (they are subcontraries).

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE TRADITIONAL SQUARE OF OPPOSITION

▶ Even though there are no libertarian presidents,All libertarian presidents are still alive

and

No libertarian presidents are still alive

cannot both be true (they are contraries).

▶ Likewise,

Some libertarian presidents are still alive

and

Some libertarian presidents are not still alive

cannot both be false (they are subcontraries).

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE TRADITIONAL SQUARE OF OPPOSITION

▶ Even though there are no libertarian presidents,All libertarian presidents are still alive

and

No libertarian presidents are still alive

cannot both be true (they are contraries).

▶ Likewise,

Some libertarian presidents are still alive

and

Some libertarian presidents are not still alive

cannot both be false (they are subcontraries).

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE TRADITIONAL SQUARE OF OPPOSITION

▶ Even though there are no libertarian presidents,All libertarian presidents are still alive

andNo libertarian presidents are still alive

cannot both be true (they are contraries).

▶ Likewise,

Some libertarian presidents are still alive

and

Some libertarian presidents are not still alive

cannot both be false (they are subcontraries).

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE TRADITIONAL SQUARE OF OPPOSITION

▶ Even though there are no libertarian presidents,All libertarian presidents are still alive

andNo libertarian presidents are still alive

cannot both be true (they are contraries).

▶ Likewise,

Some libertarian presidents are still alive

and

Some libertarian presidents are not still alive

cannot both be false (they are subcontraries).

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE TRADITIONAL SQUARE OF OPPOSITION

▶ Even though there are no libertarian presidents,All libertarian presidents are still alive

andNo libertarian presidents are still alive

cannot both be true (they are contraries).

▶ Likewise,

Some libertarian presidents are still alive

and

Some libertarian presidents are not still alive

cannot both be false (they are subcontraries).

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE TRADITIONAL SQUARE OF OPPOSITION

▶ Even though there are no libertarian presidents,All libertarian presidents are still alive

andNo libertarian presidents are still alive

cannot both be true (they are contraries).

▶ Likewise,

Some libertarian presidents are still alive

and

Some libertarian presidents are not still alive

cannot both be false (they are subcontraries).

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE TRADITIONAL SQUARE OF OPPOSITION

▶ Even though there are no libertarian presidents,All libertarian presidents are still alive

andNo libertarian presidents are still alive

cannot both be true (they are contraries).

▶ Likewise,Some libertarian presidents are still alive

and

Some libertarian presidents are not still alive

cannot both be false (they are subcontraries).

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE TRADITIONAL SQUARE OF OPPOSITION

▶ Even though there are no libertarian presidents,All libertarian presidents are still alive

andNo libertarian presidents are still alive

cannot both be true (they are contraries).

▶ Likewise,Some libertarian presidents are still alive

and

Some libertarian presidents are not still alive

cannot both be false (they are subcontraries).

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE TRADITIONAL SQUARE OF OPPOSITION

▶ Even though there are no libertarian presidents,All libertarian presidents are still alive

andNo libertarian presidents are still alive

cannot both be true (they are contraries).

▶ Likewise,Some libertarian presidents are still alive

andSome libertarian presidents are not still alive

cannot both be false (they are subcontraries).

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE TRADITIONAL SQUARE OF OPPOSITION

▶ Even though there are no libertarian presidents,All libertarian presidents are still alive

andNo libertarian presidents are still alive

cannot both be true (they are contraries).

▶ Likewise,Some libertarian presidents are still alive

andSome libertarian presidents are not still alive

cannot both be false (they are subcontraries).

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE TRADITIONAL SQUARE OF OPPOSITION

▶ The traditional view: if there are no Ss, then

▶ the I sentence is (vacuously) false, and (therefore)▶ the E sentence is (vacuously) true.

▶ Interestingly, this implies that

Some libertarian presidents are dead

is false, while

Some libertarian presidents are not still alive

is true.▶ That’s very strange.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE TRADITIONAL SQUARE OF OPPOSITION

▶ The traditional view: if there are no Ss, then▶ the I sentence is (vacuously) false, and (therefore)

▶ the E sentence is (vacuously) true.

▶ Interestingly, this implies that

Some libertarian presidents are dead

is false, while

Some libertarian presidents are not still alive

is true.▶ That’s very strange.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE TRADITIONAL SQUARE OF OPPOSITION

▶ The traditional view: if there are no Ss, then▶ the I sentence is (vacuously) false, and (therefore)▶ the E sentence is (vacuously) true.

▶ Interestingly, this implies that

Some libertarian presidents are dead

is false, while

Some libertarian presidents are not still alive

is true.▶ That’s very strange.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE TRADITIONAL SQUARE OF OPPOSITION

▶ The traditional view: if there are no Ss, then▶ the I sentence is (vacuously) false, and (therefore)▶ the E sentence is (vacuously) true.

▶ Interestingly, this implies that

Some libertarian presidents are dead

is false, while

Some libertarian presidents are not still alive

is true.▶ That’s very strange.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE TRADITIONAL SQUARE OF OPPOSITION

▶ The traditional view: if there are no Ss, then▶ the I sentence is (vacuously) false, and (therefore)▶ the E sentence is (vacuously) true.

▶ Interestingly, this implies that

Some libertarian presidents are dead

is false, while

Some libertarian presidents are not still alive

is true.▶ That’s very strange.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE TRADITIONAL SQUARE OF OPPOSITION

▶ The traditional view: if there are no Ss, then▶ the I sentence is (vacuously) false, and (therefore)▶ the E sentence is (vacuously) true.

▶ Interestingly, this implies thatSome libertarian presidents are dead

is false, while

Some libertarian presidents are not still alive

is true.▶ That’s very strange.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE TRADITIONAL SQUARE OF OPPOSITION

▶ The traditional view: if there are no Ss, then▶ the I sentence is (vacuously) false, and (therefore)▶ the E sentence is (vacuously) true.

▶ Interestingly, this implies thatSome libertarian presidents are dead

is false, while

Some libertarian presidents are not still alive

is true.▶ That’s very strange.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE TRADITIONAL SQUARE OF OPPOSITION

▶ The traditional view: if there are no Ss, then▶ the I sentence is (vacuously) false, and (therefore)▶ the E sentence is (vacuously) true.

▶ Interestingly, this implies thatSome libertarian presidents are dead

is false, whileSome libertarian presidents are not still alive

is true.▶ That’s very strange.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE TRADITIONAL SQUARE OF OPPOSITION

▶ The traditional view: if there are no Ss, then▶ the I sentence is (vacuously) false, and (therefore)▶ the E sentence is (vacuously) true.

▶ Interestingly, this implies thatSome libertarian presidents are dead

is false, whileSome libertarian presidents are not still alive

is true.

▶ That’s very strange.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE TRADITIONAL SQUARE OF OPPOSITION

▶ The traditional view: if there are no Ss, then▶ the I sentence is (vacuously) false, and (therefore)▶ the E sentence is (vacuously) true.

▶ Interestingly, this implies thatSome libertarian presidents are dead

is false, whileSome libertarian presidents are not still alive

is true.▶ That’s very strange.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE TRADITIONAL SQUARE OF OPPOSITION

AAll S are P

ENo S are P

ISome S are P

OSome S are not P

contradictories

contradictories

contraries

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE TRADITIONAL SQUARE OF OPPOSITION

AAll S are P

ENo S are P

ISome S are P

OSome S are not P

contradictories

contradictories

contraries

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

OUTLINE

CATEGORICAL PROPOSITIONS

THE COMPONENTS OF A CATEGORICAL PROPOSITION

QUALITY, QUANTITY, AND DISTRIBUTION

THE SQUARE OF OPPOSITION

THE TRADITIONAL SQUARE

THE MODERN SQUARE

CATEGORICAL SYLLOGISMS

MAJOR TERMS, MINOR TERMS, MIDDLE TERMS

MOOD AND FIGURE

VENN DIAGRAMS

REPRESENTING STANDARD FORM CATEGORICAL

PROPOSITIONS WITH VENN DIAGRAMS

TESTING THE VALIDITY OF CATEGORICAL SYLLOGISMS

WITH VENN DIAGRAMS

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE MODERN SQUARE OF OPPOSITION

AAll S are P

ENo S are P

ISome S are P

OSome S are not P

contradictories

contradictories

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE MODERN SQUARE OF OPPOSITION

▶ When the S class is empty,

Traditional ModernA: All S are P False TrueE: No S are P True TrueI: Some S are P False FalseO: Some S are not P True False

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE MODERN SQUARE OF OPPOSITION

▶ When the S class is empty,

Traditional ModernA: All S are P False TrueE: No S are P True TrueI: Some S are P False FalseO: Some S are not P True False

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE MODERN SQUARE OF OPPOSITION

▶ When the S class is empty,

Traditional ModernA: All S are P False TrueE: No S are P True TrueI: Some S are P False FalseO: Some S are not P True False

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE MODERN SQUARE OF OPPOSITION

▶ From the modern standpoint, the argument form

All S are PSome S are P

is invalid—as is the argument form:

No S are PSome S are not P

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE MODERN SQUARE OF OPPOSITION

▶ From the modern standpoint, the argument form

All S are PSome S are P

is invalid—as is the argument form:

No S are PSome S are not P

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE MODERN SQUARE OF OPPOSITION

▶ Similarly,All S are P

It is false that no S are P

No S are PIt is false that all S are P

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

THE MODERN SQUARE OF OPPOSITION

▶ Similarly,It is false that some S are P

Some S are not P

It is false that some S are not PSome S are P

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

IN CLASS EXERCISE!

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

OUTLINE

CATEGORICAL PROPOSITIONS

THE COMPONENTS OF A CATEGORICAL PROPOSITION

QUALITY, QUANTITY, AND DISTRIBUTION

THE SQUARE OF OPPOSITION

THE TRADITIONAL SQUARE

THE MODERN SQUARE

CATEGORICAL SYLLOGISMS

MAJOR TERMS, MINOR TERMS, MIDDLE TERMS

MOOD AND FIGURE

VENN DIAGRAMS

REPRESENTING STANDARD FORM CATEGORICAL

PROPOSITIONS WITH VENN DIAGRAMS

TESTING THE VALIDITY OF CATEGORICAL SYLLOGISMS

WITH VENN DIAGRAMS

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

CATEGORICAL SYLLOGISMS

▶ A syllogism is an argument containing two premises and aconclusion.

▶ A categorical syllogism is a syllogism whose premises andconclusion are categorical propositions.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

CATEGORICAL SYLLOGISMS

▶ A syllogism is an argument containing two premises and aconclusion.

▶ A categorical syllogism is a syllogism whose premises andconclusion are categorical propositions.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

CATEGORICAL SYLLOGISMS

▶ For instance,

1. All people are Kesha-lovers.2. Some Kesha-lovers are bacon-eaters.3. So, some people are bacon-eaters.

1. All hats are fashionable clothing.2. All fashionable clothing is purple.3. So, some hats are purple.

1. All hats are fashionable clothing.2. All fashionable clothing is purple.3. So, all hats are purple.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

CATEGORICAL SYLLOGISMS

▶ For instance,1. All people are Kesha-lovers.2. Some Kesha-lovers are bacon-eaters.3. So, some people are bacon-eaters.

1. All hats are fashionable clothing.2. All fashionable clothing is purple.3. So, some hats are purple.

1. All hats are fashionable clothing.2. All fashionable clothing is purple.3. So, all hats are purple.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

CATEGORICAL SYLLOGISMS

▶ For instance,1. All people are Kesha-lovers.2. Some Kesha-lovers are bacon-eaters.3. So, some people are bacon-eaters.

1. All hats are fashionable clothing.2. All fashionable clothing is purple.3. So, some hats are purple.

1. All hats are fashionable clothing.2. All fashionable clothing is purple.3. So, all hats are purple.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

CATEGORICAL SYLLOGISMS

▶ For instance,1. All people are Kesha-lovers.2. Some Kesha-lovers are bacon-eaters.3. So, some people are bacon-eaters.

1. All hats are fashionable clothing.2. All fashionable clothing is purple.3. So, some hats are purple.

1. All hats are fashionable clothing.2. All fashionable clothing is purple.3. So, all hats are purple.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

OUTLINE

CATEGORICAL PROPOSITIONS

THE COMPONENTS OF A CATEGORICAL PROPOSITION

QUALITY, QUANTITY, AND DISTRIBUTION

THE SQUARE OF OPPOSITION

THE TRADITIONAL SQUARE

THE MODERN SQUARE

CATEGORICAL SYLLOGISMS

MAJOR TERMS, MINOR TERMS, MIDDLE TERMS

MOOD AND FIGURE

VENN DIAGRAMS

REPRESENTING STANDARD FORM CATEGORICAL

PROPOSITIONS WITH VENN DIAGRAMS

TESTING THE VALIDITY OF CATEGORICAL SYLLOGISMS

WITH VENN DIAGRAMS

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

MAJOR TERMS, MINOR TERMS, MIDDLE TERMS

1. All M are P.2. Some S are M.3. So, some S are P

▶ the major term is the predicate term of the conclusion

▶ the major term is the predicate term of the conclusion▶ In a STANDARD FORM CATEGORICAL SYLLOGISM, the

middle term is a term which doesn’t show up in theconclusion, but does show up in both of the premises

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

MAJOR TERMS, MINOR TERMS, MIDDLE TERMS

1. All M are P.2. Some S are M.3. So, some S are P

▶ the major term is the predicate term of the conclusion▶ the major term is the predicate term of the conclusion

▶ In a STANDARD FORM CATEGORICAL SYLLOGISM, themiddle term is a term which doesn’t show up in theconclusion, but does show up in both of the premises

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

MAJOR TERMS, MINOR TERMS, MIDDLE TERMS

1. All M are P.2. Some S are M.3. So, some S are P

▶ the major term is the predicate term of the conclusion▶ the major term is the predicate term of the conclusion▶ In a STANDARD FORM CATEGORICAL SYLLOGISM, the

middle term is a term which doesn’t show up in theconclusion, but does show up in both of the premises

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

MAJOR TERMS, MINOR TERMS, MIDDLE TERMS

1. All M are P.2. Some S are M.3. So, some S are P

▶ the major premise is the premise which contains the majorterm.

▶ the minor premise is the premise which contains the minorterm.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

MAJOR TERMS, MINOR TERMS, MIDDLE TERMS

1. All M are P.2. Some S are M.3. So, some S are P

▶ the major premise is the premise which contains the majorterm.

▶ the minor premise is the premise which contains the minorterm.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

STANDARD FORM CATEGORICAL SYLLOGISMS

A STANDARD FORM CATEGORICAL SYLLOGISM contains:

1. Two premises and a conclusion, each a standard formcategorical proposition.

2. A major term which appears only in the first premise andthe predicate of the conclusion.

3. A minor term which appears only in the second premiseand the subject of the conclusion

4. A middle term which appears in both premises but not inthe conclusion.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

OUTLINE

CATEGORICAL PROPOSITIONS

THE COMPONENTS OF A CATEGORICAL PROPOSITION

QUALITY, QUANTITY, AND DISTRIBUTION

THE SQUARE OF OPPOSITION

THE TRADITIONAL SQUARE

THE MODERN SQUARE

CATEGORICAL SYLLOGISMS

MAJOR TERMS, MINOR TERMS, MIDDLE TERMS

MOOD AND FIGURE

VENN DIAGRAMS

REPRESENTING STANDARD FORM CATEGORICAL

PROPOSITIONS WITH VENN DIAGRAMS

TESTING THE VALIDITY OF CATEGORICAL SYLLOGISMS

WITH VENN DIAGRAMS

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

MOOD

▶ we may categorize categorical syllogisms by:

▶ the kind of standard form categorical proposition whichappears as its first (i.e., major) premise,

▶ the kind of standard form categorical proposition whichappears as its second (i.e., minor) premise, and

▶ the kind of standard form categorical propositions whichappears as its conclusion.

▶ call such a specification the MOOD of the sentence.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

MOOD

▶ we may categorize categorical syllogisms by:▶ the kind of standard form categorical proposition which

appears as its first (i.e., major) premise,

▶ the kind of standard form categorical proposition whichappears as its second (i.e., minor) premise, and

▶ the kind of standard form categorical propositions whichappears as its conclusion.

▶ call such a specification the MOOD of the sentence.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

MOOD

▶ we may categorize categorical syllogisms by:▶ the kind of standard form categorical proposition which

appears as its first (i.e., major) premise,▶ the kind of standard form categorical proposition which

appears as its second (i.e., minor) premise, and

▶ the kind of standard form categorical propositions whichappears as its conclusion.

▶ call such a specification the MOOD of the sentence.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

MOOD

▶ we may categorize categorical syllogisms by:▶ the kind of standard form categorical proposition which

appears as its first (i.e., major) premise,▶ the kind of standard form categorical proposition which

appears as its second (i.e., minor) premise, and▶ the kind of standard form categorical propositions which

appears as its conclusion.

▶ call such a specification the MOOD of the sentence.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

MOOD

▶ we may categorize categorical syllogisms by:▶ the kind of standard form categorical proposition which

appears as its first (i.e., major) premise,▶ the kind of standard form categorical proposition which

appears as its second (i.e., minor) premise, and▶ the kind of standard form categorical propositions which

appears as its conclusion.

▶ call such a specification the MOOD of the sentence.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

MOOD

All P are MAll S are MSome S are P

▶ The first premise is of the form A▶ The second premise is of the form A▶ the conclusion is of the form I.▶ Thus, the mood of this argument is AAI.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

MOOD

All P are MAll S are MSome S are P

▶ The first premise is of the form A

▶ The second premise is of the form A▶ the conclusion is of the form I.▶ Thus, the mood of this argument is AAI.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

MOOD

All P are MAll S are MSome S are P

▶ The first premise is of the form A▶ The second premise is of the form A

▶ the conclusion is of the form I.▶ Thus, the mood of this argument is AAI.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

MOOD

All P are MAll S are MSome S are P

▶ The first premise is of the form A▶ The second premise is of the form A▶ the conclusion is of the form I.

▶ Thus, the mood of this argument is AAI.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

MOOD

All P are MAll S are MSome S are P

▶ The first premise is of the form A▶ The second premise is of the form A▶ the conclusion is of the form I.▶ Thus, the mood of this argument is AAI.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

MOOD

All P are MAll S are MSome S are P

All M are PAll S are MSome S are P

▶ The mood of both arguments is AAI.▶ Yet the left-hand-side argument is invalid; whereas the

right-hand-side argument is valid.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

MOOD

All P are MAll S are MSome S are P

All M are PAll S are MSome S are P

▶ The mood of both arguments is AAI.

▶ Yet the left-hand-side argument is invalid; whereas theright-hand-side argument is valid.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

MOOD

All P are MAll S are MSome S are P

All M are PAll S are MSome S are P

▶ The mood of both arguments is AAI.▶ Yet the left-hand-side argument is invalid; whereas the

right-hand-side argument is valid.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

FIGURE

1 2 3 4M PS MS P

P MS MS P

M PM SS P

P MM SS P

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

MOOD AND FIGURE

All P are MAll S are MSome S are P

All M are PAll S are MSome S are P

▶ The left-hand-side argument has mood AAI and figure2—AAI-2

▶ The right-hand-side argument has mood AAI and figure1—AAI-1

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

MOOD AND FIGURE

All P are MAll S are MSome S are P

All M are PAll S are MSome S are P

▶ The left-hand-side argument has mood AAI and figure2—AAI-2

▶ The right-hand-side argument has mood AAI and figure1—AAI-1

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

MOOD AND FIGURE

All P are MAll S are MSome S are P

All M are PAll S are MSome S are P

▶ The left-hand-side argument has mood AAI and figure2—AAI-2

▶ The right-hand-side argument has mood AAI and figure1—AAI-1

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

MOOD AND FIGURE

▶ There are then 43 = 64 different moods

▶ And each mood has 4 different figures.▶ Thus, there are 64 × 4 = 256 different kinds of standard

form categorical syllogisms.▶ The traditional theory then just lists off the standard form

categorical syllogisms which are valid.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

MOOD AND FIGURE

▶ There are then 43 = 64 different moods▶ And each mood has 4 different figures.

▶ Thus, there are 64 × 4 = 256 different kinds of standardform categorical syllogisms.

▶ The traditional theory then just lists off the standard formcategorical syllogisms which are valid.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

MOOD AND FIGURE

▶ There are then 43 = 64 different moods▶ And each mood has 4 different figures.▶ Thus, there are 64 × 4 = 256 different kinds of standard

form categorical syllogisms.

▶ The traditional theory then just lists off the standard formcategorical syllogisms which are valid.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

MOOD AND FIGURE

▶ There are then 43 = 64 different moods▶ And each mood has 4 different figures.▶ Thus, there are 64 × 4 = 256 different kinds of standard

form categorical syllogisms.▶ The traditional theory then just lists off the standard form

categorical syllogisms which are valid.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

MOOD AND FIGURE

▶ On the traditional theory, all of the forms below are valid.On the modern theory, only the forms listed above the lineare valid.

AAA-1 EAE-2 IAI-3 AEE-4EAE-1 AEE-2 AII-3 IAI-4AII-1 EIO-2 OAO-3 EIO-4EIO-1 AOO-2 EIO-3AAI-1 AEO-2 AAI-3 AEO-4EAO-1 EAO-2 EAO-3 EAO-4

AAI-4

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

RULES FOR VALIDITY

A standard form categorical syllogism is valid on the moderntheory if and only if each of the following five propositions areall true of it. A standard form categorical syllogism is validon the traditional theory if and only if each of the first fourpropositions are true of it.

1. The middle term is distributed at least once.2. If a term is distributed in the conclusion, then that term is

distributed in one of the premises.3. There is at least one affirmative premise.4. There is a negative premise if and only if there is a

negative conclusion.5. If both premises are universal, then the conclusion is

universal.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

OUTLINE

CATEGORICAL PROPOSITIONS

THE COMPONENTS OF A CATEGORICAL PROPOSITION

QUALITY, QUANTITY, AND DISTRIBUTION

THE SQUARE OF OPPOSITION

THE TRADITIONAL SQUARE

THE MODERN SQUARE

CATEGORICAL SYLLOGISMS

MAJOR TERMS, MINOR TERMS, MIDDLE TERMS

MOOD AND FIGURE

VENN DIAGRAMS

REPRESENTING STANDARD FORM CATEGORICAL

PROPOSITIONS WITH VENN DIAGRAMS

TESTING THE VALIDITY OF CATEGORICAL SYLLOGISMS

WITH VENN DIAGRAMS

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

VENN DIAGRAMS

D

F G

x

An INTERPRETATION of a Venn diagramsays

1. what the box, or the domain, D is;and

2. which class of things each circlerepresents.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

VENN DIAGRAMS

D

F G

x

An INTERPRETATION of a Venn diagramsays

1. what the box, or the domain, D is;and

2. which class of things each circlerepresents.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

VENN DIAGRAMS

F G

H

D

x

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

OUTLINE

CATEGORICAL PROPOSITIONS

THE COMPONENTS OF A CATEGORICAL PROPOSITION

QUALITY, QUANTITY, AND DISTRIBUTION

THE SQUARE OF OPPOSITION

THE TRADITIONAL SQUARE

THE MODERN SQUARE

CATEGORICAL SYLLOGISMS

MAJOR TERMS, MINOR TERMS, MIDDLE TERMS

MOOD AND FIGURE

VENN DIAGRAMS

REPRESENTING STANDARD FORM CATEGORICAL

PROPOSITIONS WITH VENN DIAGRAMS

TESTING THE VALIDITY OF CATEGORICAL SYLLOGISMS

WITH VENN DIAGRAMS

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

STANDARD FORM CATEGORICAL PROPOSITIONS AND

VENN DIAGRAMS

▶ we will see how to represent the standard form categoricalpropositions A, E, I, and O using Venn diagrams.

▶ we will simply take D to be the class of all things. Thus, theS circles will represent all things which are S; and the Pcircles will represents all things which are P.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

STANDARD FORM CATEGORICAL PROPOSITIONS AND

VENN DIAGRAMS

▶ we will see how to represent the standard form categoricalpropositions A, E, I, and O using Venn diagrams.

▶ we will simply take D to be the class of all things. Thus, theS circles will represent all things which are S; and the Pcircles will represents all things which are P.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

VENN DIAGRAM FOR A

S P

D

Figure: All S are P

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

VENN DIAGRAM FOR E

S P

D

Figure: No S are P

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

VENN DIAGRAM FOR I

S P

D

x

Figure: Some S are P.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

VENN DIAGRAM FOR O

S P

D

x

Figure: Some S are not P.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

OUTLINE

CATEGORICAL PROPOSITIONS

THE COMPONENTS OF A CATEGORICAL PROPOSITION

QUALITY, QUANTITY, AND DISTRIBUTION

THE SQUARE OF OPPOSITION

THE TRADITIONAL SQUARE

THE MODERN SQUARE

CATEGORICAL SYLLOGISMS

MAJOR TERMS, MINOR TERMS, MIDDLE TERMS

MOOD AND FIGURE

VENN DIAGRAMS

REPRESENTING STANDARD FORM CATEGORICAL

PROPOSITIONS WITH VENN DIAGRAMS

TESTING THE VALIDITY OF CATEGORICAL SYLLOGISMS

WITH VENN DIAGRAMS

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

VENN DIAGRAMS AND VALIDITY

▶ use a Venn diagram to represent the claims of thepremises, and then check to see whether the Venn diagramcontains within it the claim of the conclusion.

▶ If it does, then the categorical syllogism is valid.▶ If it does not, then the categorical syllogism is invalid.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

VENN DIAGRAMS AND VALIDITY

▶ use a Venn diagram to represent the claims of thepremises, and then check to see whether the Venn diagramcontains within it the claim of the conclusion.

▶ If it does, then the categorical syllogism is valid.

▶ If it does not, then the categorical syllogism is invalid.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

VENN DIAGRAMS AND VALIDITY

▶ use a Venn diagram to represent the claims of thepremises, and then check to see whether the Venn diagramcontains within it the claim of the conclusion.

▶ If it does, then the categorical syllogism is valid.▶ If it does not, then the categorical syllogism is invalid.

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

VENN DIAGRAMS AND VALIDITY

▶ it will help us if we first represent the claims of universalpremises, and then represent the claims of particularpremises

▶ If there are only universal premises or only particularpremises, then it won’t matter the order in which werepresent the claims of the premises

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

VENN DIAGRAMS AND VALIDITY

▶ it will help us if we first represent the claims of universalpremises, and then represent the claims of particularpremises

▶ If there are only universal premises or only particularpremises, then it won’t matter the order in which werepresent the claims of the premises

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

VENN DIAGRAMS AND VALIDITY—AAI-1

All M are PAll S are MAll S are P

D

S M

P

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

VENN DIAGRAMS AND VALIDITY—AAI-1

All M are PAll S are MAll S are P

D

S M

P

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

VENN DIAGRAMS AND VALIDITY—AAI-1

All M are PAll S are MAll S are P

D

S M

P

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

VENN DIAGRAMS AND VALIDITY—AIA-4

All P are MSome M are SAll S are P

D

S M

P

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

VENN DIAGRAMS AND VALIDITY—AIA-4

All P are MSome M are SAll S are P

D

S M

P

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

VENN DIAGRAMS AND VALIDITY—AIA-4

All P are MSome M are SAll S are P

D

S M

P

x

CATEGORICAL PROPOSITIONS THE SQUARE OF OPPOSITION CATEGORICAL SYLLOGISMS VENN DIAGRAMS

IN CLASS EXERCISE!