introduction to logic 1 marie duží [email protected]

Introduction to Logic1

Introduction to Logic

Marie Duží

[email protected]

Introduction to Logic 2

Texts to study: http://www.cs.vsb.cz/duzi

CoursesIntroduction to Logic: Information for students

Chapters: 1. Introduction2. propositional Logic

2.1. Semantic exposition2.2. Resolution method

3. Predicate Logic3.1. Semantic exposition3.2. General resolution method

Presentation of lectures Book: Gamut L.T.F., Logic, Language and Meaning, Chicago

Press 1991, Vol. 1. Chapter 1, Chapter 2 (except of 2.7), Chapter 3, and Chapter 4 – Sections: 4.1, 4.2, 4.4.

http://www.cs.vsb.cz/duzi









Requirements to pass the course

• Accreditation: • Two written tests

– Propositional logic - max. 15 grades, min. 5 grades. – Predicate logic - max. 15 grades, min. 5 grades.– No repetitions for the tests!

+ 5 grades for being active

• Requirement: obtaining at least 15 grades.• Exam: Written test (max. 65 grades, min. 30 grades)• Total:

– at least 51 grades – good (3), – at least 66 grades – very good (2), – at least 86 grades – excellent (1)


1. Introduction

What is logic about?

What is the subject of logic?

Logic is the science of correct, valid reasoning, or, in other words, the art of a valid argumentation

What is an argument?

Argument: On the assumption of true premises P1,...,Pn it is possible to reason that the conclusion Z is true as well:

P1, ..., Pn

Z

Example: On the assumption that it is Thursday I belief that today a lecture on Introduction to logic takes place: Thursday Lecture on Logic


Introduction: valid arguments

In this course we deal only with deductively valid arguments. Notation: P1,...,Pn |= Z

The conclusion Z logically follows from the premises P1,..., Pn.

Definition 1:

The conclusion Z logically follows from the premises P1,...,Pn , notation: P1,...,Pn |= Z, iff under no circumstances it might happen that the premises were true and the conclusion false.


Introduction: valid arguments

Example: Because it is Thursday today I believe that the lecture “Introduction to Logic” takes place:

It is Thursday invalid Lecture on Logic takes place

Is it a deductively valid argument? No, it is not: It might happen that Duzi were sick and the lecture does not take place though it is Thursday (a premise is missing, for instance that Each Thursday the lecture takes place).

Each Thursday the lecture on Logic takes place.It is Thursday today valid Today the lecture on Logic takes place.


(Deductively) invalid arguments: generalization (induction), abduction

We will not deal with arguments that are not deductively valid, like: generalization (induction), abduction, and other –ductions a subject of Artificial Intelligence (non-monotonic reasoning)

Examples:Till now logic always took place on Thursday. induction, invalid(Therefore) Logic will take place also this Thursday

All swans that I have seen till now are white. induction, invalid(Therefore) All swans are white


Deductively invalid arguments: generalization (induction), abduction

Examples:All rabbits in the hat are white.These rabbits are from the hat. These rabbits are white. Deduction, valid

These rabbits are from the hat. These rabbits are white. (Probably) All rabbits in the hat are white.

Generalization, Induction, invalid

All rabbits in the hat are white. These rabbits are white. (Probably because) These rabbits are from the hat.

Abduction, invalidSeeking premises, causes of events, diagnosis of “malfunctions”


Examples of deductively valid arguments

1. He is at home or he has gone to a pub.If he is at home then he plays a piano.But he did not play a piano.------------------------------------------------ HenceHe has gone to the pub.

Sometimes the arguments are so obvious that it seems as if we did not need any logic. Well:If he did not play a piano (3. premise), then he was not at home (2. premise), and according to the first premise he must have gone to the pub.

But, we all use logic in our everyday life, we wouldn’t survive without logic:

2. All agarics (mushrooms) have a strong toxic effect.The mushroom I have picked up is an agaric.----------------------------------------------------------------------The mushroom I have picked up has a strong toxic effect.

Will you examine the mushroom by tasting it, or will you rely on logic?


Examples of deductively valid arguments

All agarics (mushrooms) have a strong toxic effect.This apple is an agaric.----------------------------------------------------------------------Hence This apple has a strong toxic effect.

The argument is valid. But the conclusion is evidently not true (false). Hence, at least one premise is false (obviously the second).

Circumstances according to Definition 1 are particular interpretations (depending on the expressive power of the logical system). Logical connectives (‘and’, ‘or’, ‘if …then …’) and quantifiers (‘all’, ‘some’, ‘every’, …) have a fixed interpretation; we interpret elementary propositions and/or their parts.

In our example, if “this apple” and “agarics” were interpreted in such a way that the second premise were true, the truth of the conclusion is guaranteed.

We also say that the argument has a valid logical form.


Deductively valid arguments

Logic is a tool that helps us to discover the relation of logical entailment, to answer questions like „What follows from particular assumptions “?, etc.

1. If the course is good then it is useful.2. The lecturer is sharply demanding studiousness or the course is not useful.3. But the lecturer is not demanding.

-------------------------------------------------------------------------- Hence4. The course is not good.

• It helps our intuition that can sometimes fail.– The assumptions can be complicated, “enmeshed in negations and other

connectives”, so that the relation of entailment is not obvious at first sight.– Similarly as all the mother-tongue speakers use intuitively rules of grammar

without knowing the grammar explicitly (often not being able to formulate the rules).

– But sometimes it is useful to consult the grammar book or a dictionary (in particular when taking part in a TV competition).


Examples of valid arguments1. All men like football and beer.

2. Some beer-lovers do not like football.

3. Xaver likes only those who like football and beer.

––––––––––––––––––––––––––––––––––––

4. Xaver does not like some women.

Necessarily, if premises are true the conclusion has to be true as well.

Is this argument valid?

Certainly, if Xaver likes only those who like football and beer (premise 3), then he does not like some beer-lovers (namely those who do not like football – according to the premise 2). Hence, (according to 1) he does not like some “no-men”, i.e., women.

But according to the Definition 1 the argument is not valid: the argument is valid if necessarily, i.e., in all the circumstances (under all interpretations) in which the premises are true the conclusion is true as well.

But: in our case those individuals that are not men would not have to be interpreted as women.

A premise is missing, viz. the premise “who is not a man is a woman”. Moreover, to be precise, we should also specify that “who is a lover of something he likes that”.


Examples of valid arguments

Hence: We have to state all the premises necessary for deriving the conclusion.

1. All men like football and beer.

2. Some beer-lovers do not like football.

3. Xaver likes only those who like football and beer.

4. Who is not a man is a woman.

5. Who is a lover of something he likes it.

––––––––––––––––––––––––––––––––––––6. Xaver does not like some women.

Now the argument is valid, it has a valid logical form. The conclusion is logically entailed by (follows from) the premises.

We also say that the conclusion is informationally (deductively) contained in the premises.


Valid arguments in mathematicsArgument A: No prime number is divisible by three.

The number 9 is divisible by three.––––––––––––––––––––––––––– validThe number 9 is not a prime.

Argument B: No prime number is divisible by six.The number eight is not a prime.––––––––––––––––––––––––––– invalidThe number eight is not divisible by six.

Though in the second case B it can never happen that the premises were true and the conclusion false, the argument is invalid. The conclusion is not logically entailed by the premises.

If the expression “eight” were interpreted as the number 12, the premises would be true and the conclusion false.(The conclusion is not deductively contained in the premises)


Theorem of deduction; semantic variant

If the argument P1,...,Pn |= Z is valid, then the statement of the form “if P1 and ... and Pn then Z”

P1 &...& Pn Zis analytically (necessarily) true. Notation: |= P1 ... Pn Z.Hence: P1,...,Pn |= Z (if and only if)

P1,...,Pn-1 |= (Pn Z) P1,...,Pn-2 |= ((Pn-1 Pn) Z) P1,...,Pn-3 |= ((Pn-2 Pn-1 Pn) Z) …

|= (P1 ... Pn) Z


Logical analysis of languageValidness of an argument is determined by the meaning (interpretation)

of particular statements that are analyzed (formalized) in a less or more fine-grained way according to the expressive power of a logical system:

• Propositional logic: makes it possible to analyze only the way in which a complex statement is composed from elementary propositions. The composition of elementary propositions is not examined, they contribute only by its truth value: True – 1, False – 0 (an algebra of truth values)

• 1st-order Predicate logic: makes it possible to analyze moreover the composition of elementary propositions, namely the way in which properties and/or relations are ascribed to (tuples of) individuals.

• 2nd-order Predicate logic: makes it possible to analyze moreover properties of properties, propertied of functions and relations between them.

• Modal logics (analyze “necessary” and “possible”), epistemic logics (knowledge), doxastic logics (of hypotheses) deontic logics (of commands), ...

• Transparent intensional logic (perhaps the most powerful system) – see the course “Principles of logical analysis“.


Properties of valid arguments A valid argument may have a false conclusion:

All primes are odd The number 2 is not odd The number 2 is not a prime

But then at least one premise has to be falseIn such a case we also say that the argument is not sound. But a valid

argument that is not sound may also be useful: a proof ad absurdum.If you want to show that your boss is not right, it is not diplomatic to

say it in an open way. Instead, you may argue by way of the proof ad absurdum: “Well, you say P – interesting, but P entails Q, and Q entails R, which is obviously false.” (Hence, P must have been false as well.)

Monotonicity: if an argument is valid then extending the set of assumptions by

another premise does not change the validity of the argument.


Properties of valid arguments• From contradictory (inconsistent) assumptions (such that it can

never happen that all of them were simultanously true) any conclusion follows.

• If I study hard then I’ll pass the exam.• I haven’t passed the exam though I studied hard.

-------------------------------------------------------------------- (e.g.) My dog plays a piano right now

• Reflexivity: If A is one of the assumptions P1,...,Pn, then P1,...,Pn |= A.

• Transitivity: If P1, …, Pn |= Z and Q1, …, Qm, Z |= Z’, thenP1, …, Pn, Q1, …, Qm |= Z’ .

Break


Naïve theory of sets What is it a set?

A set is a collection of elements, and it is determined just by its elements; a set consisting of elements a, b, c is denoted: {a, b, c}

An element of a set can be again a set, a set may consist of no elements, it may be empty (denoted by ) !

Examples: , {a, b}, {b, a}, {a, b, a}, {{a, b}}, {a, {b, a}}, {, {}, {{}}}

Sets are identical if and only if (iff) they have exactly the same elements (the principle of extensionality) Notation: x M – „x is an element of M“ a {a, b}, a {{a, b}}, {a, b} {{a, b}}, {, {}, {{}}},

{, {}}, but: x for any (i.e., all) x. {a, b} = {b, a} = {a, b, a}, but: {a, b} {{a, b}} {a, {b, a}}


Set-theoretical operations (create new sets from sets)

Union: A B = {x | x A or x B}

read: „The set of all x such that x is an element of A or x is an element of B.“ {a, b, c} {a, d} = {a, b, c, d} {odd numbers} {even numbers} = {natural numbers}

– denoted Nat

UiI Ai = {x | x Ai for some i I} Let Ai = {x | x = 2.i for some i Nat}

UiNat Ai = the set of all even numbers


Set-theoretical operations (create new sets from sets)

Intersection: A B = {x | x A and x B}

read: „The set of all x such that x is an element of A and x is an element of B as well.“ {a, b, c} {a, d} = {a} {even numbers} {odd numbers} =

iI Ai = {x | x Ai for all i I} Let Ai = {x | x Nat, x i}

Then iNat Ai =


Relations between sets A set A is a subset of a set B, denoted A B,

iff each element of A is also an element of B. A set A is a proper subset of a set B, denoted

A B, iff each element of A is also an element of B but not vice versa.{a} {a} {a, b} {{a, b}} !!!

It holds: A B, iff A B and A B It holds: A B, iff A B = B, iff A B = A


Some other set-theoretical operations

Difference: A \ B = {x | x A and x B} {a, b, c} \ {a, b} = {c}

Complement: Let A M. The complement of A with respect to M is the set A’ = M \ A

Cartesian product: A B = {a,b | aA, bB} where a,b je an ordered couple

(the ordering is important: a is the first, b is the second)

It holds: a,b = c,d iff a = c, b = d But: a,b b,a, though {a,b} = {b,a} !!! generalization: A … A the set of n-tuples,

denoted also by An


Some other set-theoretical operations

Potential set: 2A = {B | B A}, denoted also by P(A)2{a,b} = {, {a}, {b}, {a,b}}

2{a,b,c} = {, {a}, {b}, {c}, {a,b}, {a,c}, {b,c}, {a,b,c}}

How many elements are there in 2A ?

If |A| is the number of elements (cardinality) of a set A, then 2A has 2|A| elements (hence the notation: 2A)

2{a,b} {a} = {, {a,a}, {b,a}, {a,a, b,a}}


Grafical picturing (in a universe U):A: S\(PM) = (S\P)(S\M)

S(x) (P(x) M(x)) S(x) P(x) M(x)

B: P\(SM) = (P\S)(P\M)

P(x) (S(x) M(x)) P(x) S(x) M(x)

C: (S P) \ M

S(x) P(x) M(x)

D: S P M

S(x) P(x) M(x)

E: (S M) \ P

S(x) M(x) P(x)

F: (P M) \ S

P(x) M(x) S(x)

G: M\(PS) = (M\P)(M\S)

M(x) (P(x) S(x)) M(x) P(x) S(x)

H: U \ (S P M) = (U \ S U \ P U \ S)

(S(x) P(x) M(x)) S(x) P(x) M(x)

S

PM

A

B

CD

E

FG

H


Russell’s paradox Is it true that any collection of elements (i.e., a collection

defined in an arbitrary way) can be considered to be a set? It is normal that a set and its elements are entities of

different types. Hence a “normal set” is not an element of itself.

Let N is a set of all normal sets: N = {M | M M}.

Question: Is N N ? In other words, is N itself normal? Yes?

But then according to the definition of N it holds that N is normal, i.e., NN.

No? But then NN, hence N is normal, and therefore it belongs to N, i.e., NN.

Both the answers lead to a contradiction. N is not well defined. The definition does not determine a collection of elements that could be considered to be a set.


The end of lesson 1

introduction to logic 1 marie duží [email protected]

Documents

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predicate logic

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