lecture 3: propositional logic s. choimathsci.kaist.ac.kr/~schoi/logiclec3.pdf · introduction some...

Logic and the set theoryLecture 3: Propositional Logic

S. Choi

Department of Mathematical ScienceKAIST, Daejeon, South Korea

Fall semester, 2012

S. Choi (KAIST) Logic and set theory September 10, 2012 1 / 18

Introduction

About this lecture

Argument forms

Logical operatorsFormalization: well formed formula (wff)Truth tableCourse homepages:http://mathsci.kaist.ac.kr/~schoi/logic.html andthe moodle page http://moodle.kaist.ac.kr

Grading and so on in the moodle. Ask questions in moodle.

Introduction

About this lecture

Argument formsLogical operators

Formalization: well formed formula (wff)Truth tableCourse homepages:http://mathsci.kaist.ac.kr/~schoi/logic.html andthe moodle page http://moodle.kaist.ac.kr

Introduction

About this lecture

Argument formsLogical operatorsFormalization: well formed formula (wff)

Truth tableCourse homepages:http://mathsci.kaist.ac.kr/~schoi/logic.html andthe moodle page http://moodle.kaist.ac.kr

Introduction

About this lecture

Argument formsLogical operatorsFormalization: well formed formula (wff)Truth table

Course homepages:http://mathsci.kaist.ac.kr/~schoi/logic.html andthe moodle page http://moodle.kaist.ac.kr

Introduction

About this lecture

Argument formsLogical operatorsFormalization: well formed formula (wff)Truth tableCourse homepages:http://mathsci.kaist.ac.kr/~schoi/logic.html andthe moodle page http://moodle.kaist.ac.kr

Introduction

About this lecture

Argument formsLogical operatorsFormalization: well formed formula (wff)Truth tableCourse homepages:http://mathsci.kaist.ac.kr/~schoi/logic.html andthe moodle page http://moodle.kaist.ac.kr

Introduction

Some helpful references

Richard Jeffrey, Formal logic: its scope and limits, Mc Graw Hill

A mathematical introduction to logic, H. Enderton, AcademicPress.Whitehead, Russell, Principia Mathematica (our library). (Thiscould be a project idea. )http://plato.stanford.edu/contents.html has muchresource.http://ocw.mit.edu/courses/

linguistics-and-philosophy/24-241-logic-i-fall-2009/

Introduction

Richard Jeffrey, Formal logic: its scope and limits, Mc Graw HillA mathematical introduction to logic, H. Enderton, AcademicPress.

Whitehead, Russell, Principia Mathematica (our library). (Thiscould be a project idea. )http://plato.stanford.edu/contents.html has muchresource.http://ocw.mit.edu/courses/

Introduction

Richard Jeffrey, Formal logic: its scope and limits, Mc Graw HillA mathematical introduction to logic, H. Enderton, AcademicPress.Whitehead, Russell, Principia Mathematica (our library). (Thiscould be a project idea. )

http://plato.stanford.edu/contents.html has muchresource.http://ocw.mit.edu/courses/

Introduction

Richard Jeffrey, Formal logic: its scope and limits, Mc Graw HillA mathematical introduction to logic, H. Enderton, AcademicPress.Whitehead, Russell, Principia Mathematica (our library). (Thiscould be a project idea. )http://plato.stanford.edu/contents.html has muchresource.

http://ocw.mit.edu/courses/

Introduction

Richard Jeffrey, Formal logic: its scope and limits, Mc Graw HillA mathematical introduction to logic, H. Enderton, AcademicPress.Whitehead, Russell, Principia Mathematica (our library). (Thiscould be a project idea. )http://plato.stanford.edu/contents.html has muchresource.http://ocw.mit.edu/courses/

Argument forms

P, Q, R represent some sentences (not nec. atomic)

Either today is Monday or Tuesday.P or Q.If you have bad grades in KAIST, then you can get kicked out.If P, then Q.If P and Q, then R. It is not the case R. It is not the case P and Q.

Argument forms

P, Q, R represent some sentences (not nec. atomic)Either today is Monday or Tuesday.

P or Q.If you have bad grades in KAIST, then you can get kicked out.If P, then Q.If P and Q, then R. It is not the case R. It is not the case P and Q.

Argument forms

P, Q, R represent some sentences (not nec. atomic)Either today is Monday or Tuesday.P or Q.

If you have bad grades in KAIST, then you can get kicked out.If P, then Q.If P and Q, then R. It is not the case R. It is not the case P and Q.

Argument forms

P, Q, R represent some sentences (not nec. atomic)Either today is Monday or Tuesday.P or Q.If you have bad grades in KAIST, then you can get kicked out.

If P, then Q.If P and Q, then R. It is not the case R. It is not the case P and Q.

Argument forms

P, Q, R represent some sentences (not nec. atomic)Either today is Monday or Tuesday.P or Q.If you have bad grades in KAIST, then you can get kicked out.If P, then Q.

If P and Q, then R. It is not the case R. It is not the case P and Q.

Argument forms

P, Q, R represent some sentences (not nec. atomic)Either today is Monday or Tuesday.P or Q.If you have bad grades in KAIST, then you can get kicked out.If P, then Q.If P and Q, then R. It is not the case R. It is not the case P and Q.

Logical operators

It is not the case that: ¬ or˜

And: ∧ or &Or: ∨If ..., then... : →.If and only if: ↔.Seehttp://plato.stanford.edu/entries/pm-notation/.Seehttp://en.wikipedia.org/wiki/Logical_connective/.` is used to mark the conclusion.

Logical operators

It is not the case that: ¬ or˜And: ∧ or &

Or: ∨If ..., then... : →.If and only if: ↔.Seehttp://plato.stanford.edu/entries/pm-notation/.Seehttp://en.wikipedia.org/wiki/Logical_connective/.` is used to mark the conclusion.

Logical operators

It is not the case that: ¬ or˜And: ∧ or &Or: ∨

If ..., then... : →.If and only if: ↔.Seehttp://plato.stanford.edu/entries/pm-notation/.Seehttp://en.wikipedia.org/wiki/Logical_connective/.` is used to mark the conclusion.

Logical operators

It is not the case that: ¬ or˜And: ∧ or &Or: ∨If ..., then... : →.

If and only if: ↔.Seehttp://plato.stanford.edu/entries/pm-notation/.Seehttp://en.wikipedia.org/wiki/Logical_connective/.` is used to mark the conclusion.

Logical operators

It is not the case that: ¬ or˜And: ∧ or &Or: ∨If ..., then... : →.If and only if: ↔.

Seehttp://plato.stanford.edu/entries/pm-notation/.Seehttp://en.wikipedia.org/wiki/Logical_connective/.` is used to mark the conclusion.

Logical operators

It is not the case that: ¬ or˜And: ∧ or &Or: ∨If ..., then... : →.If and only if: ↔.Seehttp://plato.stanford.edu/entries/pm-notation/.

Seehttp://en.wikipedia.org/wiki/Logical_connective/.` is used to mark the conclusion.

Logical operators

It is not the case that: ¬ or˜And: ∧ or &Or: ∨If ..., then... : →.If and only if: ↔.Seehttp://plato.stanford.edu/entries/pm-notation/.Seehttp://en.wikipedia.org/wiki/Logical_connective/.

` is used to mark the conclusion.

Logical operators

It is not the case that: ¬ or˜And: ∧ or &Or: ∨If ..., then... : →.If and only if: ↔.Seehttp://plato.stanford.edu/entries/pm-notation/.Seehttp://en.wikipedia.org/wiki/Logical_connective/.` is used to mark the conclusion.

Logical operators

Precedence

The order of precedence determines which connective is the"main connective" when interpreting a non-atomic formula.

As a way of reducing the number of necessary parentheses, onemay introduce precedence rules:Operator Precedence

I ¬I ∧I ∨I →I ↔

So for example, P ∨Q ∧ ¬R → S is short for(P ∨ (Q ∧ (¬R)))→ S.

Logical operators

Precedence

The order of precedence determines which connective is the"main connective" when interpreting a non-atomic formula.As a way of reducing the number of necessary parentheses, onemay introduce precedence rules:

Operator Precedence

I ¬I ∧I ∨I →I ↔

Logical operators

Precedence

The order of precedence determines which connective is the"main connective" when interpreting a non-atomic formula.As a way of reducing the number of necessary parentheses, onemay introduce precedence rules:Operator Precedence

I ¬I ∧I ∨I →I ↔

Logical operators

Precedence

I ∧I ∨I →I ↔

Logical operators

Precedence

I ¬I ∧

I ∨I →I ↔

Logical operators

Precedence

I ¬I ∧I ∨

I →I ↔

Logical operators

Precedence

I ¬I ∧I ∨I →

I ↔So for example, P ∨Q ∧ ¬R → S is short for(P ∨ (Q ∧ (¬R)))→ S.

Logical operators

Precedence

I ¬I ∧I ∨I →I ↔

Logical operators

Precedence

I ¬I ∧I ∨I →I ↔

So for example, P ∨Q ∧ ¬R → S is short for

(P ∨ (Q ∧ (¬R)))→ S.

Logical operators

Precedence

I ¬I ∧I ∨I →I ↔

Logical operators

Formalizations

We can formalize any sentence by dividing it into atomic parts.

It is not both raining and snowing.¬(R ∧ S)

It is not raining or is not snowing.¬R ∨ ¬S

Logical operators

Formalizations

We can formalize any sentence by dividing it into atomic parts.It is not both raining and snowing.

¬(R ∧ S)

Logical operators

Formalizations

We can formalize any sentence by dividing it into atomic parts.It is not both raining and snowing.¬(R ∧ S)

Logical operators

Formalizations

It is not raining or is not snowing.

¬R ∨ ¬S

Logical operators

Formalizations

Well formed formula or wff

((∧P) ∨Q¬R) This is a nonsense

We define inductively.

I Any sentence letter is wff. (atomic one)I If φ is wff, then so is ¬φ.I If φ and ψ are wff, so is (φ ∧ ψ), (φ ∨ ψ), (φ→ ψ), and (φ↔ ψ).

A subwff is a wff within a wff.As long as atomic sentence letters are well defined, there is noambiguity in the meaning of wff.

((∧P) ∨Q¬R) This is a nonsenseWe define inductively.

I Any sentence letter is wff. (atomic one)

I If φ is wff, then so is ¬φ.I If φ and ψ are wff, so is (φ ∧ ψ), (φ ∨ ψ), (φ→ ψ), and (φ↔ ψ).

I Any sentence letter is wff. (atomic one)I If φ is wff, then so is ¬φ.

I If φ and ψ are wff, so is (φ ∧ ψ), (φ ∨ ψ), (φ→ ψ), and (φ↔ ψ).

A subwff is a wff within a wff.

As long as atomic sentence letters are well defined, there is noambiguity in the meaning of wff.

Some exercises

Either there is no Starbuck’s in Daejeon or I do not buy coffeebeans.

¬S ∨ ¬B.If I buy coffee beans, then there is no Starbuck’s in Daejeon.B → ¬S.If there were no God, then no movement is possible. But there aremovements. Hence, God exists.¬G→ ¬M, M, ` G.

Some exercises

Either there is no Starbuck’s in Daejeon or I do not buy coffeebeans.¬S ∨ ¬B.

If I buy coffee beans, then there is no Starbuck’s in Daejeon.B → ¬S.If there were no God, then no movement is possible. But there aremovements. Hence, God exists.¬G→ ¬M, M, ` G.

Some exercises

Either there is no Starbuck’s in Daejeon or I do not buy coffeebeans.¬S ∨ ¬B.If I buy coffee beans, then there is no Starbuck’s in Daejeon.

B → ¬S.If there were no God, then no movement is possible. But there aremovements. Hence, God exists.¬G→ ¬M, M, ` G.

Some exercises

Either there is no Starbuck’s in Daejeon or I do not buy coffeebeans.¬S ∨ ¬B.If I buy coffee beans, then there is no Starbuck’s in Daejeon.B → ¬S.

If there were no God, then no movement is possible. But there aremovements. Hence, God exists.¬G→ ¬M, M, ` G.

Some exercises

Either there is no Starbuck’s in Daejeon or I do not buy coffeebeans.¬S ∨ ¬B.If I buy coffee beans, then there is no Starbuck’s in Daejeon.B → ¬S.If there were no God, then no movement is possible. But there aremovements. Hence, God exists.

¬G→ ¬M, M, ` G.

Some exercises

Either there is no Starbuck’s in Daejeon or I do not buy coffeebeans.¬S ∨ ¬B.If I buy coffee beans, then there is no Starbuck’s in Daejeon.B → ¬S.If there were no God, then no movement is possible. But there aremovements. Hence, God exists.¬G→ ¬M, M, ` G.

Some exercises

Either it is raining, or it’s both snowing and raining.

R ∨ (R ∧ S).Either it is both raining and snowing or it is snowing but not raining.(R ∧ S) ∨ (S ∧ ¬R).

Some exercises

Either it is raining, or it’s both snowing and raining.R ∨ (R ∧ S).

Either it is both raining and snowing or it is snowing but not raining.(R ∧ S) ∨ (S ∧ ¬R).

Some exercises

Either it is raining, or it’s both snowing and raining.R ∨ (R ∧ S).Either it is both raining and snowing or it is snowing but not raining.

(R ∧ S) ∨ (S ∧ ¬R).

Some exercises

Either it is raining, or it’s both snowing and raining.R ∨ (R ∧ S).Either it is both raining and snowing or it is snowing but not raining.(R ∧ S) ∨ (S ∧ ¬R).

Truth table

Semantics of the logical operators

semantics: the study of meaning.

Each atomic formula has a truth or false value in a real world (orworld A).Each wff has a truth or false value in a real world (or world A).This depends on the truth values of atomic formulas.

Truth table

semantics: the study of meaning.Each atomic formula has a truth or false value in a real world (orworld A).

Each wff has a truth or false value in a real world (or world A).This depends on the truth values of atomic formulas.

Truth table

semantics: the study of meaning.Each atomic formula has a truth or false value in a real world (orworld A).Each wff has a truth or false value in a real world (or world A).

This depends on the truth values of atomic formulas.

Truth table

semantics: the study of meaning.Each atomic formula has a truth or false value in a real world (orworld A).Each wff has a truth or false value in a real world (or world A).This depends on the truth values of atomic formulas.

Truth table

Truth tables

Truth table generator:

I http://en.wikipedia.org/wiki/Truth_table,I http://logik.phl.univie.ac.at/~chris/gateway/formular-uk-zentral.html, complete (i.e. has all the steps)

I http://svn.oriontransfer.org/TruthTable/index.rhtml, hasxor, complete.

I One has to learn some notations... Sometimes use 0 and 1 insteadof F and T .

Truth table

Truth tables

Truth table generator:I http://en.wikipedia.org/wiki/Truth_table,

I http://logik.phl.univie.ac.at/~chris/gateway/formular-uk-zentral.html, complete (i.e. has all the steps)

Truth table

Truth tables

Truth table generator:I http://en.wikipedia.org/wiki/Truth_table,I http://logik.phl.univie.ac.at/~chris/gateway/formular-uk-zentral.html, complete (i.e. has all the steps)

Truth table

Truth tables

Truth table

Truth tables

Truth table

Truth tables

Elementary ones:

I ¬aI a ∧ bI a ∨ bI a→ b.I a↔ b or (a→ b) ∧ (b → a)I Every wff can be evaluated from this.I In computer science xor .

Truth table

Truth tables

Elementary ones:I ¬a

I a ∧ bI a ∨ bI a→ b.I a↔ b or (a→ b) ∧ (b → a)I Every wff can be evaluated from this.I In computer science xor .

Truth table

Truth tables

Elementary ones:I ¬aI a ∧ b

I a ∨ bI a→ b.I a↔ b or (a→ b) ∧ (b → a)I Every wff can be evaluated from this.I In computer science xor .

Truth table

Truth tables

Elementary ones:I ¬aI a ∧ bI a ∨ b

I a→ b.I a↔ b or (a→ b) ∧ (b → a)I Every wff can be evaluated from this.I In computer science xor .

Truth table

Truth tables

Elementary ones:I ¬aI a ∧ bI a ∨ bI a→ b.

I a↔ b or (a→ b) ∧ (b → a)I Every wff can be evaluated from this.I In computer science xor .

Truth table

Truth tables

Elementary ones:I ¬aI a ∧ bI a ∨ bI a→ b.I a↔ b or (a→ b) ∧ (b → a)

I Every wff can be evaluated from this.I In computer science xor .

Truth table

Truth tables

Elementary ones:I ¬aI a ∧ bI a ∨ bI a→ b.I a↔ b or (a→ b) ∧ (b → a)I Every wff can be evaluated from this.

I In computer science xor .

Truth table

Truth tables

Elementary ones:I ¬aI a ∧ bI a ∨ bI a→ b.I a↔ b or (a→ b) ∧ (b → a)I Every wff can be evaluated from this.I In computer science xor .

Truth table

Examples

To construct a truth table for a complex wff, we find the truthvalues for its smallest subwffs and then use the truth tables for thelogical operators for larger subwff and so on....

¬S ∧ ¬B.(¬G→ ¬M)→ (M → G).Also compare P → Q and ¬P ∨Q. Check(P → Q)↔ (¬P ∨Q)).This is used to compare.You can also use ¬((P → Q) xor (¬P ∨Q)).

Truth table

Examples

To construct a truth table for a complex wff, we find the truthvalues for its smallest subwffs and then use the truth tables for thelogical operators for larger subwff and so on....¬S ∧ ¬B.

(¬G→ ¬M)→ (M → G).Also compare P → Q and ¬P ∨Q. Check(P → Q)↔ (¬P ∨Q)).This is used to compare.You can also use ¬((P → Q) xor (¬P ∨Q)).

Truth table

Examples

To construct a truth table for a complex wff, we find the truthvalues for its smallest subwffs and then use the truth tables for thelogical operators for larger subwff and so on....¬S ∧ ¬B.(¬G→ ¬M)→ (M → G).

Also compare P → Q and ¬P ∨Q. Check(P → Q)↔ (¬P ∨Q)).This is used to compare.You can also use ¬((P → Q) xor (¬P ∨Q)).

Truth table

Examples

To construct a truth table for a complex wff, we find the truthvalues for its smallest subwffs and then use the truth tables for thelogical operators for larger subwff and so on....¬S ∧ ¬B.(¬G→ ¬M)→ (M → G).Also compare P → Q and ¬P ∨Q. Check(P → Q)↔ (¬P ∨Q)).

This is used to compare.You can also use ¬((P → Q) xor (¬P ∨Q)).

Truth table

Examples

To construct a truth table for a complex wff, we find the truthvalues for its smallest subwffs and then use the truth tables for thelogical operators for larger subwff and so on....¬S ∧ ¬B.(¬G→ ¬M)→ (M → G).Also compare P → Q and ¬P ∨Q. Check(P → Q)↔ (¬P ∨Q)).This is used to compare.

You can also use ¬((P → Q) xor (¬P ∨Q)).

Truth table

Examples

To construct a truth table for a complex wff, we find the truthvalues for its smallest subwffs and then use the truth tables for thelogical operators for larger subwff and so on....¬S ∧ ¬B.(¬G→ ¬M)→ (M → G).Also compare P → Q and ¬P ∨Q. Check(P → Q)↔ (¬P ∨Q)).This is used to compare.You can also use ¬((P → Q) xor (¬P ∨Q)).

Truth table

Tautology and a contradiction

Given some formula, any assignment of T and F yields T in thetruth table. Such a formula is said to be a tautology.

P ∨ ¬P.Given some formula, any assignment of T and F yields F in thetruth table. Such a formula is said to be a contradiction.(truth-functionally inconsistent)P ∧ ¬P.The formula which are not one of the above is said to betruth-functionally contingent.

Truth table

Given some formula, any assignment of T and F yields T in thetruth table. Such a formula is said to be a tautology.P ∨ ¬P.

Given some formula, any assignment of T and F yields F in thetruth table. Such a formula is said to be a contradiction.(truth-functionally inconsistent)P ∧ ¬P.The formula which are not one of the above is said to betruth-functionally contingent.

Truth table

Given some formula, any assignment of T and F yields T in thetruth table. Such a formula is said to be a tautology.P ∨ ¬P.Given some formula, any assignment of T and F yields F in thetruth table. Such a formula is said to be a contradiction.(truth-functionally inconsistent)

P ∧ ¬P.The formula which are not one of the above is said to betruth-functionally contingent.

Truth table

Given some formula, any assignment of T and F yields T in thetruth table. Such a formula is said to be a tautology.P ∨ ¬P.Given some formula, any assignment of T and F yields F in thetruth table. Such a formula is said to be a contradiction.(truth-functionally inconsistent)P ∧ ¬P.

The formula which are not one of the above is said to betruth-functionally contingent.

Truth table

Given some formula, any assignment of T and F yields T in thetruth table. Such a formula is said to be a tautology.P ∨ ¬P.Given some formula, any assignment of T and F yields F in thetruth table. Such a formula is said to be a contradiction.(truth-functionally inconsistent)P ∧ ¬P.The formula which are not one of the above is said to betruth-functionally contingent.

Truth table

Examples

(¬G→ ¬M)→ (M → G).

(P → Q)↔ (¬P ∨Q)).((P → Q)→ R)→ (P → R).

Truth table

Examples

(¬G→ ¬M)→ (M → G).(P → Q)↔ (¬P ∨Q)).

((P → Q)→ R)→ (P → R).

Truth table

Examples

(¬G→ ¬M)→ (M → G).(P → Q)↔ (¬P ∨Q)).((P → Q)→ R)→ (P → R).

Truth table

Truth table for argument forms

Here, we will have a number of premises P1,P2, . and aconclusion Q. We need to find the validity of P1,P2, ... ` Q

Pi s are complex.To check validity... We check when if every Pi is true, then so is Q.Or you can form (P1 ∧ P2 ∧ · · · ∧ Pn)→ Q.

Truth table

Here, we will have a number of premises P1,P2, . and aconclusion Q. We need to find the validity of P1,P2, ... ` QPi s are complex.

To check validity... We check when if every Pi is true, then so is Q.Or you can form (P1 ∧ P2 ∧ · · · ∧ Pn)→ Q.

Truth table

Here, we will have a number of premises P1,P2, . and aconclusion Q. We need to find the validity of P1,P2, ... ` QPi s are complex.To check validity... We check when if every Pi is true, then so is Q.

Or you can form (P1 ∧ P2 ∧ · · · ∧ Pn)→ Q.

Truth table

Here, we will have a number of premises P1,P2, . and aconclusion Q. We need to find the validity of P1,P2, ... ` QPi s are complex.To check validity... We check when if every Pi is true, then so is Q.Or you can form (P1 ∧ P2 ∧ · · · ∧ Pn)→ Q.

Truth table

Examples

P → Q,P → ¬Q ` ¬P.

((P → Q) ∧ (P → ¬Q))→ ¬P.R ` P ↔ (P ∨ (P ∧Q)).R → (P ↔ (P ∨ (P ∧Q))).

Truth table

Examples

P → Q,P → ¬Q ` ¬P.((P → Q) ∧ (P → ¬Q))→ ¬P.

R ` P ↔ (P ∨ (P ∧Q)).R → (P ↔ (P ∨ (P ∧Q))).

Truth table

Examples

P → Q,P → ¬Q ` ¬P.((P → Q) ∧ (P → ¬Q))→ ¬P.R ` P ↔ (P ∨ (P ∧Q)).

R → (P ↔ (P ∨ (P ∧Q))).

Truth table

Examples

P → Q,P → ¬Q ` ¬P.((P → Q) ∧ (P → ¬Q))→ ¬P.R ` P ↔ (P ∨ (P ∧Q)).R → (P ↔ (P ∨ (P ∧Q))).

lecture 3: propositional logic s. choimathsci.kaist.ac.kr/~schoi/logiclec3.pdf · introduction some...

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