[PPT] - 02Propositional Logic II CS 3234: Logic and Formal Systems Martin PowerPoint Presentation

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02—Propositional Logic II

CS 3234: Logic and Formal Systems

Martin Henz

August 20, 2009

Generated on Tuesday 12th January, 2010, 14:38 CS 3234: Logic and Formal Systems 02—Propositional Logic II 1

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CS 3234: Logic and Formal Systems 02—Propositional Logic II 2

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Propositions

Propositions are Declarative Sentences Sentences which one can—in principle—argue as being true or false.

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Propositions

Propositions are Declarative Sentences Sentences which one can—in principle—argue as being true or false. Examples

1

The sum of the numbers 3 and 5 equals 8.

2

Jane reacted violently to Jack’s accusations.

3

Every natural number > 2 is the sum of two prime numbers.

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Sequents as Logical Arguments

A Sequent in English If the train arrives late and there are no taxis at the station then John is late for his meeting. John is not late for his meeting. The train did arrive late. Therefore, there were taxis at the station.

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Focus on Structure, not Content

We are primarily concerned about the structure of arguments in this class, not the validity of statements in a particular domain.

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Focus on Structure, not Content

We are primarily concerned about the structure of arguments in this class, not the validity of statements in a particular domain. We therefore simply abbreviate sentences by letters such as p, q, r, p1, p2 etc.

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Focus on Structure, not Content

We are primarily concerned about the structure of arguments in this class, not the validity of statements in a particular domain. We therefore simply abbreviate sentences by letters such as p, q, r, p1, p2 etc. Instead of English words such as “if...then”, “and”, “not”, it is more convenient to use symbols such as →, ∧, ¬.

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Sequents in Symbolic Notation

A Sequent in English If the train arrives late and there are no taxis at the station then John is late for his meeting. John is not late for his meeting. The train did arrive late. Therefore, there were taxis at the station.

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Sequents in Symbolic Notation

A Sequent in English If the train arrives late and there are no taxis at the station then John is late for his meeting. John is not late for his meeting. The train did arrive late. Therefore, there were taxis at the station. The same sequent using letters and symbols p ∧ ¬q → r, ¬r, p ⊢ q Remaining task Develop proof rules that allows us to derive such sequents

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Rules for Conjunction

Introduction of Conjunction φ ψ φ ∧ ψ [∧i]

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Rules for Conjunction

Introduction of Conjunction φ ψ φ ∧ ψ [∧i] Elimination of Conjunction φ ∧ ψ φ [∧e1] φ ∧ ψ ψ [∧e2]

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Rules of Double Negation

Introduction of Double Negation φ ¬¬φ [¬¬i]

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Rules of Double Negation

Introduction of Double Negation φ ¬¬φ [¬¬i] Elimination of Double Negation ¬¬φ φ [¬¬e]

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Rules for Eliminating Implication

φ φ → ψ ψ [→ e] φ → ψ ¬ψ ¬φ [MT]

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Rule for Introduction of Implication

✄ ✂

✁

φ . . . ψ φ → ψ [→ i]

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Rules for Introduction of Disjunction

φ φ ∨ ψ [∨ii] ψ φ ∨ ψ [∨i2]

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Rule for Elimination of Disjunction

φ ∨ ψ

✄ ✂

✁

φ . . . χ

✄ ✂

✁

ψ . . . χ χ [∨e]

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Proof of p ∧ (q ∨ r) ⊢ (p ∧ q) ∨ (p ∧ r)

1 p ∧ (q ∨ r) premise 2 p ∧e1 1 3 q ∨ r ∧e2 1 4 q assumption 5 p ∧ q ∧i 2,4 6 (p ∧ q) ∨ (p ∧ r) ∨i1 5 7 r assumption 8 p ∧ r ∧i 2,7 9 (p ∧ q) ∨ (p ∧ r) ∨i2 8 10 (p ∧ q) ∨ (p ∧ r) ∨e 3, 4–6, 7–9

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A Special Proposition

Recall: We are only interested in the truth value of propositions, not the subject matter that they refer to (Martian pizzas or whatever).

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A Special Proposition

Recall: We are only interested in the truth value of propositions, not the subject matter that they refer to (Martian pizzas or whatever). Therefore, all propositions that we all agree must be true are the same!

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A Special Proposition

Recall: We are only interested in the truth value of propositions, not the subject matter that they refer to (Martian pizzas or whatever). Therefore, all propositions that we all agree must be true are the same! Example: p → p

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A Special Proposition

Recall: We are only interested in the truth value of propositions, not the subject matter that they refer to (Martian pizzas or whatever). Therefore, all propositions that we all agree must be true are the same! Example: p → p We denote the proposition that is always true using the symbol ⊤.

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Another Special Proposition

We denote the proposition that is always true using the symbol ⊤.

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Another Special Proposition

We denote the proposition that is always true using the symbol ⊤. Similarly, we denote the proposition that is always false using the symbol ⊥.

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Another Special Proposition

We denote the proposition that is always true using the symbol ⊤. Similarly, we denote the proposition that is always false using the symbol ⊥. Example: p ∧ ¬p

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Elimination of Negation

φ ¬φ ⊥ [¬e]

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Introduction of Negation

✄ ✂

✁

φ . . . ⊥ ¬φ [¬i]

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Elimination of ⊥

⊥ φ [⊥e]

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Basic Rules (conjunction and disjunction)

φ ψ φ ∧ ψ [∧i] φ ∧ ψ φ [∧e1] φ ∧ ψ ψ [∧e2] φ φ ∨ ψ [∨ii] ψ φ ∨ ψ [∨i2] φ ∨ ψ

✄ ✂

✁

φ . . . χ

✄ ✂

✁

ψ . . . χ χ [∨e]

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Basic Rules (implication)

✄ ✂

✁

φ . . . ψ φ → ψ [→ i] φ φ → ψ ψ [→ e]

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Basic Rules (negation)

φ . . . ⊥ ¬φ [¬i] φ ¬φ ⊥ [¬e]

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Basic Rules (⊥ and double negation)

⊥ φ [⊥e] ¬¬φ φ [¬¬e]

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Some Derived Rules: Introduction of Double Negation

φ ¬¬φ [¬¬i]

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Example: Deriving [¬¬ i] from [¬ i] and [¬ e]

1 φ premise 2 ¬φ assumption 3 ⊥ ¬e 1,2 4 ¬¬φ ¬i 2–3

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Some Derived Rules: Modus Tollens

φ → ψ ¬ψ ¬φ [MT]

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Some Derived Rules: Proof By Contradiction

✄ ✂

✁

¬φ . . . ⊥ φ [PBC]

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Some Derived Rules: Law of Excluded Middle

φ ∨ ¬φ [LEM]

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Motivation

Consider the following theorem. Theorem There exist irrational numbers a and b such that ab is rational. Let us call this theorem χ.

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Proof Outline for χ

Let p be the following proposition: Proposition p √ 2

√ 2 is rational.

Then the proof of χ goes like this: p ∨ ¬p [LEM]

✄ ✂

✁

p . . . χ

✄ ✂

✁

¬p . . . χ χ [∨e]

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In detail (1)

p . . . χ Assume √ 2

√ 2 is rational. Choose a and b to be

√ 2, and we have found irrational a and b such that ab is rational. Thus Theorem χ holds under the assumption p.

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In detail (2)

¬p . . . χ Assume √ 2

√ 2 is irrational. Choose a to be

√ 2

√ 2 and b to be

√

2. Then we have

ab = ( √ 2

√ 2) √ 2 =

√ 2

( √ 2· √ 2) = (

√ 2)2 = 2 2 is rational. Thus Theorem χ holds under the assumption ¬p.

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Summary of Proof for χ

Proposition p √ 2

√ 2 is rational.

p ∨ ¬p [LEM]

✄ ✂

✁

p . . . χ

✄ ✂

✁

¬p . . . χ χ [∨e]

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Summary of Proof for χ

Proposition p √ 2

√ 2 is rational.

p ∨ ¬p [LEM]

✄ ✂

✁

p . . . χ

✄ ✂

✁

¬p . . . χ χ [∨e] There exist irrational numbers a and b such that ab is rational...

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The Magic of LEM

There exist irrational numbers a and b such that ab is rational.

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The Magic of LEM

There exist irrational numbers a and b such that ab is rational. But: If they exist, do you have an example?

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The Magic of LEM

There exist irrational numbers a and b such that ab is rational. But: If they exist, do you have an example? Probably a = √ 2

√ 2 and b =

√ 2...

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The Magic of LEM

There exist irrational numbers a and b such that ab is rational. But: If they exist, do you have an example? Probably a = √ 2

√ 2 and b =

√ 2..., but we haven’t proven that √ 2

√ 2 is irrational!

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The Magic of LEM

There exist irrational numbers a and b such that ab is rational. But: If they exist, do you have an example? Probably a = √ 2

√ 2 and b =

√ 2..., but we haven’t proven that √ 2

√ 2 is irrational!

Note: √ 2

√ 2

√ 2··

= 2

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The Magic of LEM

There exist irrational numbers a and b such that ab is rational. But: If they exist, do you have an example? Probably a = √ 2

√ 2 and b =

√ 2..., but we haven’t proven that √ 2

√ 2 is irrational!

Note: √ 2

√ 2

√ 2··

= 2 Using LEM, we can make use of the “probable irrationality”

f

√ 2

√ 2 without having to prove it!

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Intuitionistic Logic

Intuitionistic logic does not accept the derived rule LEM.

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Intuitionistic Logic

Intuitionistic logic does not accept the derived rule LEM. The underlying argument for LEM is elimination of double negation. ¬¬φ φ [¬¬e]

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Deriving LEM using Basic Rules

1 ¬(φ ∨ ¬φ) assumption 2 φ assumption 3 φ ∨ ¬φ ∨i1 2 4 ⊥ ¬ e 3,1 5 ¬φ ¬ i 2–4 6 φ ∨ ¬φ ∨ i2 5 7 ⊥ ¬ e 6,1 8 ¬¬(φ ∨ ¬φ) ¬i 1–7 9 φ ∨ ¬φ ¬¬e

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Intuitionistic Logic

Intuitionistic logic is obtained from natural deduction by removing the rule ¬¬e.

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History of Intuitionistic Logic

Late 19th century: Gottlob Frege proposes to reduce mathematics to set theory Russell destroys this programme via paradox In response, L.E.J. Brouwer proposes intuitionistic mathematics, with intuitionistic logic as its formal foundation An alternative response is Hilbert’s formalistic position

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Applications of Intuitionistic Logic

Intuitionistic logic has a strong connection to computability

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Applications of Intuitionistic Logic

Intuitionistic logic has a strong connection to computability For example, if we have an intuitionistic proof of Theorem There exist irrational numbers a and b such that ab is rational. then we would know irrational a and b such that ab is rational.

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Recap: Logical Connectives

¬: negation of p is denoted by ¬p ∨: disjunction of p and r is denoted by p ∨ r, meaning at least one of the two statements is true. ∧: conjunction of p and r is denoted by p ∧ r, meaning both are true. →: implication between p and r is denoted by p → r, meaning that r is a logical consequence of p.

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Formal Description Required

Use of Meta-Language When we describe rules such as φ ∨ ¬φ [LEM] we mean that letters such as φ can be replaced by any formula.

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Formal Description Required

Use of Meta-Language When we describe rules such as φ ∨ ¬φ [LEM] we mean that letters such as φ can be replaced by any formula. But what exactly is the set of formulas that can be used for φ?

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Formal Description Required

Use of Meta-Language When we describe rules such as φ ∨ ¬φ [LEM] we mean that letters such as φ can be replaced by any formula. But what exactly is the set of formulas that can be used for φ? Allowed (p ∧ (¬q))

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Formal Description Required

Use of Meta-Language When we describe rules such as φ ∨ ¬φ [LEM] we mean that letters such as φ can be replaced by any formula. But what exactly is the set of formulas that can be used for φ? Allowed (p ∧ (¬q)) Not allowed ) ∧ p q¬(

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Definition of Well-formed Formulas

Definition Every propositional atom p, q, r, . . . and p1, p2, p3, . . . is a well-formed formula. If φ is a well-formed formula, then so is (¬φ). If φ and ψ are well-formed formulas, then so is (φ ∧ ψ). If φ and ψ are well-formed formulas, then so is (φ ∨ ψ). If φ and ψ are well-formed formulas, then so is (φ → ψ).

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Definition very restrictive

How about this formula? p ∧ ¬q ∨ r

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Definition very restrictive

How about this formula? p ∧ ¬q ∨ r Usually, this is understood to mean ((p ∧ (¬q)) ∨ r)

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Definition very restrictive

How about this formula? p ∧ ¬q ∨ r Usually, this is understood to mean ((p ∧ (¬q)) ∨ r) ...but for the formal treatment of this section and the first homework, we insist on the strict definition, and exclude such formulas.

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Backus Naur Form: A more compact definition

Backus Naur Form for propositional formulas φ ::= p|(¬φ)|(φ ∧ φ)|(φ ∨ φ)|(φ → φ) where p stands for any atomic proposition.

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Inversion principle

How can we show that a a formula such as (((¬p) ∧ q) → (p ∧ (q ∨ (¬r)))) is well-formed?

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Inversion principle

How can we show that a a formula such as (((¬p) ∧ q) → (p ∧ (q ∨ (¬r)))) is well-formed? Answer: We look for the only applicable rule in the definition (the last rule in this case), and proceed on the parts.

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Parse trees

A formula (((¬p) ∧ q) → (p ∧ (q ∨ (¬r))))

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Parse trees

A formula (((¬p) ∧ q) → (p ∧ (q ∨ (¬r)))) ...and its parse tree:

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Parse trees

A formula (((¬p) ∧ q) → (p ∧ (q ∨ (¬r)))) ...and its parse tree: → ∧ ¬ p q ∧ p ∨ q ¬ r

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Meaning of propositional formula

Meaning as mathematical object We define the meaning of formulas as a function that maps formulas and valuations to truth values.

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Meaning of propositional formula

Meaning as mathematical object We define the meaning of formulas as a function that maps formulas and valuations to truth values. Approach We define this mapping based on the structure of the formula, using the meaning of their logical connectives.

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Truth values and valuations

Definition The set of truth values contains two elements T and F, where T represents “true” and F represents “false”.

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Truth values and valuations

Definition The set of truth values contains two elements T and F, where T represents “true” and F represents “false”. Definition A valuation or model of a formula φ is an assignment of each propositional atom in φ to a truth value.

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Meaning of logical connectives

The meaning of a connective is defined as a truth table that gives the truth value of a formula, whose root symbol is the connective, based on the truth values of its components. φ ψ φ ∧ ψ T T T T F F F T F F F F

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Truth tables of formulas

Truth tables use placeholders of formulas such as φ: φ ψ φ ∧ ψ T T T T F F F T F F F F

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Truth tables of formulas

Truth tables use placeholders of formulas such as φ: φ ψ φ ∧ ψ T T T T F F F T F F F F Build the truth table for given formula: p q r (p ∧ q) ((p ∧ q) ∧ r) T T T T T T T F T F . . .

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Truth tables of other connectives

φ ψ φ ∨ ψ T T T T F T F T T F F F φ ψ φ → ψ T T T T F F F T T F F T φ ¬φ T F F T ⊤ T ⊥ F

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Constructing the truth table of a formula

p q (¬p) ¬q p → ¬q q ∨ ¬p (p → ¬q) → (q ∨ ¬p) T T F F F T T T F F T T F F F T T F T T T F F T T T T T

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Validity and Satisfiability

Validity A formula is valid if it computes T for all its valuations.

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Validity and Satisfiability

Validity A formula is valid if it computes T for all its valuations. Satisfiability A formula is satisfiable if it computes T for at least one of its valuations.

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Semantic Entailment

Definition If, for all valuations in which all φ1, φ2, . . . , φn evaluate to T, the formula ψ evaluates to T as well, we say that φ1, φ2, . . . , φn semantically entail ψ, written: φ1, φ2, . . . , φn | = ψ

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Soundness of Propositional Logic

Soundness Let φ1, φ2, . . . , φn and ψ be propositional formulas. If φ1, φ2, . . . , φn ⊢ ψ, then φ1, φ2, . . . , φn | = ψ.

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Completeness of Propositional Logic

Completeness Let φ1, φ2, . . . , φn and ψ be propositional formulas. If φ1, φ2, . . . , φn | = ψ, then φ1, φ2, . . . , φn ⊢ ψ.

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Next Week

Soundness and completeness of propositional logic Proof by induction Normal forms of propositional formulas SAT solving

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