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CS 1502 Formal Methods in CS 1502 Formal Methods in Computer ScienceComputer Science

Lecture Notes 13Lecture Notes 13Equivalences, Arguments, and Equivalences, Arguments, and

Proofs involving QuantifiersProofs involving Quantifiers

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Propositional LogicPropositional Logic

TautologyTautology

Tautological ConsequenceTautological Consequence

Tautological EquivalenceTautological Equivalence

Based on thetruth-functional

Connectives

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First-Order LogicFirst-Order Logic

Takes into consideration all of the truth-Takes into consideration all of the truth-functional connectives functional connectives (( ), ), the identity symbolthe identity symbol (=) (=), and the quantifiers , and the quantifiers ((x x y).y).

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First-Order LogicFirst-Order Logic

FO ValidityFO Validity: a sentence that can’t be false: a sentence that can’t be falseFO Consequence:FO Consequence: applies to an argument applies to an argument whose conclusion can’t be made false whose conclusion can’t be made false when all of its premises are true.when all of its premises are true.FO EquivalenceFO Equivalence applies to a pair of applies to a pair of sentences that, in all possible sentences that, in all possible circumstances, have the same truth valuescircumstances, have the same truth values

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FactsFacts

All All tautological consequencestautological consequences are are FO FO ConsequencesConsequences..

All All tautological equivalenciestautological equivalencies are are FO FO EquivalenciesEquivalencies..

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FO ConsequenceFO Consequence

x [P(x) Q(x)] Q(b)

P(b)

QP

b

x [Tet(x) Large(x)] Large(b)

Tet(b)

A B

C

C is not a tautological consequence of A and B

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Replacement MethodReplacement Method

This method is used to determine if a This method is used to determine if a sentence is an FO Validity and if an sentence is an FO Validity and if an argument is an FO Consequence.argument is an FO Consequence.

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Replacement MethodReplacement Method

Replace all predicates in the sentence or in the Replace all predicates in the sentence or in the argument with symbolic ones making sure that if a argument with symbolic ones making sure that if a predicate appears more than once it is replaced with the predicate appears more than once it is replaced with the same symbolic name.same symbolic name.

See if you can describe a circumstance where the See if you can describe a circumstance where the sentence is false, if this is impossible then the sentence sentence is false, if this is impossible then the sentence is a FO Validity.is a FO Validity.

See if you can describe a circumstance where the See if you can describe a circumstance where the conclusion is false and the premises are all true. If this is conclusion is false and the premises are all true. If this is impossible, then the conclusion is an FO Consequence impossible, then the conclusion is an FO Consequence of its premises.of its premises.

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DeMorgan’s Laws for QuantifiersDeMorgan’s Laws for Quantifiers

x P(x) x P(x) x [x [P(x)]P(x)]Nobody is P.Nobody is P.Everyone is not P.Everyone is not P.

x P(x) x P(x) x [x [P(x)]P(x)]It is not the case that everyone is P.It is not the case that everyone is P.Somebody is not P.Somebody is not P.

P

P

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Aristotelian Forms RevisitedAristotelian Forms Revisited

Negate: Negate: All P’s are Q’sAll P’s are Q’s..

~all x (P(x) ~all x (P(x) Q(x)) Q(x)) ~all x (~P(x) v Q(x)) ~all x (~P(x) v Q(x)) exist x (~(~P(x) v Q(x))) exist x (~(~P(x) v Q(x)))

exist x (P(x) ^ ~Q(x)) exist x (P(x) ^ ~Q(x))

Some P’s are not Q’sSome P’s are not Q’s

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A Special Form and its EquivalentA Special Form and its Equivalent

Only Q’s are P’sOnly Q’s are P’s

All P’s are Q’sAll P’s are Q’s

P

Q

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Other Other EquivalencesEquivalences and and Non-Non-EquivalencesEquivalences (which are which?)(which are which?)

x [P(x) x [P(x) Q(x)] Q(x)] x P(x) x P(x) x Q(x)x Q(x)

x [P(x) x [P(x) Q(x)] Q(x)] x P(x) x P(x) x Q(x)x Q(x)

x [P(x) x [P(x) Q(x)] Q(x)] x P(x) x P(x) x Q(x)x Q(x)

x [P(x) x [P(x) Q(x)] Q(x)] x P(x) x P(x) x Q(x)x Q(x)

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Other EquivalencesOther Equivalences

x P x P P, where x is not free in P P, where x is not free in P

x P x P P, where x is not free in P P, where x is not free in P

x [P x [P Q(x)] Q(x)] P P x Q(x)x Q(x)

x [P x [P Q(x)] Q(x)] P P x Q(x)x Q(x)

x P(x) x P(x) y P(y) y P(y)

x P(x) x P(x) y P(y) y P(y)

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Proofs Involving QuantifiersProofs Involving Quantifiers

Universal EliminationUniversal Elimination

x S(x)x S(x) … …

S(c) S(c) Elim Elim

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ExampleExample

ProveProve

x Cube(x)x Cube(x) x Large(x)x Large(x)

Large(d) Large(d) Cube(d)] Cube(d)]

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Proofs Involving Quantifiers

Universal Introduction c

…

S(c)

x S(x) Intro

Assume c is anarbitrary elementin the domain ofdiscourse.

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Example

Prove

x Cube(x) x Large(x)

x [Large(x) Cube(x)]

2020

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Proofs Involving Quantifiers

Existential Introduction

S(c) …

x S(x) Intro

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Example

Prove

Cube(e) Large(e) LeftOf(e,a) x [Cube(x) LeftOf(x,a)]

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Proofs Involving Quantifiers

Existential Elimination x S(x)

c S(c)

…

Q

Q Elim

Since there exists an x such that S(x),

let c designate this object.

Symbol c cannot appear outside this

subproof!

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Example

Prove

x Large(x) x Cube(x)

x [Large(x) Cube(x)]

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General Conditional Proof

Universal Introduction c P(c)

…

Q(c)

x [P(x) Q(x)] Intro

Assume c is anarbitrary elementin the domain ofDiscourse and assume P(c)

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Example

Prove

x [P(x) Q(x)] z [Q(z) R(z)]

x [P(x) R(x)]

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