CS 1502 Formal Methods in Computer Science

CS 1502 Formal Methods in Computer Science Lecture Notes 13 Equivalences, Arguments, and Proofs involving Quantifiers

Propositional Logic • Tautology • Tautological Consequence • Tautological Equivalence Based on the truth-functional Connectives

First-Order Logic • Takes into consideration all of the truth-functional connectives (     ), the identity symbol (=), and the quantifiers (x y).

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

Facts • All tautological consequences are FO Consequences. • All tautological equivalencies are FO Equivalencies.

C is not a tautological consequence of A and  B x [P(x)  Q(x)] Q(b) P(b) Q x [Tet(x)  Large(x)] Large(b) Tet(b) AB C P b FO Consequence

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

Replacement Method • Replace all predicates in the sentence or in the argument with symbolic ones making sure that if a predicate appears more than once it is replaced with the same symbolic name. • See if you can describe a circumstance where the sentence is false, if this is impossible then the sentence is a FO Validity. • See if you can describe a circumstance where the conclusion is false and the premises are all true. If this is impossible, then the conclusion is an FO Consequence of its premises.

DeMorgan’s Laws for Quantifiers • x P(x)  x [P(x)]Nobody is P.Everyone is not P. • x P(x)  x [P(x)]It is not the case that everyone is P.Somebody is not P. P P

Aristotelian Forms Revisited • Negate: All P’s are Q’s.~all x (P(x)  Q(x)) ~all x (~P(x) v Q(x)) exist x (~(~P(x) v Q(x)))  exist x (P(x) ^ ~Q(x)) Some P’s are not Q’s

P Q A Special Form and its Equivalent • Only Q’s are P’s • All P’s are Q’s

Other Equivalences and Non-Equivalences(which are which?) • x [P(x)  Q(x)]  x P(x) x Q(x) • x [P(x)  Q(x)]  x P(x) x Q(x) • x [P(x)  Q(x)]  x P(x) x Q(x) • x [P(x)  Q(x)]  x P(x) x Q(x)

Other Equivalences • x P  P, where x is not free in P • x P  P, where x is not free in P • x [P  Q(x)]  Px Q(x) • x [P  Q(x)]  P x Q(x) • x P(x)  y P(y) •  x P(x)   y P(y)

Proofs Involving Quantifiers • Universal Elimination x S(x) … S(c)  Elim

Example • Provex Cube(x)x Large(x) Large(d)  Cube(d)]

Assume c is an arbitrary element in the domain of discourse. Proofs Involving Quantifiers • Universal Introduction c … S(c) x S(x)  Intro

Example • Provex Cube(x)x Large(x)x [Large(x)  Cube(x)]

Proofs Involving Quantifiers • Existential Introduction • S(c) … x S(x)  Intro

Example • ProveCube(e) Large(e)  LeftOf(e,a)x [Cube(x)  LeftOf(x,a)]

Since there exists an x such that S(x), let c designate this object. Symbol c cannot appear outside this subproof! Proofs Involving Quantifiers • Existential Eliminationx S(x) c S(c) … Q Q  Elim

Example • Provex Large(x)x Cube(x)x [Large(x)  Cube(x)]

Assume c is an arbitrary element in the domain of Discourse and assume P(c) General Conditional Proof • Universal Introduction c P(c) … Q(c) x [P(x)  Q(x)]  Intro

Example • Provex [P(x)  Q(x)]z [Q(z)  R(z)]x [P(x)  R(x)]

CS 1502 Formal Methods in Computer Science