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.

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

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

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

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  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)]

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

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 Prove Cube(e) Large(e)  LeftOf(e,a) x [Cube(x)  LeftOf(x,a)]

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

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

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

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