Language Proof and Logic The Logic of Quantifiers

Language, Proof and Logic The Logic of Quantifiers Chapter 10

10. 0 First-order vs. second-order First-order: x [Loves(max, x) Loves(x, max)] Second-order: P [P(max) P(julia)] Mixed: P [ x. P(x)]

10. 1. a Tautologies and quantification Are the following arguments valid? Are they tautologically so? x [Cube(x) Small(x)] x Cube(x) x Small(x) x [Cube(x) Small(x)]

10. 1. b Tautologies and quantification Are the following sentences logically true? Are they also tautologies? x [Cube(x) Cube(x)] x Cube(x) x. Cube(x) [ y(P(y) R(y)) x(P(x) Q(x))] [ x(P(x) Q(x)) y(P(y) R(y))] [A B] [ B A] The truth-functional form of the previous sentence A quantified sentence of FOL is said to be a tautology iff its truth-functional form is a tautology.

10. 1. c Tautologies and quantification FO sentence x. Cube(x) [ y. Tet(y) z. Small(z)] z. Small(z) x. Cube(x) y. Tet(y) x. Cube(x) Cube(a) x(Cube(x) Cube(x)) x[Cube(x) Small(x)] x. Dodec(x) Its truth-functional form

10. 2. a First-order validity and consequence Propositional logic First-order logic General notion Tautology Tautological consequence Tautological equivalence FO validity FO consequence FO equivalence Logical truth Logical consequence Logical equivalence FO validity, consequence and equivalence mean those that hold solely in virtue of the truth-functional connectives, quantifiers and identity. Two reasons for treating identity as a logical symbol: 1. = is very common and universal 2. = allows us to express many quantified noun phrases, such as “exactly one”, “at least 5”, “at most 7”, etc.

10. 2. b First-order validity and consequence Are the following sentence FO valid? x. Same. Size(x, x) x. Cube(x) Cube(b) [Cube(b) b=c] Cube(c) [Small(b) Same. Size(b, c)] Small(c) To test FO validity, consequence or equivalence: Use nonsense Replace by meaningless symbols x. Bolshe(x, x) x. B(x, x) x. Tove(x) Tove(b) x. T(x) T(b) [Tove(b) b=c] Tove(c) [T(b) b=c] T(c) [Cheten(b) Menshe(b, c)] Cheten(c) [C(b) M(b, c)] C(c)

10. 2. c First-order validity and consequence Are the conclusions of the following arguments FO consequences of the premises? x[Tet(x) Large(x)] Large(b) Tet(b) x. Larger(x, a) x. Larger(b, x) Larger(c, d) x. Loves(x, moriarty) x. Loves(scrooge, x) Loves(romeo, juliet) Larger(a, b) Loves(moriarty, scrooge) Generally, recognizing FO (non)validity is not easy. No computer can ever be able to successfully handle every instance of this problem! You try it, p. 273

10. 3. a First-order equivalence and De. Morgan’s laws Are the following pairs FO equivalent? Are they also tautologically so? [ x. Cube(x) y. Dodec(y)] x[Cube(x) Small(x)] vs. x. Cube(x) y. Dodec(y) vs. x[ Small(x) Cube(x)] We say that two wffs are FO equivalent iff, when you replace their free variables by new names, the resulting sentences are FO equivalent. Substitution of a (sub)wff by a FO equivalent one preserves FO equivalence. Hence, e. g. , we have: x[Cube(x) Small(x)] x[ Cube(x) Small(x)] x[ Cube(x) Small(x)] x [Cube(x) Small(x)]

10. 3. b First-order equivalence and De. Morgan’s laws Remember that is a “big conjunction” and a “big disjunction”. Hence no wonder that De. Morgan’s laws extend to quantifiers: x. P(x) x P(x) Example: Bring the following formula down to an equivalent formula where negation is only applied to atoms: x[P(x) Q(x)]

10. 4 Other quantifier equivalences x[P(x) Q(x)] x. P(x) x. Q(x) When P has no free occurrences of x (“null quantification”): x. P P x[P Q(x)] P x. Q(x) When y does not occur in P(x) (“replacing bound variables”): x. P(x) y. P(y)