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Discussion #16 Validity & Equivalences Discussion #16 1/12 Topics • Validity – Tautologies with Interpretations – Contradictions with Interpretations • Logical Equivalences Involving Quantifiers • Rectification Discussion #16 2/12 Validity • An expression that is true for all interpretations is said to be valid. (A valid expression is also call a tautology.) • An expression that is true for no interpretation is said to be contradictory. (A contradictory expression is also called a contradiction.) • If A is valid, A is contradictory. (a tautology) (a contradiction) • Examples: – P(x, y) P(x, y) P(x, y) P(x, y) is valid – P(x, y) P(x, y) is contradictory Discussion #16 3/12 Laws are Valid • All laws are valid. – de Morgan’s: (P(x) Q(y)) P(x) Q(y) – Identity: P(x) T P(x) • When we replace by , the resulting expression is true for all interpretations. – de Morgan’s: (P(x) Q(y)) P(x) Q(y) – Identity: P(x) T P(x) Discussion #16 4/12 Equivalence with Variants An expression with the variable names changed is called a variant. Proper variants are equivalent, i.e. it doesn’t matter what variable name is used. Example: xA ySxyA But, we must be careful 1. We must substitute only for the x’s bound by x. 2. Further, variables must not “clash.” Strong rule: y must not be in A; weaker rule: no y in the scope of x can be free in the scope of x, and no x bound by x may be in the scope of a bound y. x(y(P(y) Q(x,z)) xP(x)) w w z z y y x(yP(y) Q(x,z)) xP(x) y y Discussion #16 Works Doesn’t work Doesn’t work Works 5/12 Equivalences Involving Quantifiers 1. xA A if x not free in A 1d. xA A if x not free in A x(xP(x, z)) xP(x, z) 1) x is already bound xP(y, z) P(y, z) 2) There are no x’s xP(x, z) P(x, z) 3) x is free in P(x, z) 2. xA ySxyA 2d. xA ySxyA Discussion #16 if x does not “clash” with y in A if x does not “clash” with y in A 6/12 Equivalences Involving Quantifiers (continued…) 3. xA SxtA xA 3d. xA SxtA xA for any term t for any term t 1. When xA is false, so is SxtA xA. 2. When xA is true for all substitutions, SxtA is true, and hence SxtA xA is true. 3. We are just “anding in” something that’s already there. 4. Dual argument for 3d (“oring in” something that’s already there). Discussion #16 7/12 Equivalences Involving Quantifiers (continued…) Distributive laws: 4. x(A B) A xB if x not free in A 4d. x(A B) A xB if x not free in A P xQ(x) P (Q(x1) Q(x2) …) (P Q(x1)) (P Q(x2)) … x(P Q(x)) Associative laws: 5. x(A B) xA xB 5d. x(A B) xA xB Discussion #16 8/12 Equivalences Involving Quantifiers (continued…) Commutative laws: 6. xyA yxA 6d. xyA yxA deMorgan’s laws: 7. xA xA 7d. xA xA xP(x) (P(x1) P(x2) …) P(x1) P(x2) … xP(x) Discussion #16 9/12 Equivalence – Example Show: xP(x) Q x(P(x) Q) xP(x) Q xP(x) Q implication law xP(x) Q de Morgan’s law x(P(x) Q) distributive law x(P(x) Q) implication law Discussion #16 10/12 Rectification Standardizing variables apart, also called rectification we can rename variables to make distinct variables have distinct names. (xP(x, y) xQ(y, x)) yR(y, x) free same free free (xP(x, y) zQ(y, z)) wR(w, v) Discussion #16 11/12 Universal Quantification of Free Variables in a Tautology • Since P(x, y) P(x, y) is a tautology, it holds for every substitution of values for its variables for every interpretation. • Thus, P(x, y) P(x, y) y(P(x, y) P(x, y)) xy(P(x, y) P(x, y)) • Hence, we can drop the quantifiers for tautologies. Discussion #16 12/12