Discussion #16 Validity & Equivalences

1. Discussion #16Validity & Equivalences

2. Topics • Validity • Tautologies with Interpretations • Contradictions with Interpretations • Logical Equivalences Involving Quantifiers • Rectification

3. 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

4. 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)

5. 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 We must substitute only for the x’s bound by x. 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)) Equivalence with Variants w w Works z z Doesn’t work y y Doesn’t work x(yP(y)  Q(x,z))  xP(x) y yWorks

6. 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 if x does not “clash” with y in A 2d. xA  ySxyA if x does not “clash” with y in A

7. Equivalences Involving Quantifiers (continued…) 3. xA  SxtA  xA for any term t 3d. xA  SxtA  xA for any term t • When xA is false, so is SxtA  xA. • 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).

8. 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

9. Equivalences Involving Quantifiers (continued…) Commutative laws: 6. xyA  yxA 6d. xyA  yxA deMorgan’s laws: 7. xA  xA 7d. xA  xA xP(x)  (P(x1)  P(x2)  …)  P(x1)  P(x2)  …  xP(x)

10. Equivalence – Example Show: xP(x)  Q  x(P(x)  Q) xP(x)  Q  xP(x)  Q implication law  xP(x)  Q de Morgan’s law  x(P(x)  Q) distributive law  x(P(x)  Q) implication law

11. free free same free 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) (xP(x, y)  zQ(y, z))  wR(w, v)

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))  xy(P(x, y)  P(x, y)) • Hence, we can drop the quantifiers for tautologies.