Discussion #16 1/12 Discussion #16 Validity & Equivalences

Discussion #16 2/12 Topics Validity –Tautologies with Interpretations –Contradictions with Interpretations Logical Equivalences Involving Quantifiers Rectification

Discussion #16 3/12 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 4/12 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 5/12 z z Doesn’t work 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   yS x y A 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 Works  x(  yP(y)  Q(x,z))   xP(x) y y Works y y Doesn’t work

Discussion #16 6/12 Equivalences Involving Quantifiers 1.  xA  Aif x not free in A 1d.  xA  Aif 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   yS x y A if x does not “clash” with y in A 2d.  xA   yS x y Aif x does not “clash” with y in A

Discussion #16 7/12 Equivalences Involving Quantifiers (continued…) 3.  xA  S x t A   xAfor any term t 3d.  xA  S x t A   xAfor any term t 1.When  xA is false, so is S x t A   xA. 2.When  xA is true for all substitutions, S x t A is true, and hence S x t A   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 8/12 Equivalences Involving Quantifiers (continued…) Distributive laws: 4.  x(A  B)  A   xBif x not free in A 4d.  x(A  B)  A   xBif x not free in A P   xQ(x)  P  (Q(x 1 )  Q(x 2 )  …)  (P  Q(x 1 ))  (P  Q(x 2 ))  …   x(P  Q(x)) Associative laws: 5.  x(A  B)   xA   xB 5d.  x(A  B)   xA   xB

Discussion #16 9/12 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(x 1 )  P(x 2 )  …)   P(x 1 )   P(x 2 )  …   x  P(x)

Discussion #16 10/12 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

Discussion #16 11/12 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) (  xP(x, y)   zQ(y, z))   wR(w, v) free same

Discussion #16 12/12 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.