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 VariantsAn 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 careful1. 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 A if x not free in A1d. 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 A2d. xA ySx

yA if x does not “clash” with y in A

Discussion #16 7/12

Equivalences Involving Quantifiers (continued…)

3. xA SxtA xA for any term t

3d. xA SxtA xA 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 8/12

Equivalences Involving Quantifiers (continued…)

Distributive laws:4. x(A B) A xB if x not free in A4d. 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 xB5d. x(A B) xA xB

Discussion #16 9/12

Equivalences Involving Quantifiers (continued…)

Commutative laws:6. xyA yxA6d. xyA yxA

deMorgan’s laws:7. xA xA7d. xA xA

xP(x) (P(x1) P(x2) …) P(x1) P(x2) … xP(x)

Discussion #16 10/12

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 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)

freefree freesame

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)) xy(P(x, y) P(x, y))

• Hence, we can drop the quantifiers for tautologies.