Discussion #16 Validity & Equivalences
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Transcript of Discussion #16 Validity & Equivalences
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.