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.
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)
Discussion #16
9/12
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
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))
 xy(P(x, y)  P(x, y))
• Hence, we can drop the quantifiers for
tautologies.
Discussion #16
12/12