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1Theorem 2.1.1 — Logical Equivalences (Epp page 35)
Given any statement variables p, q, and r, a tautology , and a contradiction , the following logical
equivalences hold:
Commutative laws:
p
q
q
p
p
q
q
p
Associative laws:
(p
q)
r
p
(q
r)
(p
q)
r
p
(q
r)
Distributive laws:
p
(q
r)
(p
q)
(p
r)
p
(q
r)
(p
q)
(p
r)
Identity laws:
p
p
p
p
Negation laws:
p
p
p
p
Double negative law:
( p)
p
Idempotent laws:
p
p
p
p
p
p
Universal bound laws:
p
p
De Morgan’s laws:
(p
q)
p
q
(p
q)
p
q
Absorption laws:
p
(p
q)
p
p
(p
q)
p
Negations of
and :
Table 2.3.1 — Rules of Inference (Epp, page 60)
Modus ponens
p
q
Disjunctive syllogism
p
q
p
q
p
q
p
q
p
q
Modus tollens
p
q
Hypothetical syllogism
p
q
q
q
r
p
p
r
Disjunctive addition
p
q
Dilemma, or
p
q
p
q
p
q
Proof by division
p
r
Conjunctive simplification
p
q
p
q
into cases
q
r
p
q
r
Conjunctive addition
p
Contradiction rule
p
q
p
p
q
Closing conditional
p (assumed)
Closing conditional
p (assumed)
world without
q (derived)
world with
q
q (derived)
contradiction
p
q
contradiction
p
Other Logical Equivalences and Rules of Inference
Definition of implication:
p
q
p
q
(p
q)
p
q
Contrapositive rule:
p
q
q
p
Definition of biconditional:
p
q
(p
q)
(q
p)
Negation of quantifiers:
( x P (x))
x
P (x)
( x P (x))
x
P (x)
Universal
x
D, P (x)
Q(x)
Universal
x
D, P (x)
Q(x)
modus ponens:
P (a) where a
D
modus tollens:
Q(a) where a
D
Q(a)
P (a)
Universal
x
D, P (x)
Existential
x
D, P (x)
instantiation:
P (a) where a
D
instantiation :
P (a) where a
D
Universal
P (a) where a
D
Existential
P (a) where a
D
generalization :
x
D, P (x)
generalization:
x
D, P (x)
Remember the special circumstances required for the rules marked by the stars.
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