The Laws of Logic
Two statements s1and s2are logically equivalent if s1↔s2is a tautology, that is, s1and
s2have the same truth table (up to the order of the rows)
If s1and s2are logically equivalent, we write s1⇔s2. Note that s1↔s2is a statement
and can in general be true or false, and s1⇔s2indicates the (higher level) fact that it is a
tautology.
Logically equivalent statements are “the same” in the sense that logically equivalent
statements can be freely substituted for each other without changing the meaning of a
compound statement.
Here are some basic logical equivalences. Each of the following can be verified (proved)
with a truth table. It is a good idea to memorize them, so that they are at your fingertips
when needed. In what follows, 1denotes a statement that is always true (i.e. a tautology),
and 0denotes a statement that is always false (i.e. a contradiction).
•Idempotence: p∨p⇔p, p ∧p⇔p
•Commutative: p∧q⇔q∧p, p ∨q⇔q∨p
•Associative: (p∧q)∧r⇔p∧(q∧r),(p∨q)∨r⇔p∨(q∨r)
•Distributative: p∨(q∧r)⇔(p∨q)∧(p∨r), p ∧(q∨r)⇔(p∧q)∨(p∧r)
•Double Negation: ¬(¬p)⇔p
•DeMorgan’s Laws: ¬(p∨q)⇔ ¬p∧ ¬q , ¬(p∧q)⇔ ¬p∨ ¬q
•Identity: p∧1⇔p, p ∨0⇔p
•Dominance: p∧0⇔0, p ∨1⇔1
The following are some other useful logical equivalences.
•p→q⇔ ¬p∨q
•p↔q⇔(p→q)∧(q→p)⇔(¬p∨q)∧(p∨ ¬q)
It is apparent that the Laws of Logic come in pairs. The dual of a statement is obtained
by replacing ∨by ∧;∧by ∨;0by 1; and 1by 0, wherever they occur. It is a theorem of
logic that if s1is logically equivalent to s2, then the dual of s1is logically equivalent to the
dual of s2.
