The Concept

Two logical expressions are logically equivalent if they have the same truth value for every possible assignment of truth values to their variables.

Notation: P ≡ Q means P and Q are logically equivalent.

Equivalence is stronger than equality of truth values in a single case. It means: there is no input combination that makes one True and the other False.

Three categories of logical expressions:

• Tautology — True for every assignment of variables. Also written ⊤ or T.
• Contradiction (also called unsatisfiable) — False for every assignment. Also written ⊥ or F.
• Contingency — True for some assignments, False for others.

A tautology and any other tautology are logically equivalent. A contradiction and any other contradiction are logically equivalent. Two contingencies are equivalent only if their truth tables match exactly.