Logical equivalence

In logic, statements p and q are logically equivalent if they have the same logical content.

Syntactically, p and q are equivalent if each can be proved from the other. Semantically, p and q are equivalent if they have the same truth value in every model.

Logical equivalence is often confused with material equivalence. The former is a statement in the metalanguage, claiming something about statements p and q in the object language. But the material equivalence of p and q (often written "p ↔ q") is itself another statement in the object language. There is a relationship, however; p and q are syntactically equivalent if and only if p ↔ q is a theorem, while p and q are semantically equivalent if and only if p ↔ q is a tautology.

The logical equivalence of p and q is sometimes expressed as p ≡ q or p ⇔ q. However, these symbols are also used for material equivalence; the proper interpretation depends on the context.

Example
The following statements are logically equivalent:


 * 1) If Lisa is in France, then she is in Europe. (In symbols, f → e.)
 * 2) If Lisa is not in Europe, then she is not in France. (In symbols, ~e → ~f.)

Syntactically, (1) and (2) are co-derivable via the rules of contraposition and double negation. Semantically, (1) and (2) are true in exactly the same models (interpretations, valuations); namely, those in which either Lisa is in France is false or Lisa is in Europe is true.

(Note that in this example classical logic is assumed. Some non-classical logics do not deem (1) and (2) logically equivalent.)