In logic, statements and are logically equivalent if they have the same logical content. This is a semantic concept; two statements are equivalent if they have the same truth value in every model (Mendelson 1979:56). The logical equivalence of and is sometimes expressed as , , or . However, these symbols are also used for material equivalence; the proper interpretation depends on the context. Logical equivalence is different from material equivalence, although the two concepts are closely related.
The following statements are logically equivalent:
If Lisa is in France, then she is in Europe. (In symbols, .)
If Lisa is not in Europe, then she is not in France. (In symbols, .)
Syntactically, (1) and (2) are derivable from each other 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.
Logical equivalence is different from material equivalence. The material equivalence of and (often written ) is itself another statement, call it , in the same object language as and . expresses the idea "' if and only if '". In particular, the truth value of can change from one model to another.
The claim that two formulas are logically equivalent is a statement in the metalanguage, expressing a relationship between two statements and . The claim that and are semantically equivalent does not depend on any particular model; it says that in every possible model, will have the same truth value as . The claim that and are syntactically equivalent does not depend on models at all; it states that there is a deduction of from and a deduction of from .
There is a close relationship between material equivalence and logical equivalence. Formulas and are syntactically equivalent if and only if is a theorem, while and are semantically equivalent if and only if is true in every model (that is, is logically valid).