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. Formulas and are logically equivalent if and only if the statement of their material equivalence () is a tautology (Copi et at. 2014:348).
The material equivalence of and (often written ) is itself another statement in the same object language as and . This statement 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 statements are logically equivalent if, in every model, they have the same truth value.