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:

#If Lisa is in France, then she is in Europe. (In symbols, "f" → "e".)
#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.)

ee also

* Logical biconditional
* Logical equality
* If and only if
* Equisatisfiability