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.
Logical equivalence is sometimes denoted p ≡ q or p ⇔ q. However, the latter notation is also used for material equivalence.
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