- Logical equivalence
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 valuein 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.
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
contrapositionand 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 logicis assumed. Some non-classical logics do not deem (1) and (2) logically equivalent.)
If and only if
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