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Syntactically , ''p'' and ''q'' are equivalent if each can be Proved from the other. Semantic ally, ''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 Logic s do not deem (1) and (2) logically equivalent.) SEE ALSO |
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