- 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.

Semantic ally, "p" and "q" are equivalent if they have the sametruth value in every model.Logical equivalence is often confused with

material equivalence .The former is a statement in themetalanguage , 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 atheorem , while "p" and "q" are semantically equivalentif 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 inEurope . (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 anddouble 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. Somenon-classical logic s do not deem (1) and (2) logically equivalent.)**ee also***

Logical biconditional

*Logical equality

*If and only if

*Equisatisfiability

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