In logic and mathematics, statements and are said to be logically equivalent, if they are provable from each other under a set of axioms,^{[1]} or have the same truth value in every model.^{[2]} The logical equivalence of and is sometimes expressed as , ,^{[3]}, or , depending on the notation being used. However, these symbols are also used for material equivalence, so proper interpretation would depend on the context. Logical equivalence is different from material equivalence, although the two concepts are intrinsically related.
In logic, many common logical equivalences exist and are often listed as laws or properties. The following tables illustrate some of these.
Equivalence | Name |
---|---|
Identity laws | |
Domination laws | |
Idempotent or tautology laws | |
Double negation law | |
Commutative laws | |
Associative laws | |
Distributive laws | |
De Morgan's laws | |
Absorption laws | |
Negation laws |
The following statements are logically equivalent:
Syntactically, (1) and (2) are derivable from each other 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 Denmark 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) to be logically equivalent.)
In mathematics, two statements and are often said to be logically equivalent, if they are provable from each other given a set of axioms and presuppositions. For example, the statement " is divisible by 6" can be regarded as equivalent to the statement " is divisible by 2 and 3", since one can prove the former from the latter (and vice versa) using some knowledge from basic number theory.^{[1]}
Logical equivalence is different from material equivalence. Formulas and are logically equivalent if and only if the statement of their material equivalence () is a tautology.^{[5]}
The material equivalence of and (often written as ) is itself another statement in the same object language as and . This statement expresses the idea "' if and only if '". In particular, the truth value of can change from one model to another.
On the other hand, the claim that two formulas are logically equivalent is a statement in the metalanguage, which expresses a relationship between two statements and . The statements are logically equivalent if, in every model, they have the same truth value.