Editing Involution (mathematics) (section)

=== Mathematical logic ===
The operation of complement in [[Boolean algebra (structure)|Boolean algebra]]s is an involution. Accordingly, [[negation]] in [[classical logic]] satisfies the ''[[Double_negation|law of double negation]]'': {{math|¬¬''A''}} is equivalent to {{math|''A''}}.

Generally in [[non-classical logic]]s, negation that satisfies the law of double negation is called ''involutive''. In [[Algebraic semantics (mathematical logic)|algebraic semantics]], such a negation is realized as an involution on the algebra of [[truth value]]s. Examples of logics that have involutive negation are Kleene and Bochvar [[three-valued logic]]s, [[Łukasiewicz logic|Łukasiewicz many-valued logic]], the [[fuzzy logic]] '[[monoidal t-norm logic|involutive monoidal t-norm logic]]' (IMTL), etc. Involutive negation is sometimes added as an additional connective to logics with non-involutive negation; this is usual, for example, in [[t-norm fuzzy logics]].

The involutiveness of negation is an important characterization property for logics and the corresponding [[variety (universal algebra)|varieties of algebras]]. For instance, involutive negation characterizes [[Boolean algebra (structure)|Boolean algebra]]s among [[Heyting algebra]]s. Correspondingly, classical [[classical logic|Boolean logic]] arises by adding the law of double negation to [[intuitionistic logic]]. The same relationship holds also between [[MV-algebra]]s and [[BL (logic)|BL-algebra]]s (and so correspondingly between [[Łukasiewicz logic]] and fuzzy logic [[BL (logic)|BL]]), IMTL and [[Monoidal t-norm logic|MTL]], and other pairs of important varieties of algebras (respectively, corresponding logics).

In the study of [[binary relation]]s, every relation has a [[converse relation]]. Since the converse of the converse is the original relation, the conversion operation is an involution on the [[category of relations]]. Binary relations are [[partial order|ordered]] through [[inclusion (set theory)|inclusion]]. While this ordering is reversed with the [[complementation (mathematics)|complementation]] involution, it is preserved under conversion.