law of double negation
double negation law

Axiom 3 says the familiar law of double negation: "Not not a is equivalent to a".

Accordingly, negation in classical logic satisfies the law of double negation: A is equivalent to A.

Generally in non-classical logics, negation that satisfies the law of double negation is called involutive.

Correspondingly, classical Boolean logic arises by adding the law of double negation to intuitionistic logic.

Involutive negation (unary) can be added as an additional negation to t-norm logics whose residual negation is not itself involutive, that is, if it does not obey the law of double negation .

The double negation law can be seen by complementing the shading in the third diagram for x, which shades the x circle.

Adding the ex falso axiom to minimal logic results in intuitionistic logic, and adding the double negation law to minimal logic results in classical logic.

In both ordinary and Boolean algebra, negation works by exchanging pairs of elements, whence in both algebras it satisfies the double negation law (also called involution law)