Double Negative Elimination

For the theorem of propositional logic based on the same concept, see double negation.

In propositional logic, double negative elimination (also called double negation elimination, double negative introduction, double negation introduction, or simply double negation) are two valid rules of replacement. They are the inferences that if A is true, then not not-A is true and its converse, that, if not not-A is true, then A is true. The rule allows one to introduce or eliminate a negation from a logical proof. The rule is based on the equivalence of, for example, It is false that it is not raining. and It is raining.

The double negation introduction rule is:

P ¬¬P

and the double negation elimination rule is:

¬¬P P

Where "" is a metalogical symbol representing "can be replaced in a proof with."