Law of double negation
The law of double negation (also the principle of double negation , or Latin duplex negatio affirmat - the double negative affirms / affirms ) is a law of classical logic according to which the negation of a negative (proposition) sentence is equivalent to the affirmation of the sentence, a doubly negative proposition ¬¬A has the same truth value as the non-negated proposition A.
The double negation in the sense of logic is to be distinguished from the negation of negation in the sense of dialectic (Hegel's).
The validity of the law of double negation is unrestricted in classical logic, since there the principle of bivalence applies. In intuitionist logic , the law is not valid. In this only A → ¬¬A, but not ¬¬A → A.
As rules of inference , the can double negation introduction and the double negation elimination lead.
The rule of introducing double negations says: If the sentence A can be inferred from a set of assumptions X, then the double negation of A can also be deduced from the same set X, i.e. ¬¬A.
The rule of double negation elimination says: If one can deduce the double negation of A - i.e. ¬¬A from a set of assumptions X, then one can also infer A from this set X.
Whether a double negative in a natural language cancels out the first negative ( German , Latin ) or reinforces it ( English , French , Spanish ) depends on the language in question.