# law of double negation

1. (logic) The statement that the negation of the negation of A implies A, for any proposition A. Stated symbolically: $\neg \neg A \to A$.
• The law of double negation is not valid intuitionistically. To show this with Heyting algebra semantics, let $A = (0,1) \cup (1,2)$. Then $\neg A = (-\infty,0) \cup (2,\infty)$, $\neg \neg A = (0,2)$,  $\neg \neg A \to A = (-\infty,1) \cup (1,\infty) \ne \mathbb{R}$.