law of double negation

law of double negation

1. (logic) The statement that the negation of the negation of A implies A, for any proposition A. Stated symbolically: ${\displaystyle \neg \neg A\to A}$.

   The law of double negation is not valid intuitionistically. To show this with Heyting algebra semantics, let ${\displaystyle A=(0,1)\cup (1,2)}$. Then ${\displaystyle \neg A=(-\infty ,0)\cup (2,\infty )}$, ${\displaystyle \neg \neg A=(0,2)}$,  ${\displaystyle \neg \neg A\to A=(-\infty ,1)\cup (1,\infty )\neq \mathbb {R} }$.