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 ${\displaystyle \neg A=(-\infty ,0)\cup (2,\infty )}$ , ${\displaystyle \neg \neg A=(0,2)}$ ,  ${\displaystyle \neg \neg A\to A=(-\infty ,1)\cup (1,\infty )\neq \mathbb {R} }$ .