Law of double negation

English

Noun

1. Template:Lb The statement that the negation of the negation of A implies A, for any proposition A. Stated symbolically: $\mathrm{\neg }\mathrm{\neg }A\to A$.

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