law of double negation
Noun
Meaning
law of double negation
(logic) The statement that the negation of the negation of A implies A, for any proposition A. Stated symbolically:
¬
¬
A
→
A
{\displaystyle \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
)
∪
(
1
,
2
)
{\displaystyle A=(0,1)\cup (1,2)}
. Then
¬
A
=
(
−
∞
,
0
)
∪
(
2
,
∞
)
{\displaystyle \neg A=(-\infty,0)\cup (2,\infty )}
,
¬
¬
A
=
(
0
,
2
)
{\displaystyle \neg \neg A=(0,2)}
,
¬
¬
A
→
A
=
(
−
∞
,
1
)
∪
(
1
,
∞
)
≠
R
{\displaystyle \neg \neg A\to A=(-\infty,1)\cup (1,\infty )\neq \mathbb {R} }
.