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} }

.

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