Double negation

In propositional logic, double negation is the theorem that states that "If a statement is true, then it is not the case that the statement is not true." This is expressed by saying that a proposition A is logically equivalent to not (not-A), or by the formula A ≡ ~(~A) where the sign ≡ expresses logical equivalence and the sign ~ expresses negation.

Double negation

Type	Theorem
Field	Propositional calculus Classical logic
Statement	If a statement is true, then it is not the case that the statement is not true."
Symbolic statement	${\displaystyle A\equiv \sim (\sim A)}$

Like the law of the excluded middle, this principle is considered to be a law of thought in classical logic, but it is disallowed by intuitionistic logic. The principle was stated as a theorem of propositional logic by Russell and Whitehead in Principia Mathematica as:

${\displaystyle \mathbf {*4\cdot 13} .\ \ \vdash .\ p\ \equiv \ \thicksim (\thicksim p)}$

"This is the principle of double negation, i.e. a proposition is equivalent of the falsehood of its negation."