Double 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, and vice versa."
Symbolic statement	${\displaystyle A\equiv \sim (\sim A)}$

In propositional logic, the double negation of a statement states that "it is not the case that the statement is not true". In classical logic, every statement is logically equivalent to its double negation, but this is not true in intuitionistic logic; this can be expressed by the formula A ≡ ~(~A) where the sign ≡ expresses logical equivalence and the sign ~ expresses negation.

Like the law of the excluded middle, this principle is considered to be a law of thought in classical logic,^[1] but it is disallowed by intuitionistic logic.^[2] The principle was stated as a theorem of propositional logic by Russell and Whitehead in Principia Mathematica as:

${\displaystyle \mathbf {*4\cdot 13} .\ \ \vdash .\ p\ \equiv \ \thicksim (\thicksim p)}$^[3]

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