Double negation

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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.

Like the law of the excluded middle, this principle is considered to be a law of thought in classical logic, (Hamilton is discussing Hegel in the following: "In the more recent systems of philosophy, the universality and necessity of the axiom of Reason has, with other logical laws, been controverted and rejected by speculators on the absolute.[On principle of Double Negation as another law of Thought, see Fries, Logik, §41, p. 190; Calker, Denkiehre odor Logic und Dialecktik, §165, p. 453; Beneke, Lehrbuch der Logic, §64, p. 41.]" (Hamilton 1860:68)</ref> but it is disallowed by intuitionistic logic.) The o of Kleene's formula *49o indicates "the demonstration is not valid for both systems [classical system and intuitionistic system]", Kleene 1952:101.</ref> The principle was stated as a theorem of propositional logic by Russell and Whitehead in Principia Mathematica as:

$\mathbf{*4\cdot13}. \ \ \vdash.\ p \ \equiv \ \thicksim(\thicksim p)$
"This is the principle of double negation, i.e. a proposition is equivalent of the falsehood of its negation."