Double negation

Double negation
TypeTheorem
Field
StatementIf a statement is true, then it is not the case that the statement is not true, and vice versa."
Symbolic statement

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:

[3]
"This is the principle of double negation, i.e. a proposition is equivalent of the falsehood of its negation."
  1. ^ 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)
  2. ^ The o of Kleene's formula *49o indicates "the demonstration is not valid for both systems [classical system and intuitionistic system]", Kleene 1952:101.
  3. ^ PM 1952 reprint of 2nd edition 1927 pp. 101–02, 117.