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".[citation needed] 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.[1]
"This is the principle of double negation, i.e. a proposition is equivalent of the falsehood of its negation."
^Or alternate symbolism such as A ↔ ¬(¬A) or Kleene's *49o: A ∾ ¬¬A (Kleene 1952:119; in the original Kleene uses an elongated tilde ∾ for logical equivalence, approximated here with a "lazy S".)
^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)
^The o of Kleene's formula *49o indicates "the demonstration is not valid for both systems [classical system and intuitionistic system]", Kleene 1952:101.
^PM 1952 reprint of 2nd edition 1927 pp. 101–02, 117.