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Double Negation Elimination
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.
"This is the principle of double negation, i.e. a proposition is equivalent of the falsehood of its negation."
Elimination and introduction
'Double negation elimination and double negation introduction are two validrules of replacement. They are the inferences that if A is true, then not not-A is true and its converse, that, if not not-A is true, then A is true. The rule allows one to introduce or eliminate a negation from a formal proof. The rule is based on the equivalence of, for example, It is false that it is not raining. and It is raining.
Because of their constructive character, a statement such as It's not the case that it's not raining is weaker than It's raining. The latter requires a proof of rain, whereas the former merely requires a proof that rain would not be contradictory. This distinction also arises in natural language in the form of litotes.
^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.