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| {{Transformation rules}}
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| {{Merge to|Double negation|date=March 2012}}
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| :''For the [[theorem]] of [[propositional calculus|propositional logic]] based on the same concept, see [[double negation]]''.
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| In [[propositional calculus|propositional logic]], '''double negative elimination''' (also called '''double negation elimination''', '''double negative introduction''', '''double negation introduction''', or simply '''double negation'''<ref>Copi and Cohen</ref><ref>Moore and Parker</ref><ref>Hurley</ref>) are two [[validity|valid]] [[rule of replacement|rules of replacement]]. They are the [[inference]]s that if ''A'' is true, then ''not not-A'' is true and its [[converse (logic)|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|logical proof]]. The rule is based on the equivalence of, for example, ''It is false that it is not raining.'' and ''It is raining.''
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| The ''double negation introduction'' rule is:
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| :''P <math>\Leftrightarrow</math> {{not}}{{not}}P''
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| and the ''double negation elimination'' rule is:
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| :''{{not}}{{not}}P <math>\Leftrightarrow</math> P''
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| Where "<math>\Leftrightarrow</math>" is a [[metalogic]]al [[Symbol (formal)|symbol]] representing "can be replaced in a proof with."
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| == Formal notation ==
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| The ''double negation introduction'' rule may be written in [[sequent]] notation:
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| :<math>P \vdash \neg \neg P</math>
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| The ''double negation elimination'' rule may be written as:
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| :<math>\neg \neg P \vdash P</math>
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| In [[inference rule|rule form]]:
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| :<math>\frac{P}{\neg \neg P}</math>
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| and
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| :<math>\frac{\neg \neg P}{P}</math>
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| or as a [[Tautology (logic)|tautology]] (plain propositional calculus sentence):
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| :<math>P \to \neg \neg P</math>
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| and
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| :<math>\neg \neg P \to P</math>
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| These can be combined together into a single biconditional formula:
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| :<math> \neg \neg P \leftrightarrow P </math>.
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| Since biconditionality is an [[equivalence relation]], any instance of ¬¬''A'' in a [[well-formed formula]] can be replaced by ''A'', leaving unchanged the [[truth-value]] of the well-formed formula.
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| Double negative elimination is a theorem of [[classical logic]], but not of weaker logics such as [[intuitionistic logic]] and [[minimal logic]]. Because of their constructive flavor, 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]].) Double negation introduction is a theorem of both intuitionistic logic and minimal logic, as is <math> \neg \neg \neg A \vdash \neg A </math>.
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| In [[naive set theory|set theory]] also we have the negation operation of the [[complement (set theory)|complement]] which obeys this property: a set A and a set (A<sup>C</sup>)<sup>C</sup> (where A<sup>C</sup> represents the complement of A) are the same.
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| ==See also==
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| *[[Gödel–Gentzen negative translation]]
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| == References ==
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| {{reflist}}
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| {{DEFAULTSORT:Double Negative Elimination}}
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| [[Category:Rules of inference]]
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Alysha Jenkin is my name and I feel relaxed when people utilize the complete name. My partner doesn't like it the way in which I do but what I really like doing is always to collect coins and I'm going to be starting something else alongside it. Nj could be the only position I Have been surviving in. I am currently an information officer.
Feel free to visit my weblog ... jaun pablo schiappacasse canepa