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{{Merge to|Double negation|date=March 2012}}
 
:''For the [[theorem]] of [[propositional calculus|propositional logic]] based on the same concept, see [[double negation]]''.
 
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.''
 
The ''double negation introduction'' rule is:
:''P <math>\Leftrightarrow</math> {{not}}{{not}}P''
and the ''double negation elimination'' rule is:
:''{{not}}{{not}}P <math>\Leftrightarrow</math> P''
 
Where "<math>\Leftrightarrow</math>" is a [[metalogic]]al [[Symbol (formal)|symbol]] representing "can be replaced in a proof with."
 
== Formal notation ==
 
The ''double negation introduction'' rule may be written in [[sequent]] notation:
:<math>P \vdash \neg \neg P</math>
 
The ''double negation elimination'' rule may be written as:
:<math>\neg \neg P \vdash P</math>
 
In [[inference rule|rule form]]:
:<math>\frac{P}{\neg \neg P}</math>
and
:<math>\frac{\neg \neg P}{P}</math>
 
or as a [[Tautology (logic)|tautology]] (plain propositional calculus sentence):
:<math>P \to \neg \neg P</math>
and
:<math>\neg \neg P \to P</math>
 
These can be combined together into a single biconditional formula:
 
:<math> \neg \neg P \leftrightarrow P </math>.
 
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.
 
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>.
 
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.
 
==See also==
*[[Gödel–Gentzen negative translation]]
 
== References ==
{{reflist}}
 
{{DEFAULTSORT:Double Negative Elimination}}
[[Category:Rules of inference]]

Latest revision as of 11:46, 27 December 2014

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There's also other styles of online jobs which can be worth your while. Medical and legal transcription pay very well, and the number of online transcription businesses is increasing. Used to do appropriate transcription for around two months, and created a great deal of cash, but writing happens to be my interest, so I returned to writing. I've a great number of on-line and off-line publishing tasks, I can barely maintain. Once my school classes end for the summer, I'll likely go back to transcription.

Full disclosure: I just needed my own advice and included the phrase technothriller at the beginning of the Amazon book description for LT. COMMANDER MOLLIE SANDERS (see http://amzn.to/N6YyFZ). I also included the phrase thriller in the beginning of the Amazon book description for CIA DROP GENTLEMAN although A Thriller is about the book cover (see http://amzn.to/L38eiP).

There's money to be produced in community marketing. The strict regulations and upfront fees are enough to prevent those who want an instantaneous means of pressing items onto visitors. These same regulations and fees, weed-out your competitors, then when visitors see links on these forums, they realize the Net marketer is needed and respected by time and commitment because of their efforts leading to clout and quality traffic.

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