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{{Transformation rules}}
{{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]]

Revision as of 18:08, 2 February 2014

Template:Transformation rules

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For the theorem of propositional logic based on the same concept, see double negation.

In propositional logic, double negative elimination (also called double negation elimination, double negative introduction, double negation introduction, or simply double negation[1][2][3]) are two valid rules 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 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 Template:NotTemplate:NotP

and the double negation elimination rule is:

Template:NotTemplate:NotP P

Where "" is a metalogical symbol representing "can be replaced in a proof with."

Formal notation

The double negation introduction rule may be written in sequent notation:

P¬¬P

The double negation elimination rule may be written as:

¬¬PP

In rule form:

P¬¬P

and

¬¬PP

or as a tautology (plain propositional calculus sentence):

P¬¬P

and

¬¬PP

These can be combined together into a single biconditional formula:

¬¬PP.

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 ¬¬¬A¬A.

In set theory also we have the negation operation of the complement which obeys this property: a set A and a set (AC)C (where AC represents the complement of A) are the same.

See also

References

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  1. Copi and Cohen
  2. Moore and Parker
  3. Hurley