Editing Negation (section)

==Rules of inference==
{{See also|Double negation}}
There are a number of equivalent ways to formulate rules for negation. One usual way to formulate classical negation in a [[natural deduction]] setting is to take as primitive rules of inference ''negation introduction'' (from a derivation of <math>P</math> to both <math>Q</math> and <math>\neg Q</math>, infer <math>\neg P</math>; this rule also being called ''[[reductio ad absurdum]]''), ''negation elimination'' (from <math>P</math> and <math>\neg P</math> infer <math>Q</math>; this rule also being called ''ex falso quodlibet''), and ''double negation elimination'' (from <math>\neg \neg P</math> infer <math>P</math>). One obtains the rules for intuitionistic negation the same way but by excluding double negation elimination.

Negation introduction states that if an absurdity can be drawn as conclusion from <math>P</math> then <math>P</math> must not be the case (i.e. <math>P</math> is false (classically) or refutable (intuitionistically) or etc.). Negation elimination states that anything follows from an absurdity. Sometimes negation elimination is formulated using a primitive absurdity sign <math>\bot</math>. In this case the rule says that from <math>P</math> and <math>\neg P</math> follows an absurdity. Together with double negation elimination one may infer our originally formulated rule, namely that anything follows from an absurdity.

Typically the intuitionistic negation <math>\neg P</math> of <math>P</math> is defined as <math>P \rightarrow \bot</math>. Then negation introduction and elimination are just special cases of implication introduction ([[conditional proof]]) and elimination (''[[modus ponens]]''). In this case one must also add as a primitive rule ''ex falso quodlibet''.