**Rules of Inference**

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 *p* to both *q* and ¬*q*, infer ¬*p*; this rule also being called *reductio ad absurdum*), *negation elimination* (from *p* and ¬*p* infer q; this rule also being called *ex falso quodlibet*), and *double negation elimination* (from ¬¬*p* infer *p*). 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 *p* then *p* must not be the case (i.e. *p* 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 ⊥. In this case the rule says that from *p* and ¬*p* 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 ¬*p* of *p* is defined as *p*→⊥. 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*.

Read more about this topic: Negation

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