Negation introduction

Negation introduction is a rule of inference, or transformation rule, in the field of propositional calculus.

Negation introduction states that if a given antecedent implies both the consequent and its complement, then the antecedent is a contradiction.^{[1] [2]}

This can be written as:

${\displaystyle {\Big (}(P\rightarrow Q)\land (P\rightarrow \neg Q){\Big )}\rightarrow \neg P}${\displaystyle {\Big (}(P\rightarrow Q)\land (P\rightarrow \neg Q){\Big )}\rightarrow \neg P}

An example of its use would be an attempt to prove two contradictory statements from a single fact. For example, if a person were to state "Whenever I hear the phone ringing I am happy" and then state "Whenever I hear the phone ringing I am not happy", one can infer that the person never hears the phone ringing.

Many proofs by contradiction use negation introduction as reasoning scheme: to prove ¬P, assume for contradiction P, then derive from it two contradictory inferences Q and ¬Q. Since the latter contradiction renders P impossible, ¬P must hold.

With ${\displaystyle \neg P}${\displaystyle \neg P} identified as ${\displaystyle P\to \bot }${\displaystyle P\to \bot }, the principle is as a special case of Frege's theorem, already in minimal logic.

Another derivation makes use of ${\displaystyle A\to \neg B}${\displaystyle A\to \neg B} as the curried, equivalent form of ${\displaystyle \neg (A\land B)}${\displaystyle \neg (A\land B)}. Using this twice, the principle is seen equivalent to the negation of ${\displaystyle {\big (}P\land (P\to Q){\big )}\land \neg (P\land Q)}${\displaystyle {\big (}P\land (P\to Q){\big )}\land \neg (P\land Q)} which, via modus ponens and rules for conjunctions, is itself equivalent to the valid noncontradiction principle for ${\displaystyle P\land Q}${\displaystyle P\land Q}.

A classical derivation passing through the introduction of a disjunction may be given as follows:

• Reductio ad absurdum

• Propositional calculus
• Rules of inference

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