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

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Negation introduction

Type	Rule of inference
Field	Propositional calculus
Statement	If a given antecedent implies both the consequent and its complement, then the antecedent is a contradiction.
Symbolic statement	${\displaystyle (P\rightarrow Q)\land (P\rightarrow \neg Q)\rightarrow \neg P}$

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

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.

• Reductio ad absurdum