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

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.