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 this implies the negated antecedent.
This can be written as:
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 ìP, assume for contradiction P, then derive from it two contradictory inferences Q and ìQ. Since the latter contradiction renders P impossible, ìP must hold.
With identified as , the principle is as a special case of Frege's theorem, already in minimal logic.
Another derivation makes use of as the curried, equivalent form of . Using this twice, the principle is seen equivalent to the negation of
which, via modus ponens and rules for conjunctions, is itself equivalent to the valid noncontradiction principle for .
A classical derivation passing through the introduction of a disjunction may be given as follows: