# Negation introduction

Short description: Logical rule of inference
Type Rule of inference Propositional calculus If a given antecedent implies both the consequent and its complement, then the antecedent is a contradiction.

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.

## Formal notation

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.

## Proof

Step Proposition Derivation
1 $\displaystyle{ (P \to Q)\land(P \to \neg Q) }$ Given
2 $\displaystyle{ (\neg P \lor Q)\land(\neg P \lor \neg Q) }$ Material implication
3 $\displaystyle{ \neg P \lor (Q \land \neg Q) }$ Distributivity
4 $\displaystyle{ \neg(Q \land \neg Q) }$ Law of noncontradiction
5 $\displaystyle{ \neg P }$ Disjunctive syllogism (3,4)