# 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. 

## Formal notation

This can be written as: $(P\rightarrow Q)\land (P\rightarrow \neg Q)\leftrightarrow \neg P$ 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 "When the phone rings I get happy" and then later state "When the phone rings I get annoyed", the logical inference which is made from this contradictory information is that the person is making a false statement about the phone ringing.

## Proof

Step Proposition Derivation
1$(P\to Q)\land (P\to \neg Q)$ Given
2$(\neg P\lor Q)\land (\neg P\lor \neg Q)$ Material implication
3$((\neg P\lor Q)\land \neg P)\lor ((\neg P\lor Q)\land \neg Q)$ Distributivity
4$((\neg P\lor Q)\lor ((\neg P\lor Q)\land \neg Q))\land (\neg P\lor ((\neg P\lor Q)\land \neg Q))$ Distributivity
5$\neg P\lor ((\neg P\lor Q)\land \neg Q)$ Conjunction elimination (4)
6$\neg P\lor ((\neg P\land \neg Q)\lor (Q\land \neg Q))$ Distributivity
7$\neg (Q\land \neg Q)$ Law of noncontradiction
8$\neg P\lor (\neg P\land \neg Q)$ Disjunctive syllogism (5,6)
9$(\neg P\lor \neg P)\land (\neg P\lor \neg Q)$ Distributivity
10$\neg P\lor \neg P$ Conjunction elimination (7)
11$\neg P$ Idempotency of disjunction