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	$\neg P\lor \neg P$	Conjunction elimination (7)
10	$\neg P$	Idempotency of disjunction