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.[1] [2]

Formal notation

This can be written as: ${\displaystyle (P\rightarrow Q)\land (P\rightarrow \neg Q)\leftrightarrow \neg P}$

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

References

1. Wansing, Heinrich, ed. (1996). Negation: A Notion in Focus. Berlin: Walter de Gruyter. ISBN 3110147696.
2. Haegeman, Lilliane (30 Mar 1995). The Syntax of Negation. Cambridge: Cambridge University Press. p. 70. ISBN 0521464927.