Negation introduction

Type	Rule of inference
Field	Propositional calculus
Statement	If a given antecedent implies both the consequent and its complement, then the antecedent is a contradiction.
Symbolic statement	${\displaystyle (P\rightarrow Q)\land (P\rightarrow \neg Q)\rightarrow \neg P}$

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: ${\displaystyle (P\rightarrow Q)\land (P\rightarrow \neg Q)\rightarrow \neg P}$

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	${\displaystyle \neg P\lor F}$	Law of noncontradiction
5	${\displaystyle \neg P}$	Disjunctive syllogism (3,4)

See also

• Reductio ad absurdum

References