Negation introduction

Short description: Logical rule of inference

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.

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: [math]\displaystyle{ (P \rightarrow Q) \land (P \rightarrow \neg Q) \rightarrow \neg P }[/math]

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	[math]\displaystyle{ (P \to Q)\land(P \to \neg Q) }[/math]	Given
2	[math]\displaystyle{ (\neg P \lor Q)\land(\neg P \lor \neg Q) }[/math]	Material implication
3	[math]\displaystyle{ \neg P \lor (Q \land \neg Q) }[/math]	Distributivity
4	[math]\displaystyle{ \neg P \lor F }[/math]	Law of noncontradiction
5	[math]\displaystyle{ \neg P }[/math]	Disjunctive syllogism (3,4)

See also

• Reductio ad absurdum