Negation introduction

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Short description: Logical rule of inference
Negation introduction
TypeRule of inference
FieldPropositional calculus
StatementIf 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.[1][2]

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

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. https://archive.org/details/syntaxofnegation0000haeg.