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Negation introduction

From Wikipedia, the free encyclopedia
Negation introduction
TypeRule of inference
FieldPropositional calculus
StatementIf a given antecedent implies both the consequent and its complement, then the antecedent is a contradiction.
Symbolic statement

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

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This can be written as:

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

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Step Proposition Derivation
1 Given
2 Material implication
3 Distributivity
4 Law of noncontradiction
5 Disjunctive syllogism (3,4)

See also

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References

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  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.