Negation introduction

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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[edit]

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 "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[edit]

Step Proposition Derivation
1 Given
2 Material implication
3 Distributivity
4 Distributivity
5 Conjunction elimination (4)
6 Distributivity
7 Law of noncontradiction
8 Disjunctive syllogism (6,7)
9 Conjunction elimination (8)
10 Idempotency of disjunction

References[edit]

  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.