Negation introduction

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 this implies the negated antecedent.

Formal notation 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 With identified as , the principle is as a special case of Frege's theorem, already in minimal logic.

Another derivation makes use of as the curried, equivalent form of . Using this twice, the principle is seen equivalent to the negation of

which, via modus ponens and rules for conjunctions, is itself equivalent to the valid noncontradiction principle for .

A classical derivation passing through the introduction of a disjunction may be given as follows: Step Proposition Derivation 1 Given 2 Classical equivalence of the material implication 3 Distributivity 4 Law of noncontradiction for 5 Disjunctive syllogism (3,4)

Negation introduction

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