negation introduction

This can be written as:

:$\backslash Big((P\; \backslash rightarrow\; Q)\; \backslash land\; (P\; \backslash rightarrow\; \backslash neg\; Q)\backslash Big)\; \backslash rightarrow\; \backslash neg\; P$

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.

With $\backslash neg\; P$ identified as $P\backslash to\backslash bot$, the principle is as a special case of Frege's theorem, already in minimal logic.

Another derivation makes use of $A\backslash to\; \backslash neg\; B$ as the curried, equivalent form of $\backslash neg\; (A\; \backslash land\; B)$. Using this twice, the principle is seen equivalent to the negation of

$\backslash big(P\backslash land(P\backslash to\; Q)\backslash big)\backslash land\; \backslash neg(P\backslash and\; Q)$

which, via modus ponens and rules for conjunctions, is itself equivalent to the valid noncontradiction principle for $P\backslash and\; Q$.

A classical derivation passing through the introduction of a disjunction may be given as follows:

• Reductio ad absurdum

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Category:Propositional calculus

Category:Rules of inference