Reductio ad absurdum

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A logical derivation rule that allows one to conclude that if a list $\Gamma$ of statements and a statement $A$ imply both a statement $B$ and the statement $\neg B$, then $\Gamma$ implies $\neg A$. The rule of reductio ad absurdum can, e.g., be written in the form $$ \frac{\Gamma, A \rightarrow B\,;\ \Gamma,A \rightarrow \neg B}{\Gamma \rightarrow \neg A} $$

Reductio ad absurdum is a sound rule in the majority of logico-mathematical calculi.


Informally, the name "reductio ad absurdum" is also used for the rule that if $\Gamma$ together with $\neg A$ implies a contradiction, then $\Gamma$ implies $A$. This is of course equivalent to the above (and therefore sound) in classical logic, but it is not a sound rule of inference in intuitionistic logic.

How to Cite This Entry:
Reductio ad absurdum. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by S.Yu. Maslov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article