# Reductio ad absurdum

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.

#### Comments

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: http://encyclopediaofmath.org/index.php?title=Reductio_ad_absurdum&oldid=41884