View source for 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]].

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