A method of proof according to which the ability to derive from a formula $\phi$ some absurd statement provides sufficient grounds to conclude $\neg \phi$.

If one accepts the principle of excluded middle, the method of reductio ad absurdum is sometimes justified as an instance of proof by cases, that is, its machinery considers the case corresponding to the truth of $\phi$ and, if this case is rejected as absurd, licenses an inference to the remaining case, i.e., the case corresponding to the falsity of $\phi$. The technique is thus closely related to the dual thesis of consequentia mirabilis, according to which if a formula $\phi$ is entailed by its own denial, i.e., if even the act of denying $\phi$ entails its truth, then $\phi$ is true.

There are alternative accounts concerning what constitutes an ‘absurd’ statement that bear on how the technique of reductio is employed. The technique of reductio ad falsum requires $\phi$ to entail either its negation $\neg \phi$ (on one interpretation) or an arbitrary falsehood (on a competing interpretation) while the reductio ad contradictionem licenses the rejection of $\phi$ whenever $\phi$ entails a contradiction. The technique of reductio ad impossibile corresponds to the assertion that to derive a metaphysically impossible proposition from $\phi$ is sufficient to reject $\phi$. If it is assumed that contradictions and statements that are otherwise impossible are absurd in the sense that they are false on their face, then these techniques are species of reductio ad absurdum, respectively. However, there are occasions in which these techniques are treated as distinct from reductiones ad contradictionem. For example, a dialetheist—who is committed to the existence of a true contradiction $\xi$—will not necessarily treat $\psi \wedge \neg \psi$ as absurd. Moreover, the strength of an argument by reductio ad contradictionem to such an individual is limited, as the fact that a statement $\psi$ entails this contradiction may not suffice to reject $\psi$.

The notion of a reductio ad absurdum figures heavily in the treatment of negation in intuitionistic logic. Where the propositional constant falsum ($\perp$) is read as a formula stipulated to be absurd or unprovable, the intuitionistic account of negation defines $\neg \phi$ as $\phi \to \perp$, that is, the existence of a derivation of an absurdity from $\phi$. Systems that distinguish rejection on the basis of reductio from constructive rejection are known as logics of constructible falsity.