Reductio ad absurdum
Reductio ad absurdum, Latin for "reduction to the absurd", traceable back to the Greek ἡ εις άτοπον απαγωγη (hê eis átopon apagogê), "reduction to the impossible", often used by Aristotle, is a form of argument where one provisionally assumes one or more claims, derives a contradiction from them, and then concludes that at least one of those claims must be false.
History
As a dialectical tool, reductio arguments date very far back. Though
Examples
Reliance on the Principle of Non-Contradiction
One of the assumptions of the reductio argument form is that claims which entail a contradiction entail an absurd or unacceptable result. This relies on the 'principle of non-contradiction,' which holds that for any claim 'p,' it cannot be the case both that p is true and p is false. With this principle, one can infer from the fact that some set of claims entail a contradictory result (p and not-p) to the fact that that set of claims entails something false (namely, the claim that p and not-p). Though the principle of non-contradiction has seemed absolutely undeniable to most philosophers (the Leibnizian 18th-century German philosopher Christian Wolff attempted to base an entire philosophical system on it), but some historical figures appear to have denied it (arguably, Heraclitus, Hegel and Meinong). In more recent years, using the name 'dialetheism,' philosophers such as Graham Priest and Richard Routley have argued that some contradictions are true (motivated by paradoxes such as that posed by the statement, "this sentence is not true").
If the law of non-contradiction is false, then it can be the case that some contradictions are true. In that case, at least some instances of reductio arguments will fail, because the assumed claims will fail to yield anything absurd. Despite this philosophical possibility, the law of non-contradiction, and so the formal legitimacy of all reductio arguments, are still almost universally accepted by logicians.
References and further reading
Relevant Stanford Encyclopedia entries:
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