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," 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. Such arguments are intimately related to the notion of 'paradox'. In both cases, one is presented with a pair of claims that cannot both be true (a contradiction), but which cannot be easily rejected. A reductio argument, however, is specifically aimed at bringing someone to reject some belief. Paradoxes, on the other hand, can be raised without there being any belief in particular that is being targeted.
As a dialectical tool, reductio arguments date very far back. The so-called 'early' dialogues of the Greek philosopher Plato are believed to have been representative of the method of his teacher Socrates (who appears in those dialogues as the main character), a method that crucially employed reductio arguments. Typically, the dialogue would represent an interaction between Socrates and someone who advanced a certain claim or claims. Socrates would then convince the person that their claims (along with certain background assumptions) led to a contradiction, thereby showing that the claims could not be sustained. The dialogues typically end with Socrates' interlocutor making a hasty retreat (for the most famous example, see the Euthyphro).
Reductio arguments were also a focus of Aristotle, who is considered the father of logic. In addition to explicitly defending the Principle of Non-Contradiction (see below), Aristotle classified reductio arguments as instances of immediate inference (as opposed to the mediate inferences formalized by syllogisms). Book 8 of Aristotle's Topics describes the use of reductio arguments as the means by which formal debates were conducted in Aristotle's Academy, suggesting that such an approach was seen as the preferred way to refute an opponent's philosophical position.
Perhaps the most well-rehearsed reductio argument concerns the existence of an omnipotent God. Here is one rendering:
Note that the last step rejects the first assumption, instead of one of the other two. The basis for doing this is that the first assumption appears less plausible than either the second or the third. This of course, can in principle be denied. George Mavrodes, for instance, has explicitly argued that 'making a rock so heavy it can't be lifted' and 'lifting a rock so heavy it can't be lifted' are not in fact tasks at all, since their description is self-contradictory.
As this illustrates, the fact that a contradiction follows from a set of assumptions is not a sufficient basis for deciding which assumption should be rejected (unless, of course, there is only one assumption). Sometimes the choice is relatively superficial (both of the above conclusions essentially amount to granting that God, if he exists, cannot perform tasks whose description is self-contradictory). But sometimes the choice is quite difficult (for an especially poignant case, see Derek Parfit's 'Mere Addition Paradox' in his Reasons and Persons).
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 eighteenth-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. In some logical systems, the reductio form has been used as a basis for introducing a negation operator.
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