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
An Example
Perhaps the most well-rehearsed reductio argument concerns the existence of an omnipotent God. Here is one rendering:
- There exists a God who can perform any task. (Assumption)
- Making a rock so heavy that it can't be lifted is a task. (Assumption)
- If there could be some rock so heavy that it can't be lifted, lifting it would be a task. (Assumption)
- God can make a rock so heavy that it can't be lifted. (From 1, 2)
- There can be a rock so heavy that it can't be lifted. (From 4)
- God can lift a rock so heavy that it can't be lifted. That is, it is true that God can lift such a rock, and false that God can lift such a rock. (From 1, 3, 5)
- Therefore, there cannot exist a God who can perform any task. (from 6, which is a contradition).
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).
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
- Mavrodes, George. (1963) "Some Puzzles Concerning Omnipotence" in Philosophical Review 72, 221-223.
- Pafit, Derek. (1984) Reasons and Persons. Oxford: Oxford University Press.
Relevant Stanford Encyclopedia entries:
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