Reductio ad absurdum
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Reductio ad absurdum is the technique of reducing an argument or hypothesis to absurdity, by pushing the argument's premises or conclusions to their logical limits and showing how ridiculous the consequences would be, thus disproving or discrediting the argument.
This has roots in the Socratic method, and has been employed throughout the history of logic, mathematics, philosophy and the philosophy of science.
As an example, one of the central tenets of homeopathy is the (unfounded) assertion that water retains a "memory" of substances dissolved in it, even when the solution becomes so weak that no trace of the original substance is present. However, if we accept this hypothesis, then any quantity of tap water would have already acquired all the benefical chemicals, and all the harmful ones too. This patently absurd scenario weakens the hypothesis of water memory.
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Validity and fallacious use
Reductio ad absurdum is only valid when it builds on assertions which are actually present in the argument it is deconstructing, and not when it misrepresents them as a straw man. For example, any creationist argument that takes the form of "if evolution were real, we'd see fish turning into monkeys and monkeys turning into people all the time" only serves to ridicule itself, since it mischaracterises the theory of evolution to an extreme degree.
In mathematics and logical systems in general, reductio ad absurdum or proof by contradiction holds if and only if the law of the excluded middle also holds in said system. The law of the excluded middle states that either a well-defined proposition is true or its converse is true, but not both or neither (in symbols: F∨¬F and ¬(F∧¬F)). Intuitionism in logic and mathematics rejects this patently obvious law and therefore proof by contradiction as a whole: thankfully, it may safely be stated that mainstream mathematics also rejects this method of logic, preserving the several hundreds of important and elegant proofs that rely upon proof by contradiction.
The argument from adverse consequences is a similar but more flawed technique. While reductio ad absurdum rejects an argument on the basis that its logical consequences / premises are so unlikely that the argument cannot possibly be sound, the argument from adverse consequences rejects an argument because its consequences are undesirable, or because accepting it could mean accepting something we would prefer not to acknowledge; this can become the moralistic fallacy.
Reductio ad absurdum should also not be confused with appeal to ridicule, although both see extensive use in satire. Appeal to ridicule simply dismisses a position as ridiculous, without explaining or arguing why, while reductio ad absurdum actually pursues the logical consequences of an argument.
In mathematics
In mathematics and formal logic, reductio ad absurdum, also known as "proof by contradiction" and "proof by assuming the opposite," is the establishing of an argument (or theory) by showing that its denial would lead to absurd consequences.
[If A is not true, then B is true (and therefore C is true' ... etc. ) which is absurd/impossible/not in accord with observation. Therefore A must be true. QED!]
As this is a logical argument, possibly with many steps between the initial premise and the ultimate conclusion, it is often open to varying interpretations and alternate steps which might not prove the conclusion. In mathematics it has been a key proof since Euclid and is a well accepted method (Often ending with a statement that some property is not equal to itself, thus showing the absurdity). In philosophy and science it is less hard and fast as there is often dispute in the causal relationship (then) between steps of the argument.
An example found in Euclidean geometry.
Hypothesis: Two distinct straight lines that intersect do so in one and only one point.
(1) Let "l" and "r" be two distinct straight lines that intersect in two or more points. (We deny the hypothesis.)
(2) Let "A" and "B" be two of the points where l and r intersect, therefore A and B are both points of l and r.
(3) (2) is absurd, as it contradicts an axiom (Two distinct points determine one and only one straight line.) therefore (1) is impossible and two distinct straight lines can't intersect in more than one point.
In Philosophy
In philosophy (although it is really the same form as in mathematics), a reductio ad absurdum is an argument formed from a valid argument (i.e there is no case where the premise is true and the conclusion false) in which the conclusion is false. The negation of this argument's premise is the conclusion of the reductio and the rest of the premises are the premises of the reductio. It is always a valid argument. Here is an example:
(1) If I am a human being, I can run up a building.
(2) I am a human being.
(3) Therefore, I can run up a building.
It is not true that I can run up a building. (3)
Therefore, reductio:
(1) entails not (2)
(2) entails not (1)
Either (1) or (2) (exclusive)