Reductio ad absurdum

Reductio ad absurdum (Latin: "reduction to the absurd") also known as an apagogical argument, reductio ad impossibile, or proof by contradiction, is a type of logical argument where one assumes a claim for the sake of argument, derives an absurd or ridiculous outcome, and then concludes that the original assumption must have been wrong as it led to an absurd result. It makes use of the law of non-contradiction — a statement cannot be both true and false. In some cases it may also make use of the law of excluded middle — a statement must be either true or false. The phrase is traceable back to the Greek η εις άτοπον απαγωγή (hê eis átopon apagogê), meaning "reduction to the impossible", often used by Aristotle.

In formal logic, reductio ad absurdum is used when a formal contradiction can be derived from a premise, allowing one to conclude that the premise is false. If a contradiction is derived from a set of premises, this shows that at least one of the premises is false, but other means must be used to determine which one.

Reductio ad absurdum is also often used to describe an argument where a conclusion is derived in the belief that everyone (or at least those being argued against) will accept that it is false or absurd. However, this is a weak form of reductio, as the decision to reject the premise requires that the conclusion is accepted as being absurd. Although a formal contradiction is by definition absurd (unacceptable), a weak reductio ad absurdum argument can be rejected simply by accepting the purportedly absurd conclusion.

There is a fairly common misconception that reductio ad absurdum simply denotes "a silly argument" and is itself a formal fallacy. However, this is not correct; a properly constructed reductio constitutes a correct argument. When reductio ad absurdum is in error, it is because of a fallacy worked into the example, somewhere, not the act of reduction itself.