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[ri¦dək·tē·ō äd ab′sərd·əm]
(mathematics)
A method of proof in which it is first supposed that the fact to be proved is false, and then it is shown that this supposition leads to the contradiction of accepted facts. Also known as indirect proof; proof by contradiction.

the type of proof in which the proving of a judgment (the thesis of the proof) is achieved by the refutation of the judgment contradicting it—its antithesis. The refutation of the antithesis is achieved by establishing the fact that it is incompatible with any judgment whose truth has been established. The following pattern of proof corresponds to this form of reductio ad absurdum: if B is true and the falsity of B follows from A, then A is false.

Another, more general, form of reductio ad absurdum is proof by refutation (establishment of the falsity) of the antithesis according to the rule: having assumed A, we deduce a contradiction, consequently not-A. Here A can be either a positive or a negative judgment, and the deduction of the contradiction can be interpreted either as the deduction of the assertion of the identity of objects known to be different, or as the deduction of the pair of judgments B and not-B, or as the deduction of the conjunction of this pair, or as the deduction of the equivalency of this pair. The different interpretations of the concepts reductio ad absurdum and “contradiction” correspond to these different cases.

The method of reductio ad absurdum is especially important in mathematics: many negative judgments of mathematics cannot be proved by any means other than reduction to a contradiction. Besides those indicated above, there is another—paradoxical—form of reductio ad absurdum, which was used by Euclid in his Elements: judgment A can be considered proven if one can show that A results even from the asumption of the falsity of A.

M. M. NOVOSELOV

References in periodicals archive ?
This indirect argument for limiting domestic presidential powers is found in The Powers of War and Peace's short and likely to be overlooked conclusion that deals with President Franklin Roosevelt's "court packing" plan that led to the "switch in time that saved nine." (74) Yoo's book is by no means a direct attack on the administrative state.
If, in addition to directly advocating an expansive foreign affairs role for the President, The Powers of War and Peace also offers an indirect argument for limiting the President's domestic powers, then it is could be even more controversial than Senator Biden suspected.
The indirect argument for the correctness of M is to the effect that it avoids the limitative theorems of classical metatheory; specifically, Church's Theorem, Tarski's Theorem, Lob's Theorem and Godel's first and second Incompleteness Theorems.
I conclude that Denyer's attack on this point, and, with it, his attack on the whole of the indirect argument for M, fails.
The possessor provides another means of expressing indirect arguments in the Mayan languages, but there are significant differences in the constraints on its use.
The use of the *-b'e suffix was optional; indirect arguments could also surface as relational noun phrases headed by the preposition *chi, or as possessors of the direct object.

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