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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 ?
The list goes on, ad absurdum, avoiding all investigations into the system of economic exploitation itself, displacing problems with the latter onto some external, contingent element, and up to, in The Trotsky, the very archetypal figure of the rival in high school melodrama: the school principal.
This is a kind of reductio ad absurdum, but it points to the problem poets are experiencing in the early 21st Century.
His model of the eucharist is a prime example, where his description sounds like a reductio ad absurdum against his own view when he states that "the soul of the believer literally eats and drinks the infinite divine spirit of Jesus, thus respectively satisfying the soul's literal hunger and quenching the soul's literal thirst" (221, italics mine).
Welch, "a sort of walking reductio ad absurdum of some of the most confident theories of his fellow resurrection-men" (16).
Essential to this proof in the solution to the argument of the Third Discussion is an implicit theory of metaphorical predication that can be pieced together from several of Ghazalis remarks as well as a reductio ad absurdum argument about the very possibility of ethical discourse.
Dancy, denies that Euthyphro 6e10-8a8 follows the structure of a reductio ad absurdum.
horribile dictu," "in nuce," "cursus honorum," "do ut des," "mirabile dictu," "lege artis," "similia similibus," "perinde ac cadaver," "traiecto ad absurdum," "communis opinio," "e contrario," "puer senex," "tremendum," "symposion," "manu militari," "plica polonica," "doktor maximus," "unio mystica," "genii loci," "arbiter elegantiarum"; "rake's progress," "temperance-movement," "juicy stories," "sex and crime"; "pairs," "lie du peuple," "en friche," "grand renfermement," etc.
To postulate the absence of a credit bubble in the middle of this decade is thus a reductio ad absurdum, irrespective of how one assesses bankers' or households' "moral compass.
The reduction ad absurdum came when they claimed there was a pro-Indonesia lobby in Canberra, which, if it exists, was only formed to counter decades of anti-Indonesian propaganda.
That is a reductio ad absurdum - it is murder of the worst possible kind.

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