# Reductio Ad Absurdum

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## reductio ad absurdum

[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.

## Reductio Ad Absurdum

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 ?
In reductio ad absurdum contexts we sacrifice hypotheses to givers; in purely hypothetical contexts we do the reverse.
Manzoni's brilliance lay in part in his ability to push this conception of art to the point of reductio ad absurdum, but in a way that underlines rather than subverts the notion.
Wilson may well have been a little too Flip back in the '60s; our society is a dangerous place for reductio ad absurdum.
Her idiosyncratic reductio ad absurdum of Gregorian calendrical notation to only forty-two denominations for each century (the date of this publication, 3/1/97 would be "20," 3 + 1 + 9 + 7), together with endless attendant extrapolations, detaches what would better be called the "experience of dates" from astronomy, from the strictures of convention, and, above all, from events.
Conceptually simple and tight, LeWitt's wall drawings were nevertheless visually complex and expansive, often with the help of colors that made both geometrical figures and lines project in a hallucinatory way--abstract expressivity carried to reductio ad absurdum from which there is no return to geometrical rationality.
Scanga uses dreamily pigmented, conical blown-glass vases, and then it's off to the resale shop where he collects a wide range of junk metal that he subsequently glazes, paints, and welds into these constructiores ad absurdum.
In a "reducio ad absurdum," some of us who were less than well behaved might have been shot rather than disciplined by a disgruntled teacher.
My favourite of the rival clubs of intelligent people the author lists is the Grail Society, which is a reductio ad absurdum of the whole business.
Fourthly, distorting someone's point of view ad absurdum, and then making fun of it, is an admission of having no rational response.
Sarcastic reductio ad absurdum aside, the question should be asked how it is that responsible European community leaders, people with a tradition that has in the past built and maintained productive civilizations in the face of attackers and foes, could seriously entertain the thought that signs, bracelets, and appeals to humanity will deter a masculine society in advanced stages of the march for world domination.
Contrary to what my own colleague, lawyer-columnist Oscar Franklin Tan, asserted in his last column, justices can criticize majority rulings and even each other without the criticism necessarily becoming, reductio ad absurdum, an argument for the impeachment of the criticized.
There are many others but (thanks to Alanna Vagianos of the Huffington Post) we were alerted that the not being ashamed mantra may have reached reductio ad absurdum status.

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