# 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 ?
Is the reductio ad absurdum no longer a legitimate form of argument?
The other notable use of the reductio ad absurdum is Lincoln's response to Douglas's being in favor of adding territory to the United States without regard to its effect on the slavery question, Carry that to its logical conclusion, Lincoln says, and "the next thing will be a grab for the territory of poor Mexico, an invasion of the rich lands of South America, then the adjoining islands will follow ..." and in each case the decision regarding slavery, following the popular sovereignty principle, "is to be left to the people of those countries for settlement." This will create a problem for Douglas, because he "has a great horror for mongrels, and I understand that the people of Mexico are most decidedly a race of mongrels" (312).
There, Burke subjects pure formalism to a reductio ad absurdum by arguing that that no formalist who draws on any knowledge of the social circumstances of a literary text or the personal circumstances of the author is being true to his or her first principles: 'Absolutely no biographical reference would be admissible.
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).
John McCririck is spot-on with his reductio ad absurdum. Creating an alternative Triple Crown with the same elements of history and challenge is impossible, and the best idea is to forego the term altogether.
The rule-following considerations provide a reductio ad absurdum of the normativity assumption itself.
Dancy, denies that Euthyphro 6e10-8a8 follows the structure of a reductio ad absurdum. (4) On their view, the argument at this passage does not generate a contradiction from its internal premises.
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."
That is a reductio ad absurdum - it is murder of the worst possible kind.
The Latin phrase "reductio ad absurdum" means "reduction to the absurd," a calamity well known to exceptional parents.
However, this is an incorrect statement, for although no calculations seem to be required, the solution of a Sudoku puzzle may require the use of logical reasoning, deduction, and "reductio ad absurdum" techniques which are all part of the rich tapestry of mathematics.
Meanwhile, Stravinsky considered Klangfarbenmelodie a reductio ad absurdum, saying it reminds him of the nineteenth-century valveless horn bands in Russia where each musician played but a single note (p.

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