# Reductio Ad Absurdum

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## reductio ad absurdum

[ri¦dək·tē·ō äd ab′sərd·əm]## 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