## Definitions

### American Heritage® Dictionary of the English Language, Fourth Edition

1. n. Disproof of a proposition by showing that it leads to absurd or untenable conclusions.

### Century Dictionary and Cyclopedia

1. n. A reduction to an absurdity; the proof of a proposition by proving tho falsity of its contradictory opposite: an indirect demonstration. In geometry the reductio ad absurdum consists in drawing a figure whose parts are supposed to have certain relations, and then showing that this leads to a conclusion contrary to a known proposition, whence it follows that the parts of the figure cannot have those relations. Thus, in Euclid's “Elements” the proposition that if a triangle has two angles equal the sides opposite those angles will be equal is proved as follows. In the triangle ABC, let the angles ABC and ACB be equal. Then, suppose AB to be greater than AC. Lay off BD=AC and join DC. Then, comparing the two triangles ACB and DBC, we have in the former the sides AC and BC and their included angle ACB equal in the latter to the sides DB and CB and their included angle DBC. Hence, these two triangles would be equal, or the part would be equal to the whole. This proof is a reductio ad absurdum. This kind of reasoning is considered somewhat objectionable as not showing the principle from which tiie proposition flows; but, it is a perfectly conclusive mode of proof, and, in fact, is in all cases readily converted into a direct proof. Thus, in the above example, we have only to compare the triangle ABC with itself, considering it as two triangles according as the angle B is named before C or vice versa. In the triangle ABC the angles Band C with the included side BC are respectively equal in the triangle ACB to the angles C and B with the included side CB; hence the other parts of the triangles are equal, and tho side AC opposite the first angle B in the first triangle is equal to the side AB opposite the tirst angle C in the second triangle.

### Wiktionary

1. n. mathematics, logic The method of proving a statement by assuming the statement is false and, with that assumption, arriving at a blatant contradiction.

### WordNet 3.0

1. n. (reduction to the absurd) a disproof by showing that the consequences of the proposition are absurd; or a proof of a proposition by showing that its negation leads to a contradiction

## Etymologies

1. Medieval Latin reductiō ad absurdum : Latin reductiō, a bringing back, reduction + Latin ad, to + Latin absurdum, absurdity, from neuter of absurdus, absurd. (American Heritage® Dictionary of the English Language, Fourth Edition)

## Lists

These user-created lists contain the word ‘reductio ad absurdum’.