Reductio ad absurdum

From RationalWiki

Reductio ad Absurdum is the establishing of an argument (or theory) by showing that its denial would lead to absurd consequences. It has been employed throughout the history of mathematics, philosophy and the philosophy of science to demonstrate the likelihood of a particular hypotheses.

[If A is not true, then B is true (and therefore C is true' ... etc. ) which is absurd/impossible/not in accord with observation. Therefore A must be true. QED!]

As this is a logical argument, possibly with many steps between the initial premise and the ultimate conclusion, it is often open to varying interpretations and alternate steps which might not prove the conclusion. In mathematics it has been a key proof since Euclid and is a well accepted method (Often ending with a statement that some property is not equal to itself, thus showing the absurdity). In philosophy and science it is less hard and fast as there is often dispute in the causal relationship (then) between steps of the argument.

An example found in euclidean geometry.

Hypothesis: Two straight lines that intersect do so in one and only one point.

(1) Let "l" and "r" be two straight lines that intersect in two or more points. (We deny the hypothesis.)

(2) Let "A" and "B" be two of the points where l and r intersect, therefore A and B are both points of l and r.

(3) (2) is absurd, as it contradicts an axiom (Two distinct points determine one and only one straight line.) therefore (1) is impossible and two straight line can't intersect in more than one point.