Ad absurdum

In logic, reductio ad absurdum (Latin for "reduction to absurdity"; or argumentum ad absurdum, "argument to absurdity") is a form of argument which attempts either to disprove a statement by showing it inevitably leads to a ridiculous, absurd, or impractical conclusion, or to prove one by showing that if it were not true, the result would be absurd or impossible. Traced back to classical Greek philosophy in Aristotle's Prior Analytics (Greek: ἡ Εις άτοπον απαγωγή, translit. hê eis atopon apagôgê, lit. 'reduction to the impossible'), this technique has been used throughout history in both formal mathematical and philosophical reasoning, as well as in debate.

Examples of arguments using reductio ad absurdum are as follows:

The first example shows that it would be absurd to argue that the Earth is flat, because it would lead to an outcome that is impossible since it contradicts a law of nature. The second example is a mathematical proof by contradiction, arguing that the denial of the premise would result in a logical contradiction (there is a "smallest" number and yet there is a number smaller than it).

This technique is used throughout Greek philosophy, beginning with Presocratic philosophers. The earliest Greek example of a reductio argument is supposedly in fragments of a satirical poem attributed to Xenophanes of Colophon (c.570 – c.475 BC). Criticizing Homer's attribution of human faults to the gods, he states that humans also believe that the gods' bodies have human form. But if horses and oxen could draw, they would draw the gods with horse and oxen bodies. The gods cannot have both forms, so this is a contradiction. Therefore the attribution of other human characteristics to the gods, such as human faults, is also false.

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