In logic, reductio ad absurdum (Latin for "reduction to absurdity"), also known as argumentum ad absurdum (Latin for "argument to absurdity"), apagogical arguments, negation introduction or the appeal to extremes, is the form of argument that attempts to establish a claim by showing that the opposite scenario would lead to absurdity or contradiction. It can be used to disprove a statement by showing that it would inevitably lead to a ridiculous, absurd, or impractical conclusion, or to prove a statement by showing that if it were false, then the result would be absurd or impossible. Traced back to classical Greek philosophy in Aristotle's Prior Analytics (Greek: ἡ εἰς τὸ ἀδύνατον ἀπόδειξις, lit. "demonstration to the impossible", 62b), this technique has been used throughout history in both formal mathematical and philosophical reasoning, as well as in debate.
The "absurd" conclusion of a reductio ad absurdum argument can take a range of forms, as these examples show:
The Earth cannot be flat; otherwise, we would find people falling off the edge.There is no smallest positive rational number because, if there were, then it could be divided by two to get a smaller one.The first example argues that denial of the premise would result in a ridiculous conclusion, against the evidence of our senses. The second example is a mathematical proof by contradiction (also known as an indirect proof), which argues 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).
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