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Reductio ad absurdum, painting by John Pettie exhibited at the Royal Academy in 1884. In logic, reductio ad absurdum (Latin for "reduction to absurdity"), also known as argumentum ad absurdum (Latin for "argument to absurdity") or apagogical arguments, is the form of argument that attempts to establish a claim by showing that the opposite scenario would lead to absurdity or contradiction.
In general usage outside mathematics and philosophy, a reductio ad absurdum is a tactic in which the logic of an argument is challenged by reducing the concept to its most absurd extreme. Translated from Aristotle's "ἡ εις άτοπον απαγωγη" (hi eis atopon apagogi, "reduction to the impossible"). reductio ad Hitlerum
Zeno's arguments may then be early examples of a method of proof called reductio ad absurdum, also known as proof by contradiction. Thus Plato has Zeno say the purpose of the paradoxes "is to show that their hypothesis that existences are many, if properly followed up, leads to still more absurd results than the hypothesis that they are one."
More broadly, proof by contradiction is any form of argument that establishes a statement by arriving at a contradiction, even when the initial assumption is not the negation of the statement to be proved. In this general sense, proof by contradiction is also known as indirect proof, proof by assuming the opposite, [2] and reductio ad ...
The heart of the dialogue opens with a challenge by Socrates to the elder and revered Parmenides and Zeno. Employing his customary method of attack, the reductio ad absurdum, Zeno has argued that if as the pluralists say things are many, then they will be both like and unlike; but this is an impossible situation, for unlike things cannot be like, nor like things unlike.
But if, when A is not white, it is necessary that B should be great, it necessarily results that if B is not great, B itself is great. But this is impossible. An. Pr. ii 4.57b3. The sense of this passage is to perform a reductio ad absurdum proof on the claim that two formulas, (A → B) and (~A → B), can be true simultaneously. The proof is:
The phrase is distinct from reductio ad absurdum, which is usually a valid logical argument. ab abusu ad usum non valet consequentia: The inference of a use from its abuse is not valid: i.e., a right is still a right even if it is abused (e.g. practiced in a morally/ethically wrong way); cf. § abusus non tollit usum. ab aeterno: from the eternal
To disprove opposing views about reality, he wrote a series of paradoxes that used reductio ad absurdum arguments, or arguments that disprove an idea by showing how it leads to illogical conclusions. [12] Furthermore, Zeno's philosophy makes use of infinitesimals, or quantities that are infinitely small while still being greater than zero. [14]