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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.
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."
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
But if it is impossible, its denial, ~(~B → B), is a logical truth. Aristotelian syllogisms (as opposed to Boolean syllogisms) appear to be based on connexive principles. For example, the contrariety of A and E statements, "All S are P," and "No S are P," follows by a reductio ad absurdum argument similar to the one given by Aristotle.
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.
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 ...
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]
The metamathematical value of the principle of explosion is that for any logical system where this principle holds, any derived theory which proves ⊥ (or an equivalent form, ) is worthless because all its statements would become theorems, making it impossible to distinguish truth from falsehood.