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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."
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 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
A common debate technique, and a method of proof in mathematics and philosophy, that proves the thesis by showing that its opposite is absurd or logically untenable. 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 ...
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. Later logicians, notably Chrysippus, are also thought to have endorsed connexive principles. By 100 BCE logicians had divided into four or five distinct schools concerning the ...
reductio ad absurdum A method of argument that demonstrates the falsity of a statement by showing that it logically leads to a contradiction or absurd conclusion. redundancy theory of truth A theory of truth that suggests stating that a proposition is true is redundant and does not add anything to the proposition's content. reference
Uncle Joe concludes his proof reductio ad absurdum, meaning in English "proof by contradiction". What Carroll calls the protasis of a conditional is now known as the antecedent, and similarly the apodosis is now called the consequent. Symbols can be used to greatly simplify logical statements such as those inherent in this story: