Reductio ad absurdum

Reductio ad absurdum is an argument that rejects a claim by showing that assuming it leads to a contradiction or an unacceptable consequence.

Law of noncontradiction: The logical principle that a proposition and its negation cannot both be true in the same sense and at the same time. A direct contradiction supplies the clearest endpoint for a reductio argument.
Proposition: A declarative statement that can be assigned a truth value. A reductio begins with a proposition whose truth or falsity is at issue.
Euclid's proof of the infinitude of primes: Euclid's argument that no finite list can contain every prime number. The proof assumes a finite complete list and constructs a number that defeats it.
Direct proof: A proof that derives a conclusion from established premises without assuming its negation. It establishes a claim by forward derivation rather than by refuting an alternative assumption.
Negation: A logical operation that turns a proposition into the claim that it is false. Reductio often tests a claim by assuming its negation and deriving an impossibility.
Inference: The process of deriving a conclusion from one or more premises. The assumed claim must lead to the absurdity through acceptable steps of inference.
Irrationality of the square root of two: The theorem that no ratio of integers has square equal to two. A parity contradiction follows from assuming the square root of two is a rational fraction.
Proof by contraposition: A proof of an implication that establishes its contrapositive instead. It can prove the same implication as direct reasoning without needing a contradiction.
Proof by contradiction: A proof method that establishes a proposition by showing its negation entails a contradiction. This is the standard mathematical form of reductio ad absurdum.
Classical logic: A family of logical systems that includes standard bivalent propositional and predicate logic. The familiar form of reductio relies on classical rules for negation and contradiction.

Linked from 9 pages