reductio

Reductio, short for reductio ad absurdum, is a classical method of argument and proof meaning “reduction to absurdity.” It involves assuming the negation of a claim and showing that this assumption leads to a contradiction or an absurd consequence; from that contradiction, the original claim is concluded to be true. In classical logic, this is a valid inference rule often described as proof by contradiction.

In mathematics, reductio is a standard tool for establishing propositions by ruling out their negations. A famous example is the proof that the square root of 2 is irrational: assuming sqrt(2) = p/q in lowest terms leads to the conclusion that both p and q are even, contradicting the lowest-term assumption, hence sqrt(2) cannot be rational.

Historically, reductio has roots in ancient Greek mathematics and philosophy. Euclid employs proofs by contradiction throughout the Elements, and Aristotle discusses the method as a legitimate form of demonstration. The technique remains widespread in various disciplines, though its use and interpretation can vary with the logical framework in play.

In logic, reductio is tied to the law of non-contradiction and the principle of the excluded middle. In some non-classical or constructive logics, proof by contradiction is more restricted; existence statements, in particular, often require constructive evidence rather than reliance on deriving a contradiction from the negation.

Overall, reductio ad absurdum remains a foundational technique for establishing truth by demonstrating the untenability of alternatives, across mathematics, philosophy, and formal logic.