proof by contradiction

反证法

Foundations & Logic · Undergraduate · Mathematics Resources editorial compilation (AI-assisted)

Other names / variants: reductio ad absurdum

Translation evidence: Literature usage

Meaning in context

In classical logic, a proof of $P$ obtained by deriving $\bot$ from the premises together with $\neg P$.

在经典逻辑中，暂设结论的否定并推出矛盾，从而确立结论的证明方法。

Example

A least positive rational $r$ would be contradicted by the smaller positive rational $r/3$.

Usage and distinctions

For positive conclusions the method uses double-negation elimination; proving a negation from a contradiction is already intuitionistically valid.

Read the encyclopedia explanation

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Avigad, Hua, Lewis and van Doorn — Logic and Proof§5.1 Proof by Contradiction; §5.2 Some Classical Principles · Checked 2026-10-14

Translation sources

1. 杨睿之（复旦大学）— 数理逻辑，第12讲PDF p.17 推论「反证法」；其形式先证明否定结论 · Checked 2026-10-14

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