Reductio ad absurdum

Establishing a claim by showing that denying it leads to a contradiction or an absurd result.

Also: reductio, reduction to absurdity, proof by contradiction, indirect proof

What it means

A reductio argument assumes the opposite of what it wants to prove, derives a contradiction or an absurdity from that assumption, and concludes that the assumption must be false. A classic example from Greek mathematics assumes that the side and diagonal of a square share a common unit of length and shows this leads to an impossibility, so they are incommensurable. Plato and Aristotle used the method, and Kant's antinomies in the Critique of Pure Reason rely on it. In ordinary speech the phrase also means pushing an idea to ridiculous extremes. Intuitionist mathematicians reject one use of it: proving that something exists merely by showing its non-existence is contradictory.

The questions it belongs to

• 4.1 Formal LogicWhat makes an argument work, regardless of its subject?
• 4.1.1 Propositional LogicWhat follows from "if this, then that"?

Thinkers

• Plato
• Aristotle
• Immanuel Kant

Traditions

• Greek & Roman
• Modern European
• Analytic

To read

• An Answer to the Question: What Is Enlightenment? — Immanuel Kant
• Apology — Plato
• Categories — Aristotle

Sources and further reading

• Reductio ad Absurdum (Internet Encyclopedia of Philosophy)
• Reductio ad absurdum (Britannica)

Written by the Coordinates of Thought editors and checked against the sources above on 29 October 2026. Not peer-reviewed. How we write and check.

All terms in the glossary