Reductio ad absurdum

form of argument in informal logic
(Redirected from Proof by contradiction)

Reductio ad absurdum is a Latin phrase which means "reduction to the absurd". The phrase describes a kind of indirect proof. It is a proof by contradiction,[1][2] and is a common form of argument. It shows that a statement is true because its denial leads to a contradiction, or a false or absurd result.[3][4] It is a way of reasoning that has been used throughout the history of mathematics and philosophy from classical antiquity onwards.[3]

The ridiculous or "absurdum" conclusion of a reductio ad absurdum argument can have many forms. For example,

  • Rocks have weight, otherwise we would see them floating in the air.
  • Society must have laws, otherwise there would be chaos.
  • There is no smallest positive rational number, because if there were, it could be divided by two to get a smaller one.



The phrase can be traced back to the Greek η εις άτοπον απαγωγή (hê eis átopon apagogê). This phrase means "reduction to the impossible".[3] It was often used by Aristotle.[5] The method is used a number of times in Euclid's Elements.



Reduction ad absurdum can be a tool of discovery.[6]

The method of proving something works by first assuming something about it. Then other things are deduced from that. If there is a contradiction, it shows that the first something cannot be correct. For example,

To prove A is true, correct, valid, credible ....
Assume the opposite -- that "not-A" is true....
Assume that if "not-A" is true, then it must mean or imply B.
Show that B is false, incorrect, invalid, incredible ...
Therefore, A must be true after all.[2]


  1. "The Definitive Glossary of Higher Mathematical Jargon: Proof by Contraction". Math Vault. 2019-08-01. Retrieved 2020-09-23.
  2. 2.0 2.1 Weston, Anthony. (2009). A Rulebook for Arguments, pp. 43-44.
  3. 3.0 3.1 3.2 "Reductio ad Absurdum | Internet Encyclopedia of Philosophy". Retrieved 2020-09-23.
  4. "Reductio ad absurdum | logic". Encyclopedia Britannica. Retrieved 2020-09-23.
  5. Heath, Thomas Little 1908. The Thirteen Books of Euclid's Elements, Vol. 1, p. 136.
  6. Polya, Goerge. (2008). How to Solve It: A New Aspect of Mathematical Method, p. 169.

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