In logic, reductio ad absurdum (Latin for "reduction to absurdity"), also known as argumentum ad absurdum (Latin for "argument to absurdity"), apagogical arguments, negation introduction or the appeal to extremes, is a form of argument that attempts to establish a claim by showing that the opposite scenario would lead to absurdity or contradiction.[1][2] It can be used to disprove a statement by showing that it would inevitably lead to a ridiculous, absurd, or impractical conclusion,[3] or to prove a statement by showing that if it were false, then the result would be absurd or impossible.[4][5] Traced back to classical Greek philosophy in Aristotle's Prior Analytics[5] (Greek: ἡ εἰς τὸ ἀδύνατον ἀπόδειξις, lit. 'demonstration to the impossible', 62b), this technique has been used throughout history in both formal mathematical and philosophical reasoning, as well as in debate.[6]

The "absurd" conclusion of a reductio ad absurdum argument can take a range of forms, as these examples show:

• The Earth cannot be flat; otherwise, we would find people falling off the edge.
• There is no smallest positive rational number because, if there were, then it could be divided by two to get a smaller one.

The first example argues that denial of the premise would result in a ridiculous conclusion, against the evidence of our senses. The second example is a mathematical proof by contradiction (also known as an indirect proof[7]), which argues that the denial of the premise would result in a logical contradiction (there is a "smallest" number and yet there is a number smaller than it).[8]

## Greek philosophy

This technique is used throughout Greek philosophy, beginning with Presocratic philosophers. The earliest Greek example of a reductio argument is supposedly in fragments of a satirical poem attributed to Xenophanes of Colophon (c. 570 – c. 475 BCE).[9] Criticizing Homer's attribution of human faults to the gods, he states that humans also believe that the gods' bodies have human form. But if horses and oxen could draw, they would draw the gods with horse and oxen bodies. The gods cannot have both forms, so this is a contradiction. Therefore, the attribution of other human characteristics to the gods, such as human faults, is also false.

The earlier dialogues of Plato (424–348 BCE), relating the debates of his teacher Socrates, raised the use of reductio arguments to a formal dialectical method (elenchus), now called the Socratic method[10] which is taught in law schools. Typically, Socrates' opponent would make an innocuous assertion, then Socrates by a step-by-step train of reasoning, bringing in other background assumptions, would make the person admit that the assertion resulted in an absurd or contradictory conclusion, forcing him to abandon his assertion.[7] The technique was also a focus of the work of Aristotle (384–322 BCE).[5]

Greek mathematicians proved fundamental propositions utilizing reductio ad absurdum. Euclid of Alexandria (mid-3rd – mid-4th centuries BCE) and Archimedes of Syracuse (c. 287 – c. 212 BCE) are two very early examples.[11]

Aristotle clarified the connection between contradiction and falsity in his principle of non-contradiction, which states that a proposition cannot be both true and false.[12][13] That is, a proposition ${\displaystyle Q}$ and its negation ${\displaystyle \lnot Q}$ (not-Q) cannot both be true. Therefore, if a proposition and its negation can both be derived logically from a premise, it can be concluded that the premise is false. This technique, known as indirect proof or proof by contradiction,[7] has formed the basis of reductio ad absurdum arguments in formal fields such as logic and mathematics.

## Logical fallacy

In common speech,[2] a reductio ad absurdum is also often deployed in a manner which results in a logical fallacy, in a similar manner to the slippery slope fallacy.[7] The fallacious nature of this deployment of the argument lies in the assumption that in order for something to be true, it must be true in all circumstances. For example, the assertion that if a reduction in working hours results in an increase in productivity[14] then working no hours would result in the highest productivity. The fallacious nature of this argument lies in the assumption that the relationship between working hours and productivity is entirely linear.

## References

1. "The Definitive Glossary of Higher Mathematical Jargon — Proof by Contradiction". Math Vault. 2019-08-01. Retrieved 2019-11-27.
2. "Reductio ad absurdum | logic". Encyclopedia Britannica. Retrieved 2019-11-27.
3. "Definition of REDUCTIO AD ABSURDUM". www.merriam-webster.com. Retrieved 2019-11-27.
4. "reductio ad absurdum", Collins English Dictionary – Complete and Unabridged (12th ed.), 2014 [1991], retrieved October 29, 2016
5. Nicholas Rescher. "Reductio ad absurdum". The Internet Encyclopedia of Philosophy. Retrieved 21 July 2009.
6. Reductio Ad Absurdum is for example frequently found in Plato's Republic, documenting Socrates' attempts to guide listeners to his conclusions about justice, democracy and friendship. It is also used by the United States Supreme Court to handle down its ruling on the 1954 case of Brown v. Board of Education. For more, see Reductio Ad Absurdum in Argument.
7. Nordquist, Richard. "Reductio Ad Absurdum in Argument". ThoughtCo. Retrieved 2019-11-27.
8. Howard-Snyder, Frances; Howard-Snyder, Daniel; Wasserman, Ryan (30 March 2012). The Power of Logic (5th ed.). McGraw-Hill Higher Education. ISBN 0078038197.
9. Daigle, Robert W. (1991). "The reductio ad absurdum argument prior to Aristotle". Master's Thesis. San Jose State Univ. Retrieved August 22, 2012.
10. Bobzien, Susanne (2006). "Ancient Logic". Stanford Encyclopedia of Philosophy. The Metaphysics Research Lab, Stanford University. Retrieved August 22, 2012.
11. Joyce, David (1996). "Euclid's Elements: Book I". Euclid's Elements. Department of Mathematics and Computer Science, Clark University. Retrieved December 23, 2017.
12. Ziembiński, Zygmunt (2013). Practical Logic. Springer. p. 95. ISBN 940175604X.
13. Ferguson, Thomas Macaulay; Priest, Graham (2016). A Dictionary of Logic. Oxford University Press. p. 146. ISBN 0192511556.
14. Kleinman, Zoe (2019-11-04). "Microsoft four-day work week 'boosts productivity'". bbc.com. Retrieved 2019-11-27.