Reductio ad absurdum
In logic, reductio ad absurdum (Latin for "reduction to absurdity"), also known as argumentum ad absurdum (Latin for "argument to absurdity") or apagogical arguments, is the form of argument that attempts to establish a claim by showing that the opposite scenario would lead to absurdity or contradiction.[1][2][3][4]
This argument form traces back to Ancient Greek philosophy and has been used throughout history in both formal mathematical and philosophical reasoning, as well as in debate. The equivalent formal rule is known as negation introduction. A related mathematical proof technique is called proof by contradiction.
Examples
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, since the Earth is assumed to be finite in extent, 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.[5] The second example is a mathematical proof by contradiction (also known as an indirect proof[6]), 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).[7]
Greek philosophy
Reductio ad absurdum was used throughout
Greek mathematicians proved fundamental propositions using reductio ad absurdum.
The earlier dialogues of
The technique was also a focus of the work of Aristotle (384–322 BCE), particularly in his Prior Analytics where he referred to it as demonstration to the impossible (Greek: ἡ εἰς τὸ ἀδύνατον ἀπόδειξις, lit. "demonstration to the impossible", 62b).[4]
Another example of this technique is found in the sorites paradox, where it was argued that if 1,000,000 grains of sand formed a heap, and removing one grain from a heap left it a heap, then a single grain of sand (or even no grains) forms a heap.[12]
Buddhist philosophy
Much of
Example from Nāgārjuna's Mūlamadhyamakakārikā
In 13.5, Nagarjuna wishes to demonstrate consequences of the presumption that things essentially, or inherently, exist, pointing out that if a "young man" exists in himself then it follows he cannot grow old (because he would no longer be a "young man"). As we attempt to separate the man from his properties (youth), we find that everything is subject to momentary change, and are left with nothing beyond the merely arbitrary convention that such entities as "young man" depend upon.
13:5
- A thing itself does not change.
- Something different does not change.
- Because a young man does not grow old.
- And because an old man does not grow old either.[14]
Principle of non-contradiction
Aristotle clarified the connection between contradiction and falsity in his
See also
Sources
- Hyde, Dominic; Raffman, Diana (2018). "Sorites Paradox". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy (Summer 2018 ed.).
- Garfield, Jay L. (1995), The Fundamental Wisdom of the Middle Way, Oxford: Oxford University Press
- Pasti, Mary. Reductio Ad Absurdum: An Exercise in the Study of Population Change. United States, Cornell University, Jan., 1977.
- Daigle, Robert W.. The Reductio Ad Absurdum Argument Prior to Aristotle. N.p., San Jose State University, 1991.
References
- ^ "Reductio ad absurdum | logic". Encyclopedia Britannica. Retrieved 2019-11-27.
- ^ "Definition of REDUCTIO AD ABSURDUM". www.merriam-webster.com. Retrieved 2019-11-27.
- ^ "reductio ad absurdum", Collins English Dictionary – Complete and Unabridged (12th ed.), 2014 [1991], retrieved October 29, 2016
- ^ a b Nicholas Rescher. "Reductio ad absurdum". The Internet Encyclopedia of Philosophy. Retrieved 21 July 2009.
- ^ DeLancey, Craig (2017-03-27), "8. Reductio ad Absurdum", A Concise Introduction to Logic, Open SUNY Textbooks, retrieved 2021-08-31
- ^ a b c Nordquist, Richard. "Reductio Ad Absurdum in Argument". ThoughtCo. Retrieved 2019-11-27.
- ISBN 978-0078038198.
- ^ Daigle, Robert W. (1991). "The reductio ad absurdum argument prior to Aristotle". Master's Thesis. San Jose State Univ. Retrieved August 22, 2012.
- ^ "Reductio ad Absurdum - Definition & Examples". Literary Devices. 2014-05-18. Retrieved 2021-08-31.
- ^ Joyce, David (1996). "Euclid's Elements: Book I". Euclid's Elements. Department of Mathematics and Computer Science, Clark University. Retrieved December 23, 2017.
- ^ Bobzien, Susanne (2006). "Ancient Logic". Stanford Encyclopedia of Philosophy. The Metaphysics Research Lab, Stanford University. Retrieved August 22, 2012.
- ^ Hyde & Raffman 2018.
- ^ Wasler, Joseph. Nagarjuna in Context. New York: Columibia University Press. 2005, pgs. 225-263.
- ^ Garfield 1995, p. 210.
- ISBN 978-9401756044.
- ISBN 978-0192511553.
External links
- The dictionary definition of per impossibile at Wiktionary
- "Reductio ad absurdum". Internet Encyclopedia of Philosophy.