Reductio ad absurdum

File:John Pettie - Reductio Ad Absurdum.jpg exhibited at the Royal Academy in 1884|alt=A bearded white Christian cleric in red argues towards an older pensive white Christian cleric in black.]]

In logic, {{lang|la|reductio ad absurdum}} (Latin for "reduction to absurdity"), also known as {{lang|la|argumentum ad absurdum}} (Latin for "argument to absurdity") or apagogical argument, is the form of argument that attempts to establish a claim by showing that the opposite scenario would lead to absurdity or contradiction.{{Cite web|url=https://www.britannica.com/topic/reductio-ad-absurdum|title=Reductio ad absurdum {{!}} logic|website=Encyclopedia Britannica|language=en|access-date=2019-11-27}}{{Cite web|url=https://www.merriam-webster.com/dictionary/reductio+ad+absurdum|title=Definition of REDUCTIO AD ABSURDUM|website=www.merriam-webster.com|language=en|access-date=2019-11-27}}{{citation|title=reductio ad absurdum|work=Collins English Dictionary – Complete and Unabridged|edition=12th|year=2014|orig-year=1991|access-date=October 29, 2016|url=http://www.thefreedictionary.com/reductio+ad+absurdum}}{{cite encyclopedia |url=http://www.utm.edu/research/iep/r/reductio.htm |encyclopedia = The Internet Encyclopedia of Philosophy |title = Reductio ad absurdum |author = Nicholas Rescher |access-date = 21 July 2009}}

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. In mathematics, the technique is called proof by contradiction. In formal logic, this technique is captured by an axiom for "Reductio ad Absurdum", normally given the abbreviation RAA, which is expressible in propositional logic. This axiom is the introduction rule for negation (see negation introduction).

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 {{tmath|q}}. If there were, then {{tmath|q/2}} would also be a rational number, it would be positive, and we would have {{tmath|q/2 < q}}. This contradicts the hypothetical minimality of {{tmath|q}} among positive rational numbers, so we conclude that there is no such smallest positive rational number.

The first example argues that denial of the premise would result in a ridiculous conclusion, against the evidence of our senses (empirical evidence).{{Citation|last=DeLancey|first=Craig|title=8. Reductio ad Absurdum|date=2017-03-27|url=https://milnepublishing.geneseo.edu/concise-introduction-to-logic/chapter/8-reductio-ad-absurdum/|work=A Concise Introduction to Logic|publisher=Open SUNY Textbooks|language=en|access-date=2021-08-31}} The second example is a mathematical proof by contradiction (also known as an indirect proof{{Cite web|url=https://www.thoughtco.com/reductio-ad-absurdum-argument-1691903|title=Reductio Ad Absurdum in Argument|last=Nordquist|first=Richard|website=ThoughtCo|language=en|access-date=2019-11-27}}), 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).{{cite book|last1=Howard-Snyder|first1=Frances|last2=Howard-Snyder|first2=Daniel|last3=Wasserman|first3=Ryan|title=The Power of Logic|date=30 March 2012|publisher=McGraw-Hill Higher Education|isbn=978-0078038198|edition=5th}}

Greek philosophy

Reductio ad absurdum was used throughout Greek philosophy. The earliest example of a {{lang|la|reductio}} argument can be found in a satirical poem attributed to Xenophanes of Colophon (c. 570 – c. 475 BCE).{{cite web | last = Daigle | first = Robert W. | title = The reductio ad absurdum argument prior to Aristotle | work = Master's Thesis | publisher = San Jose State Univ. | date = 1991 | url = http://scholarworks.sjsu.edu/cgi/viewcontent.cgi?article=1228&context=etd_theses | access-date =August 22, 2012 }} Criticizing Homer's attribution of human faults to the gods, Xenophanes 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 ox bodies.{{Cite web|date=2014-05-18|title=Reductio ad Absurdum - Definition & Examples|url=https://literarydevices.net/reductio-ad-absurdum/|access-date=2021-08-31|website=Literary Devices|language=en-US}} 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.

Greek mathematicians proved fundamental propositions using reductio ad absurdum. Euclid of Alexandria (mid-4th – mid-3rd centuries BCE) and Archimedes of Syracuse (c. 287 – c. 212 BCE) are two very early examples.{{cite web | last= Joyce | first= David |author-link=David E. Joyce (mathematician)| title = Euclid's Elements: Book I | work = Euclid's Elements | publisher = Department of Mathematics and Computer Science, Clark University | date = 1996 | url = https://mathcs.clarku.edu/~djoyce/elements/bookI/propI6.html | access-date = December 23, 2017}}

The earlier dialogues of Plato (424–348 BCE), relating the discourses of Socrates, raised the use of {{lang|la|reductio}} arguments to a formal dialectical method ({{lang|la|elenchus}}), also called the Socratic method.{{cite encyclopedia | last = Bobzien | first = Susanne | title = Ancient Logic | encyclopedia = Stanford Encyclopedia of Philosophy | publisher = The Metaphysics Research Lab, Stanford University | date = 2006 | url = http://plato.stanford.edu/entries/logic-ancient/#NonModSyl | access-date = August 22, 2012}} Typically, Socrates' opponent would make what would seem to be an innocuous assertion. In response, Socrates, via 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 and adopt a position of aporia.

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 ({{langx|grc|ἡ εἰς τὸ ἀδύνατον ἀπόδειξις||demonstration to the impossible}}, 62b).

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.{{sfn|Hyde|Raffman|2018}}

Buddhist philosophy

Much of Madhyamaka Buddhist philosophy centers on showing how various essentialist ideas have absurd conclusions through reductio ad absurdum arguments (known as prasaṅga, "consequence" in Sanskrit). In the Mūlamadhyamakakārikā, Nāgārjuna's reductio ad absurdum arguments are used to show that any theory of substance or essence was unsustainable and therefore, phenomena (dharmas) such as change, causality, and sense perception were empty (sunya) of any essential existence. Nāgārjuna's main goal is often seen by scholars as refuting the essentialism of certain Buddhist Abhidharma schools (mainly Vaibhasika) which posited theories of svabhava (essential nature) and also the Hindu Nyāya and Vaiśeṣika schools which posited a theory of ontological substances (dravyatas).Wasler, Joseph. Nagarjuna in Context. New York: Columibia University Press. 2005, pgs. 225-263.

=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.{{sfn|Garfield|1995|p=210}}

Principle of non-contradiction

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.{{cite book

| last1 = Ziembiński

| first1 = Zygmunt

| title = Practical Logic

| publisher = Springer

| date = 2013

| pages = 95

| url = https://books.google.com/books?id=LOfsCAAAQBAJ&q=%22principle+of+non-contradiction%22&pg=PA95

| isbn = 978-9401756044

}}{{cite book

| last1 = Ferguson

| first1 = Thomas Macaulay

| last2 = Priest

| first2 = Graham

| title = A Dictionary of Logic

| publisher = Oxford University Press

| date = 2016

| pages = 146

| url = https://books.google.com/books?id=2Q5nDAAAQBAJ&q=%22principle+of+non-contradiction%22&pg=PT146

| isbn = 978-0192511553

}} That is, a proposition $Q$ and its negation $\lnot Q$ (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, has formed the basis of {{lang|la|reductio ad absurdum}} arguments in formal fields such as logic and mathematics.

References

Sources

{{cite SEP|last=Hyde|first=Dominic|title=Sorites Paradox|date=2018|url=https://plato.stanford.edu/archives/sum2018/entries/sorites-paradox/|work=The Stanford Encyclopedia of Philosophy|editor-last=Zalta|editor-first=Edward N.|edition=Summer 2018|last2=Raffman|first2=Diana|url-id=sorites-paradox}}
{{Citation | last =Garfield | first =Jay L. | year =1995 | title =The Fundamental Wisdom of the Middle Way | place =Oxford | publisher =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.