Reductio ad absurdum

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Reductio ad absurdum , painting by John Pettie exhibited at the Royal Academy in 1884 John Pettie - Reductio Ad Absurdum.jpg
Reductio ad absurdum, painting by John Pettie exhibited at the Royal Academy in 1884

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]

Contents

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. Formally, the proof 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 ) and it is sometimes named to make this connection clear. It is a consequence of the related mathematical proof technique 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 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 philosophy. The earliest example of a reductio argument can be found in a satirical poem attributed to Xenophanes of Colophon (c. 570 – c. 475 BCE). [8] 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. [9] 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. [10]

The earlier dialogues of Plato (424–348 BCE), relating the discourses of Socrates, raised the use of reductio arguments to a formal dialectical method ( elenchus ), also called the Socratic method. [11] 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. [6]

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 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). [13]

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 principle of non-contradiction, which states that a proposition cannot be both true and false. [15] [16] That is, a proposition and its negation (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, [6] has formed the basis of reductio ad absurdum arguments in formal fields such as logic and mathematics.

See also

Sources

Related Research Articles

In classical logic, disjunctive syllogism is a valid argument form which is a syllogism having a disjunctive statement for one of its premises.

In logic, the law of non-contradiction (LNC) states that contradictory propositions cannot both be true in the same sense at the same time, e. g. the two propositions "p is the case" and "p is not the case" are mutually exclusive. Formally, this is expressed as the tautology ¬(p ∧ ¬p). The law is not to be confused with the law of excluded middle which states that at least one, "p is the case" or "p is not the case", holds.

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In logic, proof by contradiction is a form of proof that establishes the truth or the validity of a proposition, by showing that assuming the proposition to be false leads to a contradiction. Although it is quite freely used in mathematical proofs, not every school of mathematical thought accepts this kind of nonconstructive proof as universally valid.

<span class="mw-page-title-main">Syllogism</span> Type of logical argument that applies deductive reasoning

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<span class="mw-page-title-main">Contradiction</span> Logical incompatibility between two or more propositions

In traditional logic, a contradiction occurs when a proposition conflicts either with itself or established fact. It is often used as a tool to detect disingenuous beliefs and bias. Illustrating a general tendency in applied logic, Aristotle's law of noncontradiction states that "It is impossible that the same thing can at the same time both belong and not belong to the same object and in the same respect."

<span class="mw-page-title-main">Mathematical proof</span> Reasoning for mathematical statements

A mathematical proof is a deductive argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning which establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning which establish "reasonable expectation". Presenting many cases in which the statement holds is not enough for a proof, which must demonstrate that the statement is true in all possible cases. A proposition that has not been proved but is believed to be true is known as a conjecture, or a hypothesis if frequently used as an assumption for further mathematical work.

<span class="mw-page-title-main">Sorites paradox</span> Logical paradox from vague predicates

The sorites paradox is a paradox that results from vague predicates. A typical formulation involves a heap of sand, from which grains are removed individually. With the assumption that removing a single grain does not cause a heap to become a non-heap, the paradox is to consider what happens when the process is repeated enough times that only one grain remains: is it still a heap? If not, when did it change from a heap to a non-heap?

In classical logic, intuitionistic logic and similar logical systems, the principle of explosion, or the principle of Pseudo-Scotus, is the law according to which any statement can be proven from a contradiction. That is, from a contradiction, any proposition can be inferred; this is known as deductive explosion.

<span class="mw-page-title-main">Euclid of Megara</span> Greek philosopher (c. 435 – c. 365 BC)

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In logic, specifically in deductive reasoning, an argument is valid if and only if it takes a form that makes it impossible for the premises to be true and the conclusion nevertheless to be false. It is not required for a valid argument to have premises that are actually true, but to have premises that, if they were true, would guarantee the truth of the argument's conclusion. Valid arguments must be clearly expressed by means of sentences called well-formed formulas.

Stoic logic is the system of propositional logic developed by the Stoic philosophers in ancient Greece.

<span class="mw-page-title-main">Logic</span> Study of correct reasoning

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References

  1. ↑ "Reductio ad absurdum | logic". Encyclopedia Britannica. Retrieved 2019-11-27.
  2. ↑ "Definition of REDUCTIO AD ABSURDUM". www.merriam-webster.com. Retrieved 2019-11-27.
  3. ↑ "reductio ad absurdum", Collins English Dictionary – Complete and Unabridged (12th ed.), 2014 [1991], retrieved October 29, 2016
  4. 1 2 Nicholas Rescher. "Reductio ad absurdum". The Internet Encyclopedia of Philosophy. Retrieved 21 July 2009.
  5. ↑ DeLancey, Craig (2017-03-27), "8. Reductio ad Absurdum", A Concise Introduction to Logic, Open SUNY Textbooks, retrieved 2021-08-31
  6. 1 2 3 Nordquist, Richard. "Reductio Ad Absurdum in Argument". ThoughtCo. Retrieved 2019-11-27.
  7. ↑ Howard-Snyder, Frances; Howard-Snyder, Daniel; Wasserman, Ryan (30 March 2012). The Power of Logic (5th ed.). McGraw-Hill Higher Education. ISBN   978-0078038198.
  8. ↑ Daigle, Robert W. (1991). "The reductio ad absurdum argument prior to Aristotle". Master's Thesis. San Jose State Univ. Retrieved August 22, 2012.
  9. ↑ "Reductio ad Absurdum - Definition & Examples". Literary Devices. 2014-05-18. Retrieved 2021-08-31.
  10. ↑ Joyce, David (1996). "Euclid's Elements: Book I". Euclid's Elements. Department of Mathematics and Computer Science, Clark University. Retrieved December 23, 2017.
  11. ↑ Bobzien, Susanne (2006). "Ancient Logic". Stanford Encyclopedia of Philosophy. The Metaphysics Research Lab, Stanford University. Retrieved August 22, 2012.
  12. ↑ Hyde & Raffman 2018.
  13. ↑ Wasler, Joseph. Nagarjuna in Context. New York: Columibia University Press. 2005, pgs. 225-263.
  14. ↑ Garfield 1995, p. 210.
  15. ↑ Ziembiński, Zygmunt (2013). Practical Logic. Springer. p. 95. ISBN   978-9401756044.
  16. ↑ Ferguson, Thomas Macaulay; Priest, Graham (2016). A Dictionary of Logic. Oxford University Press. p. 146. ISBN   978-0192511553.
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