Paradox

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A paradox is a logically self-contradictory statement or a statement that runs contrary to one's expectation. [1] [2] It is a statement that, despite apparently valid reasoning from true or apparently true premises, leads to a seemingly self-contradictory or a logically unacceptable conclusion. [3] [4] A paradox usually involves contradictory-yet-interrelated elements that exist simultaneously and persist over time. [5] [6] [7] They result in "persistent contradiction between interdependent elements" leading to a lasting "unity of opposites". [8]

Contents

In logic, many paradoxes exist that are known to be invalid arguments, yet are nevertheless valuable in promoting critical thinking, [9] while other paradoxes have revealed errors in definitions that were assumed to be rigorous, and have caused axioms of mathematics and logic to be re-examined. One example is Russell's paradox, which questions whether a "list of all lists that do not contain themselves" would include itself and showed that attempts to found set theory on the identification of sets with properties or predicates were flawed. [10] [11] Others, such as Curry's paradox, cannot be easily resolved by making foundational changes in a logical system. [12]

Examples outside logic include the ship of Theseus from philosophy, a paradox that questions whether a ship repaired over time by replacing each and all of its wooden parts one at a time would remain the same ship. [13] Paradoxes can also take the form of images or other media. For example, M.C. Escher featured perspective-based paradoxes in many of his drawings, with walls that are regarded as floors from other points of view, and staircases that appear to climb endlessly. [14]

Informally, the term paradox is often used to describe a counterintuitive result.

Common elements

Self-reference, contradiction and infinite regress are core elements of many paradoxes. [15] Other common elements include circular definitions, and confusion or equivocation between different levels of abstraction.

Self-reference

Self-reference occurs when a sentence, idea or formula refers to itself. Although statements can be self referential without being paradoxical ("This statement is written in English" is a true and non-paradoxical self-referential statement), self-reference is a common element of paradoxes. One example occurs in the liar paradox, which is commonly formulated as the self-referential statement "This statement is false". [16] Another example occurs in the barber paradox, which poses the question of whether a barber who shaves all and only those men who do not shave themselves will shave himself. In this paradox, the barber is a self-referential concept.

Contradiction

Contradiction, along with self-reference, is a core feature of many paradoxes. [15] The liar paradox, "This statement is false," exhibits contradiction because the statement cannot be false and true at the same time. [17] The barber paradox is contradictory because it implies that the barber shaves himself if and only if the barber does not shave himself.

As with self-reference, a statement can contain a contradiction without being a paradox. "This statement is written in French" is an example of a contradictory self-referential statement that is not a paradox. [15]

Vicious circularity, or infinite regress

Vicious circularity illustrated. Liars paradox.svg
Vicious circularity illustrated.

Another core aspect of paradoxes is non-terminating recursion, in the form of circular reasoning or infinite regress. [15] When this recursion creates a metaphysical impossibility through contradiction, the regress or circularity is vicious. Again, the liar paradox is an instructive example: "This statement is false"—if the statement is true, then the statement is false, thereby making the statement true, thereby making the statement false, and so on. [15] [18]

The barber paradox also exemplifies vicious circularity: The barber shaves those who do not shave themselves, so if the barber does not shave himself, then he shaves himself, then he does not shave himself, and so on.

Other elements

Other paradoxes involve false statements and half-truths ("'impossible' is not in my vocabulary") or rely on hasty assumptions (A father and his son are in a car crash; the father is killed and the boy is rushed to the hospital. The doctor says, "I can't operate on this boy. He's my son." There is no contradiction, the doctor is the boy's mother.).

Paradoxes that are not based on a hidden error generally occur at the fringes of context or language, and require extending the context or language in order to lose their paradoxical quality. Paradoxes that arise from apparently intelligible uses of language are often of interest to logicians and philosophers. "This sentence is false" is an example of the well-known liar paradox: it is a sentence that cannot be consistently interpreted as either true or false, because if it is known to be false, then it can be inferred that it must be true, and if it is known to be true, then it can be inferred that it must be false. Russell's paradox, which shows that the notion of the set of all those sets that do not contain themselves leads to a contradiction, was instrumental in the development of modern logic and set theory. [10]

Thought-experiments can also yield interesting paradoxes. The grandfather paradox, for example, would arise if a time-traveler were to kill his own grandfather before his mother or father had been conceived, thereby preventing his own birth. [19] This is a specific example of the more general observation of the butterfly effect, or that a time-traveller's interaction with the past—however slight—would entail making changes that would, in turn, change the future in which the time-travel was yet to occur, and would thus change the circumstances of the time-travel itself.

Often a seemingly paradoxical conclusion arises from an inconsistent or inherently contradictory definition of the initial premise. In the case of that apparent paradox of a time-traveler killing his own grandfather, it is the inconsistency of defining the past to which he returns as being somehow different from the one that leads up to the future from which he begins his trip, but also insisting that he must have come to that past from the same future as the one that it leads up to.

Quine's classification

W. V. O. Quine (1962) distinguished between three classes of paradoxes: [20] [21]

According to Quine's classification of paradoxes:

A fourth kind, which may be alternatively interpreted as a special case of the third kind, has sometimes been described since Quine's work:

Ramsey's classification

Frank Ramsey drew a distinction between logical paradoxes and semantic paradoxes, with Russell's paradox belonging to the former category, and the liar paradox and Grelling's paradoxes to the latter. [22] Ramsey introduced the by-now standard distinction between logical and semantical contradictions. Logical contradictions involve mathematical or logical terms like class and number, and hence show that our logic or mathematics is problematic. Semantical contradictions involve, besides purely logical terms, notions like thought, language, and symbolism, which, according to Ramsey, are empirical (not formal) terms. Hence these contradictions are due to faulty ideas about thought or language, and they properly belong to epistemology. [23]

In philosophy

A taste for paradox is central to the philosophies of Laozi, Zeno of Elea, Zhuangzi, Heraclitus, Bhartrhari, Meister Eckhart, Hegel, Kierkegaard, Nietzsche, and G.K. Chesterton, among many others. Søren Kierkegaard, for example, writes in the Philosophical Fragments that:

But one must not think ill of the paradox, for the paradox is the passion of thought, and the thinker without the paradox is like the lover without passion: a mediocre fellow. But the ultimate potentiation of every passion is always to will its own downfall, and so it is also the ultimate passion of the understanding to will the collision, although in one way or another the collision must become its downfall. This, then, is the ultimate paradox of thought: to want to discover something that thought itself cannot think. [24]

In medicine

A paradoxical reaction to a drug is the opposite of what one would expect, such as becoming agitated by a sedative or sedated by a stimulant. Some are common and are used regularly in medicine, such as the use of stimulants such as Adderall and Ritalin in the treatment of attention deficit hyperactivity disorder (also known as ADHD), while others are rare and can be dangerous as they are not expected, such as severe agitation from a benzodiazepine. [25]

The actions of antibodies on antigens can rarely take paradoxical turns in certain ways. One example is antibody-dependent enhancement (immune enhancement) of a disease's virulence; another is the hook effect (prozone effect), of which there are several types. However, neither of these problems is common, and overall, antibodies are crucial to health, as most of the time they do their protective job quite well.

In the smoker's paradox, cigarette smoking, despite its proven harms, has a surprising inverse correlation with the epidemiological incidence of certain diseases.

See also

Related Research Articles

The Berry paradox is a self-referential paradox arising from an expression like "The smallest positive integer not definable in under sixty letters".

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In logic, the law of excluded middle or the principle of excluded middle states that for every proposition, either this proposition or its negation is true. It is one of the three laws of thought, along with the law of noncontradiction, and the law of identity; however, no system of logic is built on just these laws, and none of these laws provides inference rules, such as modus ponens or De Morgan's laws. The law is also known as the law / principleof the excluded third, in Latin principium tertii exclusi. Another Latin designation for this law is tertium non datur or "no third [possibility] is given". In classical logic, the law is a tautology.

In philosophy and logic, the classical liar paradox or liar's paradox or antinomy of the liar is the statement of a liar that they are lying: for instance, declaring that "I am lying". If the liar is indeed lying, then the liar is telling the truth, which means the liar just lied. In "this sentence is a lie" the paradox is strengthened in order to make it amenable to more rigorous logical analysis. It is still generally called the "liar paradox" although abstraction is made precisely from the liar making the statement. Trying to assign to this statement, the strengthened liar, a classical binary truth value leads to a contradiction.

Logical positivism, later called logical empiricism, and both of which together are also known as neopositivism, is a movement whose central thesis is the verification principle. This theory of knowledge asserts that only statements verifiable through direct observation or logical proof are meaningful in terms of conveying truth value, information or factual content. Starting in the late 1920s, groups of philosophers, scientists, and mathematicians formed the Berlin Circle and the Vienna Circle, which, in these two cities, would propound the ideas of logical positivism.

<span class="mw-page-title-main">Self-reference</span> Sentence, idea or formula that refers to itself

Self-reference is a concept that involves referring to oneself or one's own attributes, characteristics, or actions. It can occur in language, logic, mathematics, philosophy, and other fields.

Truth or verity is the property of being in accord with fact or reality. In everyday language, truth is typically ascribed to things that aim to represent reality or otherwise correspond to it, such as beliefs, propositions, and declarative sentences.

<span class="mw-page-title-main">Willard Van Orman Quine</span> American philosopher and logician (1908–2000)

Willard Van Orman Quine was an American philosopher and logician in the analytic tradition, recognized as "one of the most influential philosophers of the twentieth century". He served as the Edgar Pierce Chair of Philosophy at Harvard University from 1956 to 1978.

In mathematical logic, Russell's paradox is a set-theoretic paradox published by the British philosopher and mathematician Bertrand Russell in 1901. Russell's paradox shows that every set theory that contains an unrestricted comprehension principle leads to contradictions. The paradox had already been discovered independently in 1899 by the German mathematician Ernst Zermelo. However, Zermelo did not publish the idea, which remained known only to David Hilbert, Edmund Husserl, and other academics at the University of Göttingen. At the end of the 1890s, Georg Cantor – considered the founder of modern set theory – had already realized that his theory would lead to a contradiction, as he told Hilbert and Richard Dedekind by letter.

<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."

Curry's paradox is a paradox in which an arbitrary claim F is proved from the mere existence of a sentence C that says of itself "If C, then F". The paradox requires only a few apparently innocuous logical deduction rules. Since F is arbitrary, any logic having these rules allows one to prove everything. The paradox may be expressed in natural language and in various logics, including certain forms of set theory, lambda calculus, and combinatory logic.

In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic has only two possible values.

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Logical truth is one of the most fundamental concepts in logic. Broadly speaking, a logical truth is a statement which is true regardless of the truth or falsity of its constituent propositions. In other words, a logical truth is a statement which is not only true, but one which is true under all interpretations of its logical components. Thus, logical truths such as "if p, then p" can be considered tautologies. Logical truths are thought to be the simplest case of statements which are analytically true. All of philosophical logic can be thought of as providing accounts of the nature of logical truth, as well as logical consequence.

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The analytic–synthetic distinction is a semantic distinction used primarily in philosophy to distinguish between propositions that are of two types: analytic propositions and synthetic propositions. Analytic propositions are true or not true solely by virtue of their meaning, whereas synthetic propositions' truth, if any, derives from how their meaning relates to the world.

A priori and a posteriori are Latin phrases used in philosophy to distinguish types of knowledge, justification, or argument by their reliance on experience. A priori knowledge is independent from any experience. Examples include mathematics, tautologies and deduction from pure reason. A posteriori knowledge depends on empirical evidence. Examples include most fields of science and aspects of personal knowledge.

<span class="mw-page-title-main">Stephen Yablo</span> Canadian-born American philosopher

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Logical consequence is a fundamental concept in logic which describes the relationship between statements that hold true when one statement logically follows from one or more statements. A valid logical argument is one in which the conclusion is entailed by the premises, because the conclusion is the consequence of the premises. The philosophical analysis of logical consequence involves the questions: In what sense does a conclusion follow from its premises? and What does it mean for a conclusion to be a consequence of premises? All of philosophical logic is meant to provide accounts of the nature of logical consequence and the nature of logical truth.

References

Notes

  1. ↑ Weisstein, Eric W. "Paradox". mathworld.wolfram.com. Retrieved 2019-12-05.
  2. ↑ "By “paradox” one usually means a statement claiming something that goes beyond (or even against) ‘common opinion’ (what is usually believed or held)." Cantini, Andrea; Bruni, Riccardo (2017-02-22). "Paradoxes and Contemporary Logic". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy (Fall 2017 ed.).
  3. ↑ "paradox". Oxford Dictionary. Oxford University Press. Archived from the original on February 5, 2013. Retrieved 21 June 2016.
  4. ↑ Bolander, Thomas (2013). "Self-Reference". The Metaphysics Research Lab, Stanford University. Retrieved 21 June 2016.
  5. ↑ Smith, W. K.; Lewis, M. W. (2011). "Toward a theory of paradox: A dynamic equilibrium model of organizing". Academy of Management Review. 36 (2): 381–403. doi:10.5465/amr.2009.0223. JSTOR   41318006.
  6. ↑ Zhang, Y.; Waldman, D. A.; Han, Y.; Li, X. (2015). "Paradoxical leader behaviors in people management: Antecedents and consequences" (PDF). Academy of Management Journal. 58 (2): 538–566. doi:10.5465/amj.2012.0995.
  7. ↑ Waldman, David A.; Bowen, David E. (2016). "Learning to Be a Paradox-Savvy Leader". Academy of Management Perspectives. 30 (3): 316–327. doi:10.5465/amp.2015.0070. S2CID   2034932.
  8. ↑ Schad, Jonathan; Lewis, Marianne W.; Raisch, Sebastian; Smith, Wendy K. (2016-01-01). "Paradox Research in Management Science: Looking Back to Move Forward" (PDF). Academy of Management Annals. 10 (1): 5–64. doi:10.5465/19416520.2016.1162422. ISSN   1941-6520.
  9. ↑ Eliason, James L. (March–April 1996). "Using Paradoxes to Teach Critical Thinking in Science". Journal of College Science Teaching. 15 (5): 341–44. Archived from the original on 2013-10-23.
  10. 1 2 Irvine, Andrew David; Deutsch, Harry (2016), "Russell's Paradox", in Zalta, Edward N. (ed.), The Stanford Encyclopedia of Philosophy (Winter 2016 ed.), Metaphysics Research Lab, Stanford University, retrieved 2019-12-05
  11. ↑ Crossley, J.N.; Ash, C.J.; Brickhill, C.J.; Stillwell, J.C.; Williams, N.H. (1972). What is mathematical logic?. London-Oxford-New York: Oxford University Press. pp. 59–60. ISBN   0-19-888087-1. Zbl   0251.02001.
  12. ↑ Shapiro, Lionel; Beall, Jc (2018), "Curry's Paradox", in Zalta, Edward N. (ed.), The Stanford Encyclopedia of Philosophy (Summer 2018 ed.), Metaphysics Research Lab, Stanford University, retrieved 2019-12-05
  13. ↑ "Identity, Persistence, and the Ship of Theseus". faculty.washington.edu. Retrieved 2019-12-05.
  14. ↑ Skomorowska, Amira (ed.). "The Mathematical Art of M.C. Escher". Lapidarium notes. Retrieved 2013-01-22.
  15. 1 2 3 4 5 Hughes, Patrick; Brecht, George (1975). Vicious Circles and Infinity - A Panoply of Paradoxes. Garden City, New York: Doubleday. pp. 1–8. ISBN   0-385-09917-7. LCCN   74-17611.
  16. ↑ S.J. Bartlett; P. Suber (2012). Self-Reference: Reflections on Reflexivity (illustrated ed.). Springer Science & Business Media. p. 32. ISBN   978-94-009-3551-8. Extract of page 32
  17. ↑ C.I. Lewis: The Last Great Pragmatist. SUNY Press. 2005. p. 376. ISBN   978-0-7914-8282-7. Extract of page 376
  18. ↑ Myrdene Anderson; Floyd Merrell (2014). On Semiotic Modeling (reprinted ed.). Walter de Gruyter. p. 268. ISBN   978-3-11-084987-5. Extract of page 268
  19. ↑ "Introduction to paradoxes | Brilliant Math & Science Wiki". brilliant.org. Retrieved 2019-12-05.
  20. ↑ Quine, W.V. (1966). "The ways of paradox". The Ways of Paradox, and other essays. New York: Random House. ISBN   9780674948358.
  21. ↑ W.V. Quine (1976). The Ways of Paradox and Other Essays (REVISED AND ENLARGED ed.). Cambridge, Massachusetts and London, England: Harvard University Press.
  22. ↑ Fraser MacBride; Mathieu Marion; María José Frápolli; Dorothy Edgington; Edward Elliott; Sebastian Lutz; Jeffrey Paris (2020). "Frank Ramsey". Chapter 2. The Foundations of Logic and Mathematics, Frank Ramsey, < Stanford Encyclopedia of Philosophy>. Metaphysics Research Lab, Stanford University.
  23. ↑ Cantini, Andrea; Riccardo Bruni (2021). "Paradoxes and Contemporary Logic". Paradoxes and Contemporary Logic (Fall 2017), <Stanford Encyclopedia of Philosophy>. Metaphysics Research Lab, Stanford University.
  24. ↑ Kierkegaard, Søren (1844). Hong, Howard V.; Hong, Edna H. (eds.). Philosophical Fragments. Princeton University Press (published 1985). p. 37. ISBN   9780691020365.
  25. ↑ Wilson MP, Pepper D, Currier GW, Holloman GH, Feifel D (February 2012). "The Psychopharmacology of Agitation: Consensus Statement of the American Association for Emergency Psychiatry Project BETA Psychopharmacology Workgroup". Western Journal of Emergency Medicine. 13 (1): 26–34. doi: 10.5811/westjem.2011.9.6866 . PMC   3298219 . PMID   22461918.

Bibliography

  • Frode Alfson Bjørdal, Librationist Closures of the Paradoxes, Logic and Logical Philosophy, Vol. 21 No. 4 (2012), pp. 323–361.
  • Mark Sainsbury, 1988, Paradoxes, Cambridge: Cambridge University Press
  • William Poundstone, 1989, Labyrinths of Reason: Paradox, Puzzles, and the Frailty of Knowledge, Anchor
  • Roy Sorensen, 2005, A Brief History of the Paradox: Philosophy and the Labyrinths of the Mind, Oxford University Press
  • Patrick Hughes, 2011, Paradoxymoron: Foolish Wisdom in Words and Pictures, Reverspective
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