Aparadoxis alogicallyself-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]
Inlogic,many paradoxes exist that are known to beinvalidarguments, yet are nevertheless valuable in promotingcritical thinking,[9]while other paradoxes have revealed errors in definitions that were assumed to be rigorous, and have causedaxiomsof mathematics and logic to be re-examined. One example isRussell's paradox,which questions whether a "list of all lists that do not contain themselves" would include itself and showed that attempts to foundset theoryon the identification of sets withpropertiesorpredicateswere flawed.[10][11]Others, such asCurry's paradox,cannot be easily resolved by making foundational changes in a logical system.[12]
Examples outside logic include theship of Theseusfrom 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. Escherfeaturedperspective-basedparadoxes 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 termparadoxis often used to describe a counterintuitive result.
Common elements
editSelf-reference,contradictionandinfinite regressare core elements of many paradoxes.[15]Other common elements includecircular definitions,and confusion or equivocation between different levels ofabstraction.
Self-reference
editSelf-referenceoccurs when asentence,idea orformularefers 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 theliar paradox,which is commonly formulated as the self-referential statement "This statement is false".[16]Another example occurs in thebarber paradox,which poses the question of whether abarberwho 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
editContradiction,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 and is instead false.[15]
Vicious circularity, or infinite regress
editAnother core aspect of paradoxes is non-terminatingrecursion,in the form ofcircular reasoningorinfinite regress.[15]When this recursion creates a metaphysical impossibility through contradiction, the regress or circularity isvicious.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
editOther paradoxes involve false statements andhalf-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 orlanguage,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 tologiciansandphilosophers."This sentence is false" is an example of the well-knownliar 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 ofthesetof all those sets that do not contain themselvesleads to a contradiction, was instrumental in the development of modern logic and set theory.[10]
Thought-experimentscan also yield interesting paradoxes. Thegrandfather paradox,for example, would arise if atime-travelerwere 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 thebutterfly 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
edit
W. V. O. Quine(1962) distinguished between three classes of paradoxes:[20][21]
According toQuine's classification of paradoxes:
- Averidical paradoxproduces a result that appears counter tointuition,but is demonstrated to be true nonetheless:
- That the Earth is anapproximately spherical objectthat isrotating and in rapid motion around the Sun,rather than the apparently obvious and common-sensical appearance of the Earth as a stationaryapproximately flat planeilluminated by a Sun thatrises and falls throughout the day.
- Condorcet's paradoxdemonstrates the surprising result thatmajority rulecan be self-contradictory, i.e. it is possible for a majority of voters to support some outcome other than the one chosen (regardless of the outcome itself).
- TheMonty Hall paradox(or equivalentlythree prisoners problem) demonstrates that a decision that has an intuitive fifty–fifty chance can instead have a provably different probable outcome. Another veridical paradox with a concise mathematical proof is thebirthday paradox.
- In 20th-century science,Hilbert's paradox of the Grand Hotelor theUgly duckling theoremare famously vivid examples of a theory being taken to a logical but paradoxical end.
- The divergence of theharmonic series:
- Afalsidical paradoxestablishes a result thatappearsfalse and alsoisfalse, due to afallacyin the demonstration. Therefore, falsidical paradoxes can be classified asfallacious arguments:
- The variousinvalid mathematical proofs(e.g., that 1 = 2) are classic examples of this, often relying on a hiddendivision by zero.
- Thehorse paradox,which falsely generalises from true specific statements
- Zeno's paradoxesare 'falsidical', concluding, for example, that a flying arrow never reaches its target or that a speedy runner cannot catch up to a tortoise with a small head-start.
- A paradox that is in neither class may be anantinomy,which reaches a self-contradictory result by properly applying accepted ways of reasoning. For example, theGrelling–Nelson paradoxpoints out genuine problems in our understanding of the ideas of truth and description.
A fourth kind, which may be alternatively interpreted as a special case of the third kind, has sometimes been described since Quine's work:
- A paradox that is both true and false at the same time and in the same sense is called adialetheia.In logic, it is often assumed, followingAristotle,that nodialetheiaexist, but they are allowed in someparaconsistent logics.
Ramsey's classification
editFrank Ramseydrew a distinction between logical paradoxes and semantic paradoxes, withRussell's paradoxbelonging to the former category, and theliar paradoxand 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 likeclassandnumber,and hence show that our logic or mathematics is problematic. Semantical contradictions involve, besides purely logical terms, notions likethought,language,andsymbolism,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 toepistemology.[23]
In philosophy
editA taste for paradox is central to the philosophies ofLaozi,Zeno of Elea,Zhuangzi,Heraclitus,Bhartrhari,Meister Eckhart,Hegel,Kierkegaard,Nietzsche,andG.K. Chesterton,among many others. Søren Kierkegaard, for example, writes in thePhilosophical Fragmentsthat:
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
editAparadoxical reactionto adrugis the opposite of what one would expect, such as becoming agitated by asedativeor sedated by astimulant.Some are common and are used regularly in medicine, such as the use of stimulants such asAdderallandRitalinin the treatment ofattention 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 abenzodiazepine.[25]
The actions ofantibodiesonantigenscan rarely take paradoxical turns in certain ways. One example isantibody-dependent enhancement(immune enhancement) of a disease's virulence; another is thehook 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 thesmoker's paradox,cigarette smoking, despite itsproven harms,has a surprising inverse correlation with the epidemiological incidence of certain diseases.
See also
edit- Absurdism– Theory that life in general is meaningless
- Animalia Paradoxa– Mythical, magical or otherwise suspect animals mentioned in Systema Naturae
- Aporia– State of puzzlement or expression of doubt, in philosophy and rhetoric
- Dilemma– Problem requiring a choice between equally undesirable alternatives
- Ethical dilemma– Type of dilemma in philosophy
- Fallacy– Argument that uses faulty reasoning
- Formal fallacy– Faulty deductive reasoning due to a logical flaw
- Four-valued logic– Any logic with four truth values
- Impossible object– Type of optical illusion
- Category:Mathematical paradoxes
- List of paradoxes– List of statements that appear to contradict themselves
- Mu (negative)– Term meaning 'not', 'without', or 'lack'
- Oxymoron– Figure of speech
- Paradox of tolerance– Logical paradox in decision-making theory
- Paradox of value– Contradiction between utility and price
- Paradoxes of material implication– logical contradictions centred on the difference between natural language and logic theory
- Plato's beard– Example of a paradoxical argument
- Revision theory
- Self-refuting idea– Idea that refutes itself
- Syntactic ambiguity– Sentences with structures permitting multiple possible interpretations
- Temporal paradox– Theoretical paradox resulting from time travel
- Twin paradox– Thought experiment in special relativity
- Zeno's paradoxes– Set of philosophical problems
References
editNotes
edit- ^Weisstein, Eric W."Paradox".mathworld.wolfram.com.Retrieved2019-12-05.
- ^"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".InZalta, Edward N.(ed.).Stanford Encyclopedia of Philosophy(Fall 2017 ed.).
- ^"paradox".Oxford Dictionary.Oxford University Press. Archived fromthe originalon February 5, 2013.Retrieved21 June2016.
- ^Bolander, Thomas (2013)."Self-Reference".The Metaphysics Research Lab, Stanford University.Retrieved21 June2016.
- ^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.JSTOR41318006.
- ^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.
- ^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.S2CID2034932.
- ^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.ISSN1941-6520.
- ^Eliason, James L. (March–April 1996)."Using Paradoxes to Teach Critical Thinking in Science".Journal of College Science Teaching.15(5): 341–44. Archived fromthe originalon 2013-10-23.
- ^abIrvine, 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,retrieved2019-12-05
- ^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.ISBN0-19-888087-1.Zbl0251.02001.
- ^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,retrieved2019-12-05
- ^"Identity, Persistence, and the Ship of Theseus".faculty.washington.edu.Retrieved2019-12-05.
- ^Skomorowska, Amira (ed.)."The Mathematical Art of M.C. Escher".Lapidarium notes.Retrieved2013-01-22.
- ^abcdeHughes, Patrick;Brecht, George(1975).Vicious Circles and Infinity - A Panoply of Paradoxes.Garden City, New York: Doubleday. pp. 1–8.ISBN0-385-09917-7.LCCN74-17611.
- ^S.J. Bartlett; P. Suber (2012).Self-Reference: Reflections on Reflexivity(illustrated ed.). Springer Science & Business Media. p. 32.ISBN978-94-009-3551-8.Extract of page 32
- ^C.I. Lewis: The Last Great Pragmatist.SUNY Press. 2005. p. 376.ISBN978-0-7914-8282-7.Extract of page 376
- ^Myrdene Anderson; Floyd Merrell (2014).On Semiotic Modeling(reprinted ed.). Walter de Gruyter. p. 268.ISBN978-3-11-084987-5.Extract of page 268
- ^"Introduction to paradoxes | Brilliant Math & Science Wiki".brilliant.org.Retrieved2019-12-05.
- ^Quine, W.V.(1966)."The ways of paradox".The Ways of Paradox, and other essays.New York: Random House.ISBN9780674948358.
- ^W.V. Quine (1976).The Ways of Paradox and Other Essays(REVISED AND ENLARGED ed.). Cambridge, Massachusetts and London, England: Harvard University Press.
- ^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.
- ^Cantini, Andrea; Riccardo Bruni (2021)."Paradoxes and Contemporary Logic".Paradoxes and Contemporary Logic (Fall 2017), <Stanford Encyclopedia of Philosophy>.Metaphysics Research Lab, Stanford University.
- ^Kierkegaard, Søren (1844). Hong, Howard V.; Hong, Edna H. (eds.).Philosophical Fragments.Princeton University Press (published 1985). p. 37.ISBN9780691020365.
- ^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.PMC3298219.PMID22461918.
Bibliography
edit- 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
External links
edit- Cantini, Andrea (Winter 2012)."Paradoxes and Contemporary Logic".In Zalta, Edward N. (ed.).Stanford Encyclopedia of Philosophy.
- Spade, Paul Vincent (Fall 2013)."Insolubles".In Zalta, Edward N. (ed.).Stanford Encyclopedia of Philosophy.
- ParadoxesatCurlie
- "Zeno and the Paradox of Motion".MathPages.com.
- ""Logical Paradoxes"".Internet Encyclopedia of Philosophy.
- Smith, Wendy K.; Lewis, Marianne W.;Jarzabkowski, Paula;Langley, Ann (2017).The Oxford Handbook of Organizational Paradox.Oxford University Press.ISBN9780198754428.