Aparadox is alogically 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]
Inlogic, many paradoxes exist that are known to beinvalid arguments, yet are nevertheless valuable in promotingcritical thinking,[9] while other paradoxes have revealed errors in definitions that were assumed to be rigorous, and have causedaxioms of 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 theory on the identification of sets withproperties orpredicates were 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 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 featuredperspective-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 termparadox is often used to describe a counterintuitive result.
Self-reference occurs when asentence, idea orformula 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 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 abarber who shaves all and only those who do not shave themselves will shave himself. In this paradox, the barber is a self-referential concept.
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 and is instead false.[15]
Another core aspect of paradoxes is non-terminatingrecursion, in the form ofcircular reasoning orinfinite 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 paradoxes involve false statements andhalf-truths 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 tologicians andphilosophers. "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 oftheset 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. Thegrandfather paradox, for example, would arise if atime traveler were to kill his own grandfather before his mother or father had been conceived, thereby preventing his own birth. This is a specific instance of thebutterfly effect – in that any interaction a time traveler has with the past would alter conditions such that divergent events "propagate" through the world over time, ultimately altering the circumstances in which the time travel initially takes place.
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.
Condorcet's paradox demonstrates the surprising result thatmajority rule can 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.
Afalsidical paradox establishes a result that appears false and actually is false, due to afallacy in the demonstration. Therefore, falsidical paradoxes can be classified asfallacious arguments:
Thehorse paradox, which falsely generalises from true specific statements
Zeno's paradoxes are '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.
Anantinomy is a paradox which reaches a self-contradictory result by properly applying accepted ways of reasoning. For example, theGrelling–Nelson paradox points out genuine problems in our understanding of the ideas of truth and description.
Sometimes described since Quine's work, adialetheia is a paradox that is both true and false at the same time. It may be regarded as a fourth kind, or alternatively as a special case of antinomy. In logic, it is often assumed, followingAristotle, that nodialetheia exist, but they are allowed in someparaconsistent logics.
Frank Ramsey drew a distinction between logical paradoxes and semantic paradoxes, withRussell's paradox belonging to the former category, and theliar paradox and Grelling's paradoxes to the latter.[21] Ramsey introduced the by-now standard distinction between logical and semantical contradictions. Logical contradictions involve mathematical or logical terms likeclass andnumber, 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.[22]
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.[23]
The actions ofantibodies onantigens can 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.
Paradoxes of material implication – logical contradictions centred on the difference between natural language and logic theoryPages displaying wikidata descriptions as a fallback
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^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
^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
^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.
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