Paradox
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]
In
Examples outside logic include the
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
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
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
According to Quine's classification of paradoxes:
- A veridical paradox produces a result that appears counter to intuition, but is demonstrated to be true nonetheless:
- That the Earth is an rises and falls throughout the day.
- Condorcet's paradox demonstrates the surprising result that majority 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).
- The Monty Hall paradox (or equivalently three 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 the birthday paradox.
- In 20th-century science, Hilbert's paradox of the Grand Hotel or the Ugly duckling theorem are famously vivid examples of a theory being taken to a logical but paradoxical end.
- That the Earth is an
- A falsidical paradox establishes a result that appears false and also is false, due to a fallacy in the demonstration. Therefore, falsidical paradoxes can be classified as fallacious arguments:
- The various invalid mathematical proofs (e.g., that 1 = 2) are classic examples of this, often relying on a hidden division by zero.
- The horse 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.
- The various
- A paradox that is in neither class may be an antinomy, which reaches a self-contradictory result by properly applying accepted ways of reasoning. For example, the Grelling–Nelson paradox points 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 a dialetheia. In logic, it is often assumed, following Aristotle, that no dialetheia exist, but they are allowed in some paraconsistent logics.
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
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
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
See also
- 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
Notes
- ^ Weisstein, Eric W. "Paradox". mathworld.wolfram.com. Retrieved 2019-12-05.
- ^ "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". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy (Fall 2017 ed.).
- ^ "paradox". Oxford Dictionary. Oxford University Press. Archived from the original on February 5, 2013. Retrieved 21 June 2016.
- ^ Bolander, Thomas (2013). "Self-Reference". The Metaphysics Research Lab, Stanford University. Retrieved 21 June 2016.
- JSTOR 41318006.
- .
- S2CID 2034932.
- ISSN 1941-6520.
- ^ Eliason, James L. (March–April 1996). "Using Paradoxes to Teach Critical Thinking in Science". Journal of College Science Teaching. 15 (5): 341–44. Archived from the original on 2013-10-23.
- ^ a b 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
- Zbl 0251.02001.
- ^ 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, retrieved 2019-12-05
- ^ "Identity, Persistence, and the Ship of Theseus". faculty.washington.edu. Retrieved 2019-12-05.
- ^ Skomorowska, Amira (ed.). "The Mathematical Art of M.C. Escher". Lapidarium notes. Retrieved 2013-01-22.
- ^ LCCN 74-17611.
- ^ "Introduction to paradoxes | Brilliant Math & Science Wiki". brilliant.org. Retrieved 2019-12-05.
- ISBN 9780674948358.
- ^ 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.
- ISBN 9780691020365.
- 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
External links
- 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.
- Paradoxes at Curlie
- "Zeno and the Paradox of Motion". MathPages.com.
- ""Logical Paradoxes"". Internet Encyclopedia of Philosophy.
- Smith, Wendy K.; Lewis, Marianne W.; ISBN 9780198754428.