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Paradox

March 06, 2023

For other uses, see Paradox (disambiguation).
Not to be confused with parody.

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 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 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]

In common usage, the word "paradox" often refers to statements that are ironic or unexpected, such as "the paradox that standing is more tiring than walking".[15]

Introduction

See also: List of paradoxes

Common themes in paradoxes include self-reference, infinite regress, circular definitions, and confusion or equivocation between different levels of abstraction.

Patrick Hughes outlines three laws of the paradox:[16]

Self-reference
An example is the statement "This statement is false", a form of the liar paradox. The statement is referring to itself. Another example of self-reference is the question of whether the barber shaves himself in the barber paradox. Yet another example involves the question "Is the answer to this question 'No'?"
Contradiction
"This statement is false"; the statement cannot be false and true at the same time. Another example of contradiction is if a man talking to a genie wishes that wishes couldn't come true. This contradicts itself because if the genie grants their wish, they did not grant their wish, and if the genie refuses to grant their wish, then he did indeed grant their wish, therefore making it impossible either to grant or not grant their wish without leading to a contradiction.
Vicious circularity, or infinite regress
"This statement is false"; if the statement is true, then the statement is false, thereby making the statement true. Another example of vicious circularity is the following group of statements:
"The following sentence is true."
"The previous sentence is false."

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 paradox, 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.[17] 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

See also: Veridicality

W. V. O. Quine (1962) distinguished between three classes of paradoxes:[18][19]

According to Quine's classification of paradoxes:

  • A veridical paradox produces a result that appears absurd, but is demonstrated to be true nonetheless. The paradox of Frederic's birthday in The Pirates of Penzance establishes the surprising fact that a twenty-one-year-old would have had only five birthdays had he been born on a leap day. Likewise, Arrow's impossibility theorem demonstrates difficulties in mapping voting results to the will of the people. Monty Hall paradox (or equivalently three prisoners problem) demonstrates that a decision that has an intuitive fifty–fifty chance is in fact heavily biased towards making a decision that, given the intuitive conclusion, the player would be unlikely to make. In 20th-century science, Hilbert's paradox of the Grand Hotel, Schrödinger's cat, Wigner's friend or Ugly duckling theorem are famously vivid examples of a theory being taken to a logical but paradoxical end.
  • A falsidical paradox establishes a result that not only appears false but actually is false, due to a fallacy in the demonstration. The various invalid mathematical proofs (e.g., that 1 = 2) are classic examples of this, often relying on a hidden division by zero. Another example is the inductive form of 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. Therefore, falsidical paradoxes can be classified as fallacious arguments.
  • 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 Western logics, it is often assumed, following Aristotle, that no dialetheia exist, but they are sometimes accepted in Eastern traditions (e.g. in the Mohists,[20] the Gongsun Longzi,[21] and in Zen[22]) and in paraconsistent logics. It would be mere equivocation or a matter of degree, for example, to both affirm and deny that "John is here" when John is halfway through the door, but it is self-contradictory simultaneously to affirm and deny the event.

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.[23] 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.[24]

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.[25]

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.[26]

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

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. ^ . 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". Academy of Management Annals. 10 (1): 5–64. doi:10.5465/19416520.2016.1162422. ISSN 1941-6520.
  9. ^ Eliason, James L. (March–April 1996). . Journal of College Science Teaching. 15 (5): 341–44. Archived from the original on 2013-10-23.
  10. ^ 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
  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. ^ "Paradox". Free Online Dictionary, Thesaurus and Encyclopedia. Retrieved 2013-01-22.
  16. ^ 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.
  17. ^ "Introduction to paradoxes | Brilliant Math & Science Wiki". brilliant.org. Retrieved 2019-12-05.
  18. ^ Quine, W.V. (1966). "The ways of paradox". The Ways of Paradox, and other essays. New York: Random House. ISBN 9780674948358.
  19. ^ W.V. Quine (1976). The Ways of Paradox and Other Essays (REVISED AND ENLARGED ed.). Cambridge, Massachusetts and London, England: Harvard University Press.
  20. ^ The Logicians (Warring States period),"Miscellaneous paradoxes" Stanford Encyclopedia of Philosophy
  21. ^ Graham, Angus Charles. (1990). Studies in Chinese Philosophy and Philosophical Literature, p. 334., p. 334, at Google Books
  22. ^ Chung-ying Cheng (1973) "On Zen (Ch’an) Language and Zen Paradoxes" Journal of Chinese Philosophy, V. 1 (1973) pp. 77-102
  23. ^ MacBride, Fraser, etc. (2020). "Frank Ramsey". Chapter 2. The Foundations of Logic and Mathematics, Frank Ramsey, < Stanford Encyclopedia of Philosophy>. Metaphysics Research Lab, Stanford University.{{cite book}}: CS1 maint: multiple names: authors list (link)
  24. ^ Cantini, Andrea; Riccardo Bruni (2021). "Paradoxes and Contemporary Logic". Paradoxes and Contemporary Logic (Fall 2017), <Stanford Encyclopedia of Philosophy>. Metaphysics Research Lab, Stanford University.
  25. ^ Kierkegaard, Søren (1844). Hong, Howard V.; Hong, Edna H. (eds.). Philosophical Fragments. Princeton University Press (published 1985). p. 37. ISBN 9780691020365.
  26. ^ 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

External links

 
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March 06, 2023
paradox, other, uses, disambiguation, confused, with, parody, paradox, logically, self, contradictory, statement, statement, that, runs, contrary, expectation, statement, that, despite, apparently, valid, reasoning, from, true, premises, leads, seemingly, self. For other uses see Paradox disambiguation Not to be confused with parody 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 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 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 In common usage the word paradox often refers to statements that are ironic or unexpected such as the paradox that standing is more tiring than walking 15 Contents 1 Introduction 2 Quine s classification 3 Ramsey s classification 4 In philosophy 5 In medicine 6 See also 7 References 7 1 Notes 7 2 Bibliography 8 External linksIntroduction EditSee also List of paradoxes This section needs additional citations for verification Please help improve this article by adding citations to reliable sources Unsourced material may be challenged and removed Find sources Paradox news newspapers books scholar JSTOR August 2019 Learn how and when to remove this template message Common themes in paradoxes include self reference infinite regress circular definitions and confusion or equivocation between different levels of abstraction Patrick Hughes outlines three laws of the paradox 16 Self reference An example is the statement This statement is false a form of the liar paradox The statement is referring to itself Another example of self reference is the question of whether the barber shaves himself in the barber paradox Yet another example involves the question Is the answer to this question No Contradiction This statement is false the statement cannot be false and true at the same time Another example of contradiction is if a man talking to a genie wishes that wishes couldn t come true This contradicts itself because if the genie grants their wish they did not grant their wish and if the genie refuses to grant their wish then he did indeed grant their wish therefore making it impossible either to grant or not grant their wish without leading to a contradiction Vicious circularity or infinite regress This statement is false if the statement is true then the statement is false thereby making the statement true Another example of vicious circularity is the following group of statements The following sentence is true The previous sentence is false dd 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 paradox 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 17 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 EditSee also Veridicality W V O Quine 1962 distinguished between three classes of paradoxes 18 19 According to Quine s classification of paradoxes A veridical paradox produces a result that appears absurd but is demonstrated to be true nonetheless The paradox of Frederic s birthday in The Pirates of Penzance establishes the surprising fact that a twenty one year old would have had only five birthdays had he been born on a leap day Likewise Arrow s impossibility theorem demonstrates difficulties in mapping voting results to the will of the people Monty Hall paradox or equivalently three prisoners problem demonstrates that a decision that has an intuitive fifty fifty chance is in fact heavily biased towards making a decision that given the intuitive conclusion the player would be unlikely to make In 20th century science Hilbert s paradox of the Grand Hotel Schrodinger s cat Wigner s friend or Ugly duckling theorem are famously vivid examples of a theory being taken to a logical but paradoxical end A falsidical paradox establishes a result that not only appears false but actually is false due to a fallacy in the demonstration The various invalid mathematical proofs e g that 1 2 are classic examples of this often relying on a hidden division by zero Another example is the inductive form of 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 Therefore falsidical paradoxes can be classified as fallacious arguments 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 Western logics it is often assumed following Aristotle that no dialetheia exist but they are sometimes accepted in Eastern traditions e g in the Mohists 20 the Gongsun Longzi 21 and in Zen 22 and in paraconsistent logics It would be mere equivocation or a matter of degree for example to both affirm and deny that John is here when John is halfway through the door but it is self contradictory simultaneously to affirm and deny the event Ramsey s classification EditFrank 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 23 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 24 In philosophy EditA 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 Soren 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 25 In medicine EditA 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 26 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 Edit Philosophy portalAbsurdism Theory that life in general is absurd 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 Buddhist term meaning not without or un negative prefix Oxymoron Figure of speech Paradox of tolerance Logical paradox in decision making theory Paradox of value The value of a good is determined by the value of its most important use to someone Pages displaying wikidata descriptions as a fallback Paradoxes of material implication Logical contradictions centred on the difference between natural language and Logic theoryPages displaying wikidata descriptions as a fallback 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 problemsReferences EditNotes Edit 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 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 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 S2CID 2034932 Schad Jonathan Lewis Marianne W Raisch Sebastian Smith Wendy K 2016 01 01 Paradox Research in Management Science Looking Back to Move Forward Academy of Management Annals 10 1 5 64 doi 10 5465 19416520 2016 1162422 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 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 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 Paradox Free Online Dictionary Thesaurus and Encyclopedia Retrieved 2013 01 22 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 Introduction to paradoxes Brilliant Math amp Science Wiki brilliant org Retrieved 2019 12 05 Quine W V 1966 The ways of paradox The Ways of Paradox and other essays New York Random House 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 The Logicians Warring States period Miscellaneous paradoxes Stanford Encyclopedia of Philosophy Graham Angus Charles 1990 Studies in Chinese Philosophy and Philosophical Literature p 334 p 334 at Google Books Chung ying Cheng 1973 On Zen Ch an Language and Zen Paradoxes Journal of Chinese Philosophy V 1 1973 pp 77 102 MacBride Fraser etc 2020 Frank Ramsey Chapter 2 The Foundations of Logic and Mathematics Frank Ramsey lt Stanford Encyclopedia of Philosophy gt Metaphysics Research Lab Stanford University a href Template Cite book html title Template Cite book cite book a CS1 maint multiple names authors list link Cantini Andrea Riccardo Bruni 2021 Paradoxes and Contemporary Logic Paradoxes and Contemporary Logic Fall 2017 lt Stanford Encyclopedia of Philosophy gt Metaphysics Research Lab Stanford University Kierkegaard Soren 1844 Hong Howard V Hong Edna H eds Philosophical Fragments Princeton University Press published 1985 p 37 ISBN 9780691020365 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 Edit Frode Alfson Bjordal 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 ReverspectiveExternal links Edit Wikiquote has quotations related to Paradox Look up paradox in Wiktionary the free dictionary Wikimedia Commons has media related to Paradoxes Listen to this article 23 minutes source source This audio file was created from a revision of this article dated 7 July 2005 2005 07 07 and does not reflect subsequent edits Audio help More spoken articles 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 Jarzabkowski Paula Langley Ann 2017 The Oxford Handbook of Organizational Paradox ISBN 9780198754428 Retrieved from https en wikipedia org w index php title Paradox amp oldid 1142602991, wikipedia, wiki, book, books, library,

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