View source for Paradox

{{Short description|Logically self-contradictory statement}}
{{Other uses}}

A '''paradox''' is a [[logic]]ally self-contradictory statement or a statement that runs contrary to one's expectation.<ref>{{Cite web|url=http://mathworld.wolfram.com/Paradox.html|title=Paradox|last=Weisstein|first=Eric W.|website=mathworld.wolfram.com|language=en|access-date=2019-12-05}}</ref><ref>"By “paradox” one usually means a statement claiming something that goes beyond (or even against) ‘common opinion’ (what is usually believed or held)." {{cite SEP
|title   = Paradoxes and Contemporary Logic
|url-id  = paradoxes-contemporary-logic
|date    = 2017-02-22
|edition = Fall 2017
|last    = Cantini
|first   = Andrea
|last2   = Bruni
|first2  = Riccardo
}}</ref> 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.<ref>{{cite web|title=paradox|url=http://www.oxforddictionaries.com/us/definition/american_english/paradox|archive-url=https://web.archive.org/web/20130205104405/http://oxforddictionaries.com/us/definition/american_english/paradox|url-status=dead|archive-date=February 5, 2013|website=Oxford Dictionary|publisher=Oxford University Press|access-date=21 June 2016}}</ref><ref>{{cite news|last1=Bolander|first1=Thomas|title=Self-Reference|url=http://plato.stanford.edu/entries/self-reference/|access-date=21 June 2016|publisher=The Metaphysics Research Lab, Stanford University|date=2013}}</ref> A paradox usually involves contradictory-yet-interrelated elements that exist simultaneously and persist over time.<ref>{{cite journal | last1 = Smith | first1 = W. K. | last2 = Lewis | first2 = M. W. | year = 2011 | title = Toward a theory of paradox: A dynamic equilibrium model of organizing | journal = Academy of Management Review | volume = 36 | issue = 2| pages = 381–403 | doi=10.5465/amr.2009.0223| jstor = 41318006 }}</ref><ref>{{cite journal | last1 = Zhang | first1 = Y. | last2 = Waldman | first2 = D. A. | last3 = Han | first3 = Y. | last4 = Li | first4 = X. | year = 2015 | title = Paradoxical leader behaviors in people management: Antecedents and consequences | url = https://www.researchgate.net/publication/275720775 | format = PDF | journal = Academy of Management Journal | volume = 58 | issue = 2| pages = 538–566 | doi=10.5465/amj.2012.0995}}</ref><ref>{{cite journal | last1 = Waldman | first1 = David A. | last2 = Bowen | first2 = David E. | year = 2016 | title = Learning to Be a Paradox-Savvy Leader | journal = Academy of Management Perspectives | volume = 30 | issue = 3| pages = 316–327 | doi = 10.5465/amp.2015.0070 | s2cid = 2034932 }}</ref> They result in "persistent contradiction between interdependent elements" leading to a lasting "unity of opposites".<ref>{{Cite journal |last1=Schad |first1=Jonathan |last2=Lewis |first2=Marianne W. |last3=Raisch |first3=Sebastian |last4=Smith |first4=Wendy K. |date=2016-01-01 |title=Paradox Research in Management Science: Looking Back to Move Forward |url=https://openaccess.city.ac.uk/id/eprint/15616/3/ANNALS-final.pdf |journal=Academy of Management Annals |volume=10 |issue=1 |pages=5–64 |doi=10.5465/19416520.2016.1162422 |issn=1941-6520}}</ref>

In [[logic]], many paradoxes exist that are known to be [[Validity (logic)|invalid]] arguments, yet are nevertheless valuable in promoting [[critical thinking]],<ref>{{cite journal |last=Eliason |first=James L. |url=http://connection.ebscohost.com/c/articles/9604072434/using-paradoxes-teach-critical-thinking-science |archive-url=https://web.archive.org/web/20131023061500/http://connection.ebscohost.com/c/articles/9604072434/using-paradoxes-teach-critical-thinking-science |url-status=dead |archive-date=2013-10-23 |title=Using Paradoxes to Teach Critical Thinking in Science |journal=Journal of College Science Teaching |volume=15 |issue=5 |pages=341–44 |date=March–April 1996 |url-access=subscription }}</ref> 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 [[Property (philosophy)|properties]] or [[Predicate (mathematical logic)|predicates]] were flawed.<ref name=":1">{{Citation|last1=Irvine|first1=Andrew David|title=Russell's Paradox|date=2016|url=https://plato.stanford.edu/archives/win2016/entries/russell-paradox/|encyclopedia=The Stanford Encyclopedia of Philosophy|editor-last=Zalta|editor-first=Edward N.|edition=Winter 2016|publisher=Metaphysics Research Lab, Stanford University|access-date=2019-12-05|last2=Deutsch|first2=Harry}}</ref><ref>{{cite book | last1=Crossley | first1=J.N. | last2=Ash | first2=C.J. | last3=Brickhill | first3=C.J. | last4=Stillwell | first4=J.C. | last5=Williams | first5=N.H. | title=What is mathematical logic? | zbl=0251.02001 | location=London-Oxford-New York | publisher=[[Oxford University Press]] | year=1972 | isbn=0-19-888087-1 | pages=59–60}}</ref> Others, such as [[Curry's paradox]], cannot be easily resolved by making foundational changes in a logical system.<ref>{{Citation|last1=Shapiro|first1=Lionel|title=Curry's Paradox|date=2018|url=https://plato.stanford.edu/archives/sum2018/entries/curry-paradox/|encyclopedia=The Stanford Encyclopedia of Philosophy|editor-last=Zalta|editor-first=Edward N.|edition=Summer 2018|publisher=Metaphysics Research Lab, Stanford University|access-date=2019-12-05|last2=Beall|first2=Jc}}</ref>

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.<ref>{{Cite web|url=https://faculty.washington.edu/smcohen/320/theseus.html|title=Identity, Persistence, and the Ship of Theseus|website=faculty.washington.edu|access-date=2019-12-05}}</ref> Paradoxes can also take the form of images or other media. For example, [[M.C. Escher]] featured [[Perspective (visual)|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.<ref>{{cite web |url=http://aminotes.tumblr.com/post/653017235/the-mathematical-art-of-m-c-escher-for-me-it |title=The Mathematical Art of M.C. Escher |website=Lapidarium notes |editor-first=Amira |editor-last=Skomorowska |access-date=2013-01-22}}</ref>

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

== Common elements ==
<!-- This Anchor tag serves to provide a permanent target for incoming section links. Please do not remove it, nor modify it, except to add another appropriate anchor.  If you modify the section title, please anchor the old title. It is always best to anchor an old section header that has been changed so that links to it will not be broken. See [[Template:Anchor]] for details. This template is {{subst:Anchor comment}} -->[[Self-reference]], [[contradiction]] and [[infinite regress]] are core elements of many paradoxes.<ref name=":0">{{cite book |last1=Hughes |first1=Patrick |url=https://archive.org/details/viciouscirclesin0000hugh_r3o0 |title=Vicious Circles and Infinity - A Panoply of Paradoxes |last2=Brecht |first2=George |publisher=Doubleday |year=1975 |isbn=0-385-09917-7 |location=Garden City, New York |pages=1–8 |lccn=74-17611 |author1-link=Patrick Hughes (artist) |author2-link=George Brecht}}</ref> Other common elements include [[circular definition]]s, and confusion or equivocation between different levels of [[abstraction]].

=== Self-reference ===
[[Self-reference]] occurs when a [[Sentence (linguistics)|sentence]], idea or [[Well-formed formula|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".<ref>{{cite book |title=Self-Reference: Reflections on Reflexivity |author1=S.J. Bartlett |author2=P. Suber |edition=illustrated |publisher=Springer Science & Business Media |year=2012 |isbn=978-94-009-3551-8 |page=32 |url=https://books.google.com/books?id=NkDyBwAAQBAJ}} [https://books.google.com/books?id=NkDyBwAAQBAJ&pg=PA32 Extract of page 32]</ref>  Another example occurs in the [[barber paradox]], which poses the question of whether a [[barber]] 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 ===
[[Contradiction]], along with self-reference, is a core feature of many paradoxes.<ref name=":0" />  The liar paradox, "This statement is false," exhibits contradiction because the statement cannot be false and true at the same time.<ref>{{cite book |author1= |url=https://books.google.com/books?id=LbPRUBorL-sC |title=C.I. Lewis: The Last Great Pragmatist |publisher=SUNY Press |year=2005 |isbn=978-0-7914-8282-7 |edition= |page=376}} [https://books.google.com/books?id=LbPRUBorL-sC&pg=PA376 Extract of page 376]</ref>  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.<ref name=":0" />

=== Vicious circularity, or infinite regress ===
[[File:Liars paradox.svg|thumb|Vicious circularity illustrated]]
Another core aspect of paradoxes is non-terminating [[recursion]], in the form of [[circular reasoning]] or [[infinite regress]].<ref name=":0" />  When this recursion creates a metaphysical impossibility through contradiction, the regress or circularity is [[Infinite regress#Viciousness|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.<ref name=":0" /><ref>{{cite book |author1=Myrdene Anderson |url=https://books.google.com/books?id=SlW1BwAAQBAJ |title=On Semiotic Modeling |author2=Floyd Merrell |publisher=Walter de Gruyter |year=2014 |isbn=978-3-11-084987-5 |edition=reprinted |page=268}} [https://books.google.com/books?id=SlW1BwAAQBAJ&pg=PA268 Extract of page 268]</ref>

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-truth]]s ("'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 [[logic]]ians and [[philosopher]]s. "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 (mathematics)|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.<ref name=":1" />

[[Thought-experiment]]s can also yield interesting paradoxes. The [[grandfather paradox]], for example, would arise if a [[time-travel]]er were to kill his own grandfather before his mother or father had been conceived, thereby preventing his own birth.<ref>{{Cite web|url=https://brilliant.org/wiki/introduction-to-paradoxes/|title=Introduction to paradoxes {{!}} Brilliant Math & Science Wiki|website=brilliant.org|language=en-us|access-date=2019-12-05}}</ref> 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<!--"Quine's classification of paradoxes", "Veridical paradox", and "Falsidical paradox" redirect here-->==
{{anchor|Veridical paradox|Falsidical paradox}}

[[W. V. O. Quine]] (1962) distinguished between three classes of paradoxes:<ref>{{cite book | title = The Ways of Paradox, and other essays | last1 = Quine | first1 = W.V. | author-link = W.V. Quine | year = 1966 | publisher = Random House | location = New York | chapter = The ways of paradox | isbn = 9780674948358 |chapter-url=https://books.google.com/books?id=YReOv31gdVIC&q=%22The+ways+of+paradox%22&pg=PA1}}</ref><ref name=Quine>{{Cite book | author=W.V. Quine |title=The Ways of Paradox and Other Essays | location= Cambridge, Massachusetts and London, England | publisher= Harvard University Press | date= 1976 | url=https://math.dartmouth.edu/~matc/Readers/HowManyAngels/WaysofParadox/WaysofParadox.html | edition=REVISED AND ENLARGED}}</ref>

===Veridical paradox===
{{See also|Veridicality}}
A ''veridical paradox'' produces a result that appears counter to [[intuition]], but is demonstrated to be true nonetheless:
* That the Earth is an [[Spherical Earth|approximately spherical object]] that is [[heliocentrism|rotating and in rapid motion around the Sun]], rather than the apparently obvious and common-sensical appearance of the Earth as a stationary [[flat Earth|approximately flat plane]] illuminated by a Sun that [[geocentrism|rises and falls throughout the day]]. 
* [[Condorcet paradox|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 problem|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.
* The divergence of the [[harmonic series (mathematics)|harmonic series]]:<math>\sum_{n=1}^\infty\frac{1}{n} = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \cdots.</math>

===Falsidical paradox===

A ''falsidical paradox'' establishes a result that appears false and actually is false, due to a [[fallacy]] in the demonstration. Therefore,  falsidical paradoxes can be classified as [[Fallacy|fallacious arguments]]:
* The various [[invalid proof|invalid mathematical proofs]] (e.g., that 1 = 2) are classic examples of this, often relying on a hidden [[division by zero]].
* The [[All horses are the same color|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.

===Antinomy===

An ''[[antinomy]]'' is a paradox 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.

Sometimes described since Quine's work, a ''[[dialetheia]]'' 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, following [[Aristotle]], that no ''dialetheia'' exist, but they are allowed in some [[paraconsistent logic]]s.

==Ramsey's classification ==
[[Frank Ramsey (mathematician)|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.<ref  name=SEP_ramsey>{{cite book |title=Chapter 2. The Foundations of Logic and Mathematics, Frank Ramsey, < Stanford Encyclopedia of Philosophy>|chapter-url= https://plato.stanford.edu/entries/ramsey/|author1=Fraser MacBride|author2=Mathieu Marion|author3=María José Frápolli|author4=Dorothy Edgington|author5=Edward Elliott|author6=Sebastian Lutz|author7=Jeffrey Paris|chapter= Frank Ramsey|year= 2020|publisher= Metaphysics Research Lab, Stanford University}}</ref>  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]].<ref  name=SEP_Paradoxes>{{cite book |title=Paradoxes and Contemporary Logic (Fall 2017), <Stanford Encyclopedia of Philosophy>|chapter-url=https://plato.stanford.edu/entries/paradoxes-contemporary-logic|author1=Cantini, Andrea | author2= Riccardo Bruni|chapter=Paradoxes and Contemporary Logic|year=2021|publisher=Metaphysics Research Lab, Stanford University}}</ref>

==In philosophy==

A taste for paradox is central to the philosophies of [[Laozi]], [[Zeno of Elea]], [[Zhuang Zhou|Zhuangzi]], [[Heraclitus]], [[Bhartrhari]], [[Meister Eckhart]], [[Georg Wilhelm Friedrich Hegel|Hegel]], [[Søren Kierkegaard|Kierkegaard]], [[Friedrich Nietzsche|Nietzsche]], and [[G.K. Chesterton]], among many others. Søren Kierkegaard, for example, writes in the ''[[Philosophical Fragments]]'' that:
<blockquote>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.<ref>{{cite book |last=Kierkegaard |first=Søren |editor1-first=Howard V. |editor1-last=Hong |editor2-first=Edna H. |editor2-last=Hong |title=Philosophical Fragments |year=1844 |page=37 |url=https://books.google.com/books?id=kuMoXUAaEr0C&pg=PA37 |publisher=Princeton University Press |publication-date=1985 |isbn=9780691020365}}</ref></blockquote>

==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]].<ref>{{cite journal |vauthors=Wilson MP, Pepper D, Currier GW, Holloman GH, Feifel D |title=The Psychopharmacology of Agitation: Consensus Statement of the American Association for Emergency Psychiatry Project BETA Psychopharmacology Workgroup |journal=Western Journal of Emergency Medicine |volume=13 |issue=1 |pages=26–34 |date=February 2012 |doi=10.5811/westjem.2011.9.6866 |doi-access=free |pmc=3298219 |pmid=22461918}}</ref>

The actions of [[antibody|antibodies]] on [[antigen]]s 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.<!--Acknowledging here for the reader to comprehend these facts within the proper framework of perspective; see the comment nearby below for another example of the same kind of helping the reader with [[critical thinking]].-->

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

== See also ==

{{Portal|Philosophy}}
{{cmn|
* {{annotated link|Absurdism}}
* {{annotated link|Animalia Paradoxa}}
* {{annotated link|Aporia}}
* [[Contronym]]
* {{annotated link|Dilemma}}
* {{annotated link|Ethical dilemma}}
* {{annotated link|Fallacy}}
* {{annotated link|Formal fallacy}}
* {{annotated link|Four-valued logic}}
* {{annotated link|Impossible object}}
* [[:Category:Mathematical paradoxes|Category:Mathematical paradoxes]]
* {{annotated link|List of paradoxes}}
* {{annotated link|Mu (negative)}}
* {{annotated link|Oxymoron}}
* {{annotated link|Paradox of tolerance}}
* {{annotated link|Paradox of value}}
* {{annotated link|Paradoxes of material implication}}
* {{annotated link|Plato's beard}}
* {{annotated link|Revision theory}}
* {{annotated link|Self-refuting idea}}
* {{annotated link|Syntactic ambiguity}}
* {{annotated link|Temporal paradox}}
* {{annotated link|Twin paradox}}
* {{annotated link|Zeno's paradoxes}}
}}

==References==
===Notes===

{{Reflist}}

===Bibliography===

{{Refbegin}}
* {{Cite journal |last=Bjørdal |first=Frode |author-link=Frode Alfson Bjørdal |date=2012 |title=Librationist closures of the paradoxes |journal=Logic and Logical Philosophy |volume=21 |issue=4 |pages=323–361 |doi=10.12775/LLP.2012.016 |hdl=10852/24479 |issn=1425-3305|hdl-access=free }}
* {{Cite book |last=Sainsbury |first=R. M. |url=https://books.google.com/books?id=vIU2NK1WNdgC |title=Paradoxes |date=2009 |publisher=[[Cambridge University Press]] |isbn=978-0-521-89632-0 |edition=3rd |location=Cambridge |oclc=244652614 |orig-date=1987}}
* {{Cite book |last=Poundstone |first=William |title=Labyrinths of Reason: Paradox, Puzzles, and the Frailty of Knowledge |date=2011 |publisher=[[Knopf Doubleday Publishing Group]] |isbn=978-0-385-24271-4 |location=Westminster |orig-date=1989}}
* {{Cite book |last=Sorensen |first=Roy A. |url=https://books.google.com/books?id=i_vhBwAAQBAJ |title=A Brief History of the Paradox: Philosophy and the Labyrinths of the Mind |date=2003 |publisher=[[Oxford University Press]] |isbn=978-0-19-515903-5 |location=Oxford; New York}}
* {{Cite book |last=Hughes |first=Patrick |author-link=Patrick Hughes (artist) |title=Paradoxymoron: Foolish Wisdom in Words and Pictures |date=2011 |publisher=Reverspective |isbn=978-0-956-80610-9 |location=London}}
{{Refend}}

==External links==

{{wikiquote}}
{{Wiktionary|paradox}}
{{Commons category|Paradoxes}}
{{Spoken Wikipedia|En-Paradox-article.oga|date=2005-07-07|SubCat=}}
* {{cite encyclopedia |url=http://plato.stanford.edu/entries/paradoxes-contemporary-logic/ |title=Paradoxes and Contemporary Logic |first=Andrea |last=Cantini |encyclopedia=[[Stanford Encyclopedia of Philosophy]] |date=Winter 2012 |editor-first=Edward N. |editor-last=Zalta}}
* {{cite encyclopedia |url=http://plato.stanford.edu/entries/insolubles |title=Insolubles |first=Paul Vincent |last=Spade |encyclopedia=[[Stanford Encyclopedia of Philosophy]] |date=Fall 2013 |editor-first=Edward N. |editor-last=Zalta}}
* {{MathPages|id=rr/s3-07/3-07|title=Zeno and the Paradox of Motion}}
* {{cite IEP |url-id=par-log |title="Logical Paradoxes"}}
* {{cite book |last1= Smith|first1=Wendy K. |last2=Lewis |first2=Marianne W. |author3-link=Paula Jarzabkowski |last3=Jarzabkowski |first3=Paula |last4=Langley |first4=Ann |date= 2017|title=The Oxford Handbook of Organizational Paradox |publisher=Oxford University Press |url=http://www.oxfordhandbooks.com/view/10.1093/oxfordhb/9780198754428.001.0001/oxfordhb-9780198754428 |isbn= 9780198754428}}

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