Editing Paradox (section)

== 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 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. This is a specific instance of the [[butterfly effect]]{{snd}}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.