A paradox can be an apparently true statement or group of statements that leads to a contradiction or a situation which defies intuition; or it can be, seemingly opposite, an apparent contradiction that actually expresses a non-dual truth (cf. The meaning of the word truth extends from Honesty, Good faith, and Sincerity in general to agreement with Fact or Reality In Logic and Philosophy, proposition refers to either (a the content or Meaning of a meaningful Declarative sentence In Classical logic, a contradiction consists of a logical incompatibility between two or more Propositions It occurs when the propositions taken together yield Intuition is apparent ability to acquire knowledge without a clear inference or the use of reason In Classical logic, a contradiction consists of a logical incompatibility between two or more Propositions It occurs when the propositions taken together yield Nondualism implies that things appear distinct while not being separate Koan). Typically, either the statements in question do not really imply the contradiction, the puzzling result is not really a contradiction, or the premises themselves are not all really true or cannot all be true together. In Discourse and Logic, a premise is a claim that is a reason (or element of a set of reasons for or objection against some other claim The word paradox is often used interchangeably with contradiction. In Classical logic, a contradiction consists of a logical incompatibility between two or more Propositions It occurs when the propositions taken together yield Often, mistakenly, it is used to describe situations that are ironic. Irony is a literary or Rhetorical device, in which there is an incongruity or Discordance between what one says or does and what one means or

The recognition of ambiguities, equivocations, and unstated assumptions underlying known paradoxes has led to significant advances in science, philosophy and mathematics. Ambiguity (Am-big-u-i-ty is the property of being ambiguous, where a Word, term notation sign Symbol, Phrase, sentence, or any Equivocation is classified as both a formal and Informal fallacy. Science (from the Latin scientia, meaning " Knowledge " or "knowing" is the effort to discover, and increase human understanding Philosophy is the study of general problems concerning matters such as existence knowledge truth beauty justice validity mind and language Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and But many paradoxes, such as Curry's paradox, do not yet have universally accepted resolutions. Curry's paradox is a Paradox that occurs in Naive set theory or naive Logics and allows the derivation of an arbitrary sentence from a self-referring sentence

Sometimes the term paradox is used for situations that are merely surprising. The birthday paradox, for instance, is unexpected but perfectly logical. In Probability theory, the birthday problem, pertains to the Probability that in a set of Randomly chosen people some pair of them will have the same The logician Willard V. O. Quine distinguishes falsidical paradoxes, which are seemingly valid, logical demonstrations of absurdities, from veridical paradoxes, such as the birthday paradox, which are seeming absurdities that are nevertheless true. Willard Van Orman Quine (June 25 1908 Akron, Ohio &ndash December 25 2000 (known to intimates as "Van" ^[1] Paradoxes in economics tend to be the veridical type, typically counterintuitive outcomes of economic theory. Economics is the social science that studies the production distribution, and consumption of goods and services. In literature a paradox can be any contradictory or obviously untrue statement, which resolves itself upon later inspection. Literature is the Art of written works Literally translated the word means "acquaintance with letters" (from Latin littera letter In Classical logic, a contradiction consists of a logical incompatibility between two or more Propositions It occurs when the propositions taken together yield An inspection is most generally an organised examination or formal evaluation exercise

Contents 1 Logical paradox 2 Paradox in literature 3 Moral paradox 4 Genetic Paradox 5 See also 6 References 7 Footnotes 8 External links

Logical paradox

See also: List of paradoxes

Common themes in paradoxes include self-reference, the infinite, circular definitions, and confusion of levels of reasoning. This is a list of Paradoxes, grouped thematically Note that many of the listed paradoxes have a clear resolution—see Quine's Classification of Paradoxes. Self-reference is a phenomenon in natural or Formal languages consisting of a sentence or Formula referring to itself directly or In Set theory, an infinite set is a set that is not a Finite set. In Logic, begging the question has traditionally described a type of Logical fallacy (also called petitio principii) in which the proposition Gödel Escher Bach an Eternal Golden Braid (commonly GEB) is a Pulitzer Prize -winning book by Douglas Hofstadter, described Reason involves the ability to think understand and draw Conclusions in an Abstract way as in Human thinking Other paradoxes involve false statements or half-truths and the resulting biased assumptions. A false statement is a statement that can be either willfully or unknowingly untrue A half-truth comes in several forms and is a deceptive statement, that includes some element of Truth. For an article about the conceptual problems of the mind see Cognitive closure (philosophy.

For example, consider a situation in which a father and son are driving down the road. The car collides with a tree and the father is killed. The boy is rushed to the nearest hospital where he is prepared for emergency surgery. Surgery (from the χειρουργική cheirourgikē, via chirurgiae meaning "hand work" is a medical specialty that uses operative manual and instrumental On entering the surgery suite, the surgeon says, "I can't operate on this boy. He's my son. "

The apparent paradox is caused by a hasty generalization. Hasty generalization is a Logical fallacy of Faulty generalization by reaching an inductive Generalization based on insufficient Evidence The reader, upon seeing the word surgeon, applies a poll of their knowledge of surgeons (regardless of its depth) and reasons that since the majority of surgeons are male, the surgeon is a man, hence the contradiction: the father of the child, a man, was killed in the crash. In Medicine, a surgeon is a person who performs Surgery. Surgery is a broad category of invasive Medical treatment that involves the cutting of a The paradox is resolved if it is revealed that the surgeon is a woman, the boy's mother. Other assumptions whose resolution would also resolve the paradox are based on cognitive bias; the reader, reading terms like "father" and "son" and thinking of a familial relationship, may assume a traditional family (biological father, biological mother, and son) because other combinations are unknown or disregarded out of prejudicial views. For an article about the conceptual problems of the mind see Cognitive closure (philosophy. The paradox would resolve itself if it were revealed that the child was adopted and therefore had a biological and adopted father, or if a divorce resulted in the boy having a father and stepfather, or if a homosexual male couple had adopted a son or entered a committed relationship after one had already fathered a son. Divorce or dissolution of marriage is the termination of a Marriage. Another solution is that the father and son in the car are indeed not related at all - the father being parent to another individual distinct to the one in the car with him. This is because most people read the words "father and son" and immediately conclude that they are referring to two people in the same family, which is not necessarily true.

Paradoxes which are not based on a hidden error generally happen at the fringes of context or language, and require extending the context or language to lose their paradoxical quality. A language is a dynamic set of visual auditory or tactile Symbols of Communication and the elements used to manipulate them Paradoxes that arise from apparently intelligible uses of language are often of interest to logicians and philosophers. Logic is the study of the principles of valid demonstration and Inference. Philosophy is the study of general problems concerning matters such as existence knowledge truth beauty justice validity mind and language This sentence is false is an example of the famous liar paradox: it is a sentence which cannot be consistently interpreted as true or false, because if it is false it must be true, and if it is true it must be false. In Philosophy and Logic, the liar paradox, known to the ancients as the pseudomenon, encompasses Paradoxical statements such as "This Therefore, it can be concluded the sentence is neither true nor 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. Part of the Foundations of mathematics, Russell's paradox (also known as Russell's antinomy) discovered by Bertrand Russell in 1901 showed that the

Thought experiments can also yield interesting paradoxes. A thought experiment (from the German Gedankenexperiment) is a proposal for an Experiment that would test a Hypothesis or Theory The grandfather paradox, for example, would arise if a time traveler were to kill his own grandfather before his father was conceived, thereby preventing his own birth. This article refers to the Time travel paradox In novels based on the television series Doctor Who, "Grandfather Paradox" is the semi-mythical This article details time travel itself For other uses see Time Traveler. Under the 'traditional' definition of a paradox, the Grandfather Paradox (and other similar situations) are typically thought to cause spacetime to rip itself apart under the strain of attempting to resolve an 'unresolvable' conclusion (ie, the time traveller killed his grandfather, therefore the time traveller wouldn't be born, therefore his grandfather could not have been killed, therefore he (and the time traveller) are still alive - and so on). However, if the many worlds theory is correct, the death of the man does not cause the father of the time traveller and the time traveller to never be born because he is an alternate version of the grandfather. The many-worlds interpretation or MWI (also known as relative state formulation, theory of the universal wavefunction, parallel universes,

W. V. Quine (1962) distinguished between three classes of paradoxes. Willard Van Orman Quine (June 25 1908 Akron, Ohio &ndash December 25 2000 (known to intimates as "Van"

• A veridical paradox produces a result that appears absurd but is demonstrated to be true nevertheless. Thus, the paradox of Frederic's birthday in The Pirates of Penzance establishes the surprising fact that a person's fifth birthday is the day he turns twenty, if born on a leap day. The Pirates of Penzance, or The Slave of Duty is a Comic opera in two acts with music by Arthur Sullivan and Libretto by W Leap years Although the modern calendar counts a year as 365 days a complete revolution around the sun takes approximately 365 days and 6 hours Likewise, Arrow's impossibility theorem involves behaviour of voting systems that is surprising but true. In Social choice theory, Arrow’s impossibility theorem, or Arrow’s paradox, demonstrates that no voting system can convert the ranked preferences of individuals
• A falsidical paradox establishes a result that not only appears false but actually is false; there is a fallacy in the supposed demonstration. The various invalid proofs (e. In Mathematics, there are a variety of spurious proofs of obvious Contradictions Although the proofs are flawed the errors usually by design are comparatively subtle g. that 1 = 2) are classic examples, generally relying on a hidden division by zero. In Another example would be the inductive form of the Horse paradox.
• A paradox which is in neither class may be an antinomy, which reaches a self-contradictory result by properly applying accepted ways of reasoning. Antinomia redirects here For the Brachiopod Genus, see Antinomia (brachiopod. For example, the Grelling-Nelson paradox points out genuine problems in our understanding of the ideas of truth and description. The Grelling–Nelson paradox is a Semantic Self-referential Paradox formulated in 1908 by Kurt Grelling and Leonard Nelson and

A fourth kind has sometimes been asserted since Quine's work.

• A paradox which is both true and false at the same time in the same sense is called a dialetheia. Dialetheism is the view that there are true contradictions or dialetheias In Western logics it is often assumed, following Aristotle, that no dialetheia exist, but they are sometimes accepted in Eastern traditions and in paraconsistent logics. Aristotle (Greek Aristotélēs) (384 BC – 322 BC was a Greek philosopher a student of Plato and teacher of Alexander the Great. A paraconsistent logic is a Logical system that attempts to deal with Contradictions in a discriminating way An example might be to affirm or deny the statement "John is in the room" when John is standing precisely halfway through the doorway. It is reasonable (by human thinking) to both affirm and deny it ("well, he is, but he isn't"), and it is also reasonable to say that he is neither ("he's halfway in the room, which is neither in nor out"), despite the fact that the statement is to be exclusively proven or disproven.

Paradox in literature

Main article: Paradox (literature)

The paradox as a literary device has been defined as an anomalous juxtaposition of incongruous ideas for the sake of striking exposition or unexpected insight. In Literature, the paradox is an anomalous juxtaposition of incongruous ideas for the sake of striking exposition or unexpected insight It functions as a method of literary analysis which involves examining apparently contradictory statements and drawing conclusions either to reconcile them or to explain their presence. ^[2]

Literary or rhetorical paradoxes abound in the works of Oscar Wilde and G. K. Chesterton; other literature deals with paradox of situation. Oscar Fingal O'Flahertie Wills Wilde (16 October 1854 – 30 November 1900 was an Irish Playwright, Novelist, poet and Author of Gilbert Keith Chesterton (29 May 1874 – 14 June 1936 was an influential English writer of the early 20th century Rabelais, Cervantes, Sterne, Borges, and Chesterton are all concerned with episodes and narratives designed around paradoxes. Miguel de Cervantes Saavedra ( in modern Spanish; September 29, 1547 &ndash April 22, 1616) was a Spanish Novelist Statements such as Wilde’s “I can resist anything except temptation” and Chesterton’s “spies do not look like spies” are examples of rhetorical paradox. Further back, Polonius’ observation in Hamlet that “though this be madness, yet there is method in’t” is a memorable third. ^[2]

Moral paradox

In moral philosophy, paradox in a loose sense plays a role in ethics debates. Ethics is a major branch of Philosophy, encompassing right conduct and good life Ethics is a major branch of Philosophy, encompassing right conduct and good life For instance, it may be considered that an ethical admonition to "love thy neighbour" is not just in contrast with, but in contradiction to armed neighbours actively trying to kill you: if they succeed, you will not be able to love them. But to preemptively attack them or restrain them is not usually understood as loving. This might be better termed an ethical dilemma rather than a paradox in the strict sense. An ethical dilemma is a situation that will often involve an apparent conflict between Moral imperatives in which to obey one would result in transgressing another Another example is the conflict between an injunction not to steal and one to care for a family that you cannot afford to feed without stolen money (like the character of Robin Hood). Robin Hood is an archetypal figure in English folklore, whose story originates from medieval times but who remains significant in popular culture where Such a conflict between two maxims is normally resolved through weakening one or the other of them, e. g. the need for survival is greater than the need to avoid harm to your neighbor). However, as maxims are added for consideration, the questions of which to weaken in the general case and by how much pose issues related to Arrow's theorem (see above); it may be impossible to formulate a single system of ethics rules with a definite order of preference in the general case, a so-called "ethical calculus". Paradoxes in a more strict sense have been relatively neglected in philosophical discussion within ethics, as compared to their role in other philosophical fields such as logic, epistemology, metaphysics or even the philosophy of science. Important book-length discussions appear in Derek Parfit's Reasons and Persons and in Saul Smilansky's 10 Moral Paradoxes.

Genetic Paradox

In zoology, a paradox albino is an animal that exhibits trait of an albino, with little melanin and albino eyes, but may have black or pigmented blotches or half albino, half normal eyes. This trait is most commonly seen with reptiles, though it is still a rare occurrence. It has been seen with boa constrictors, leopard geckos, iguanas and corn snakes as a few examples. These specimens are prized pets for collectors and prices are much higher than the average albino.

See also

• List of paradoxes
• Temporal paradox
• Anachronism
• Impossible object
• Formal fallacy
• Dilemma
• Puzzle
• Zeno's paradoxes
• Self refuting ideas
• Paradoxes of set theory

References

• R. M. Sainsbury (1988). This is a list of Paradoxes, grouped thematically Note that many of the listed paradoxes have a clear resolution—see Quine's Classification of Paradoxes. A temporal paradox is a Paradoxical situation in which a Time traveler causes through actions in the past the exclusion of the possibility of the time travel that An anachronism (from the Greek "ana" " ανά " "against anti-" and "chronos" " χρόνος " An impossible object (also known as an impossible figure or an undecidable figure) is a type of Optical illusion consisting of a two- Dimensional In philosophy, a formal fallacy or a logical fallacy is a pattern of reasoning which is always wrong A dilemma (Greek δί-λημμα "double proposition " is a Problem offering at least two solutions or possibilities of which none are practically A puzzle is a Problem or Enigma that challenges Ingenuity. In a basic puzzle one is intended to piece together objects in a logical way in order to Self-refuting ideas are ideas or statements whose falsehood is a Logical consequence of the act or situation of holding them to be true This article contains a discussion of paradoxes of set theory. R Mark Sainsbury (born 1943 is a Philosopher from the United Kingdom who has worked in the areas of Philosophical logic, Philosophy of language Paradoxes. Cambridge.
• W. V. Quine (1962). Willard Van Orman Quine (June 25 1908 Akron, Ohio &ndash December 25 2000 (known to intimates as "Van" "Paradox". Scientific American, April 1962, pp. Scientific American is a Popular science magazine, published (first weekly and later monthly since August 28, 1845, making it 84–96.
• Michael Clarke (2002). Michael Clarke may refer to Michael Clarke (cricketer, Australian cricketer Michael Clarke (musician, American musician Paradoxes from A to Z. London: Routledge.
• Derek Parfit (1984). Derek Parfit (born December 11, 1942) is a British Philosopher who specializes in problems of personal identity, Rationality Reasons and Persons. Oxford: Oxford University Press.
• Saul Smilansky (2007). 10 Moral Paradoxes. Malden, MA: Blackwell Publishing.

Footnotes

1. ^ Van Orman Quine, Willard (1966). Willard Van Orman Quine (June 25 1908 Akron, Ohio &ndash December 25 2000 (known to intimates as "Van" "The Ways of Paradox", The Ways of Paradox and Other Essays. Random House, 5.  
2. ^ ^{a b} Rescher, Nicholas (2001). Paradoxes: Their Roots, Range, and Resolution. Open Court. ISBN 0812694368.  .

External links

• Some paradoxes - an anthology
• Stanford Encyclopedia of Philosophy:
  • "Paradoxes and Contemporary Logic" -- by Andrea Cantini. The Stanford Encyclopedia of Philosophy (SEP is a freely-accessible Online encyclopedia of Philosophy maintained by Stanford University.
  • "Insolubles" -- by Paul Vincent Spade.

• Paradoxes at the Open Directory Project
• "MathPages - Zeno and the Paradox of Motion"

The Open Directory Project ( ODP) also known as dmoz (from directory

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