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Quine's Paradox
Quine's paradox is a paradox concerning truth values, stated by Willard Van Orman Quine. It is related to the liar paradox as a problem, and it purports to show that a sentence can be paradoxical even if it is not self-referring and does not use demonstratives or indexicals (i.e. it does not explicitly refer to itself). The paradox can be expressed as follows: :"yields falsehood when preceded by its quotation" yields falsehood when preceded by its quotation. If the paradox is not clear, consider each part of the above description of the paradox incrementally: :it = ''yields falsehood when preceded by its quotation'' :its quotation = ''"yields falsehood when preceded by its quotation"'' :it preceded by its quotation = ''"yields falsehood when preceded by its quotation" yields falsehood when preceded by its quotation.'' With these tools, the description of the paradox may now be reconsidered; it can be seen to assert the following: :The statement "''yields falsehood when precede ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paradox
A paradox is a logically self-contradictory statement or a statement that runs contrary to one's expectation. It is a statement that, despite apparently valid reasoning from true premises, leads to a seemingly self-contradictory or a logically unacceptable conclusion. A paradox usually involves contradictory-yet-interrelated elements that exist simultaneously and persist over time. They result in "persistent contradiction between interdependent elements" leading to a lasting "unity of opposites". In logic, many paradoxes exist that are known to be invalid arguments, yet are nevertheless valuable in promoting critical thinking, 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quotation Marks
Quotation marks (also known as quotes, quote marks, speech marks, inverted commas, or talking marks) are punctuation marks used in pairs in various writing systems to set off direct speech, a quotation, or a phrase. The pair consists of an opening quotation mark and a closing quotation mark, which may or may not be the same character. Quotation marks have a variety of forms in different languages and in different media. History The single quotation mark is traced to Ancient Greek practice, adopted and adapted by monastic copyists. Isidore of Seville, in his seventh century encyclopedia, , described their use of the Greek ''diplé'' (a chevron): 3⟩ Diplé. Our copyists place this sign in the books of the people of the Church, to separate or to indicate the quotations drawn from the Holy Scriptures. The double quotation mark derives from a marginal notation used in fifteenth-century manuscript annotations to indicate a passage of particular importance (not necessar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
An Eternal Golden Braid
An, AN, aN, or an may refer to: Businesses and organizations * Airlinair (IATA airline code AN) * Alleanza Nazionale, a former political party in Italy * AnimeNEXT, an annual anime convention located in New Jersey * Anime North, a Canadian anime convention * Ansett Australia, a major Australian airline group that is now defunct (IATA designator AN) * Apalachicola Northern Railroad (reporting mark AN) 1903–2002 ** AN Railway, a successor company, 2002– * Aryan Nations, a white supremacist religious organization * Australian National Railways Commission, an Australian rail operator from 1975 until 1987 * Antonov, a Ukrainian (formerly Soviet) aircraft manufacturing and services company, as a model prefix Entertainment and media * Antv, an Indonesian television network * ''Astronomische Nachrichten'', or ''Astronomical Notes'', an international astronomy journal * ''Avisa Nordland'', a Norwegian newspaper * ''Sweet Bean'' (あん), a 2015 Japanese film also known as ''An'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yablo's Paradox
Stephen Yablo is a Canadian-born American philosopher. He is David W. Skinner Professor of Philosophy at the Massachusetts Institute of Technology (MIT), and taught previously at the University of Michigan, Ann Arbor. He specializes in the philosophy of logic, philosophy of mind, metaphysics, philosophy of language, and philosophy of mathematics. Biography He was born in Toronto, on 30 September 1957, to a Polish father Saul Yablo and Romanian-Canadian mother Gloria Yablo (née Herman), both Jewish. He is married to fellow MIT philosopher Sally Haslanger. His Ph.D. is from University of California, Berkeley, where he worked with Donald Davidson and George Myro. In 2012, he was elected a Fellow of the American Academy of Arts and Sciences. Philosophical work In 1993, he published a short paper showing that a liar-like paradox can be generated without self-reference. He has published a number of influential papers in philosophy of mind, philosophy of language, and metaphysics, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Russell Paradox
In mathematical logic, Russell's paradox (also known as Russell's antinomy) is a set-theoretic paradox discovered by the British philosopher and mathematician Bertrand Russell in 1901. Russell's paradox shows that every set theory that contains an unrestricted comprehension principle leads to contradictions. The paradox had already been discovered independently in 1899 by the German mathematician Ernst Zermelo. However, Zermelo did not publish the idea, which remained known only to David Hilbert, Edmund Husserl, and other academics at the University of Göttingen. At the end of the 1890s, Georg Cantor – considered the founder of modern set theory – had already realized that his theory would lead to a contradiction, which he told Hilbert and Richard Dedekind by letter. According to the unrestricted comprehension principle, for any sufficiently well-defined property, there is the set of all and only the objects that have that property. Let ''R'' be the set of all sets that are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Self-reference
Self-reference occurs in natural or formal languages when a sentence, idea or formula refers to itself. The reference may be expressed either directly—through some intermediate sentence or formula—or by means of some encoding. In philosophy, it also refers to the ability of a subject to speak of or refer to itself, that is, to have the kind of thought expressed by the first person nominative singular pronoun "I" in English. Self-reference is studied and has applications in mathematics, philosophy, computer programming, second-order cybernetics, and linguistics, as well as in humor. Self-referential statements are sometimes paradoxical, and can also be considered recursive. In logic, mathematics and computing In classical philosophy, paradoxes were created by self-referential concepts such as the omnipotence paradox of asking if it was possible for a being to exist so powerful that it could create a stone that it could not lift. The Epimenides paradox, 'All Cretans are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Source Code
In computing, source code, or simply code, is any collection of code, with or without comments, written using a human-readable programming language, usually as plain text. The source code of a program is specially designed to facilitate the work of computer programmers, who specify the actions to be performed by a computer mostly by writing source code. The source code is often transformed by an assembler or compiler into binary machine code that can be executed by the computer. The machine code is then available for execution at a later time. Most application software is distributed in a form that includes only executable files. If the source code were included it would be useful to a user, programmer or a system administrator, any of whom might wish to study or modify the program. Alternatively, depending on the technology being used, source code may be interpreted and executed directly. Definitions Richard Stallman's definition, formulated in his 1989 seminal li ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quine (computing)
A quine is a computer program which takes no input and produces a copy of its own source code as its only output. The standard terms for these programs in the computability theory and computer science literature are "self-replicating programs", "self-reproducing programs", and "self-copying programs". A quine is a fixed point of an execution environment, when the execution environment is viewed as a function transforming programs into their outputs. Quines are possible in any Turing-complete programming language, as a direct consequence of Kleene's recursion theorem. For amusement, programmers sometimes attempt to develop the shortest possible quine in any given programming language. The name "quine" was coined by Douglas Hofstadter, in his popular science book ''Gödel, Escher, Bach'', in honor of philosopher Willard Van Orman Quine (1908–2000), who made an extensive study of indirect self-reference, and in particular for the following paradox-producing expression, known as Q ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Paradoxes
This list includes well known paradoxes, grouped thematically. The grouping is approximate, as paradoxes may fit into more than one category. This list collects only scenarios that have been called a paradox by at least one source and have their own article in this encyclopedia. Although considered paradoxes, some of these are simply based on fallacious reasoning ( falsidical), or an unintuitive solution (veridical). Informally, the term ''paradox'' is often used to describe a counter-intuitive result. However, some of these paradoxes qualify to fit into the mainstream perception of a paradox, which is a self-contradictory result gained even while properly applying accepted ways of reasoning. These paradoxes, often called ''antinomy,'' point out genuine problems in our understanding of the ideas of truth and description. Logic * : The supposition that, 'if one of two simultaneous assumptions leads to a contradiction, the other assumption is also disproved' leads to paradoxical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gödel's Incompleteness Theorems
Gödel's incompleteness theorems are two theorems of mathematical logic Mathematical logic is the study of logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of for ... that are concerned with the limits of in formal axiomatic theories. These results, published by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics. The theorems are widely, but not universally, interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all mathematics is impossible. The first incompleteness theorem states that no consistency, consistent system of axioms whose theorems can be listed by an effective procedure (i.e., an algorithm) is capable of proving all truths about the arithmetic of natural numbers. For any such consistent formal system, there will always b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Indirect Self-reference
Indirect self-reference describes an object referring to itself ''indirectly''. For example, define the function f such that f(x) = x(x). Any function passed as an argument to f is invoked with itself as an argument, and thus in any use of that argument is indirectly referring to itself. This example is similar to the Scheme expression "((lambda(x)(x x)) (lambda(x)(x x)))", which is expanded to itself by beta reduction, and so its evaluation loops indefinitely despite the lack of explicit looping constructs. An equivalent example can be formulated in lambda calculus. Indirect self-reference is special in that its self-referential quality is not explicit, as it is in the sentence "this sentence is false." The phrase "this sentence" refers directly to the sentence as a whole. An indirectly self-referential sentence would replace the phrase "this sentence" with an expression that effectively still referred to the sentence, but did not use the pronoun "this." An example will help to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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