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Grelling–Nelson Paradox
The Grelling–Nelson paradox is an antinomy, or a semantic self-referential paradox, concerning the applicability to itself of the word "heterological", meaning "inapplicable to itself". It was formulated in 1908 by Kurt Grelling and Leonard Nelson, and is sometimes mistakenly attributed to the German philosopher and mathematician Hermann Weyl. It is thus occasionally called Weyl's paradox and Grelling's paradox. It is closely related to several other well-known paradoxes, in particular, the barber paradox and Russell's paradox. The paradox Suppose one interprets the adjectives "autological" and "heterological" as follows: # An adjective is ''autological'' (sometimes ''homological'') if it describes itself. For example, the English word "English" is autological, as are "unhyphenated" and "pentasyllabic". # An adjective is ''heterological'' if it does not describe itself. Hence "long" is a heterological word (because it is not a long word), as are "hyphenated" (because it has no ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Antinomy
Antinomy (Greek ἀντί, ''antÃ'', "against, in opposition to", and νόμος, ''nómos'', "law") refers to a real or apparent mutual incompatibility of two laws. It is a term used in logic and epistemology, particularly in the philosophy of Immanuel Kant. There are many examples of antinomy. A self-contradictory phrase such as "There is no absolute truth" can be considered an antinomy because this statement is suggesting in itself to be an absolute truth, and therefore denies itself any truth in its statement. A paradox such as " this sentence is false" can also be considered to be an antinomy; for the sentence to be true, it must be false, and vice versa. Kant's use The term acquired a special significance in the philosophy of Immanuel Kant (1724–1804), who used it to describe the equally rational but contradictory results of applying to the universe of pure thought the categories or criteria of reason that are proper to the universe of sensible perception or experience ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homological
Homology may refer to: Sciences Biology *Homology (biology), any characteristic of biological organisms that is derived from a common ancestor *Sequence homology, biological homology between DNA, RNA, or protein sequences *Homologous chromosomes, chromosomes in a biological cell that pair up (synapse) during meiosis *Homologous recombination, genetic recombination in which nucleotide sequences are exchanged between molecules of DNA *Homologous desensitization, a receptor decreases its response to a signalling molecule when that agonist is in high concentration *Homology modeling, a method of protein structure prediction Chemistry *Homology (chemistry), the relationship between compounds in a homologous series *Homologous series, a series of organic compounds having different quantities of a repeated unit *Homologous temperature, the temperature of a material as a fraction of its absolute melting point *Homologation reaction, a chemical reaction which produces the next lo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proceedings Of The London Mathematical Society
The London Mathematical Society (LMS) is one of the United Kingdom's learned societies for mathematics (the others being the Royal Statistical Society (RSS), the Institute of Mathematics and its Applications (IMA), the Edinburgh Mathematical Society and the Operational Research Society (ORS). History The Society was established on 16 January 1865, the first president being Augustus De Morgan. The earliest meetings were held in University College, but the Society soon moved into Burlington House, Piccadilly. The initial activities of the Society included talks and publication of a journal. The LMS was used as a model for the establishment of the American Mathematical Society in 1888. Mary Cartwright was the first woman to be President of the LMS (in 1961–62). The Society was granted a royal charter in 1965, a century after its foundation. In 1998 the Society moved from rooms in Burlington House into De Morgan House (named after the society's first president), at 57†... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Liar Paradox
In philosophy and logic, the classical liar paradox or liar's paradox or antinomy of the liar is the statement of a liar that they are lying: for instance, declaring that "I am lying". If the liar is indeed lying, then the liar is telling the truth, which means the liar just lied. In "this sentence is a lie" the paradox is strengthened in order to make it amenable to more rigorous logical analysis. It is still generally called the "liar paradox" although abstraction is made precisely from the liar making the statement. Trying to assign to this statement, the strengthened liar, a classical binary truth value leads to a contradiction. If "this sentence is false" is true, then it is false, but the sentence states that it is false, and if it is false, then it must be true, and so on. History The Epimenides paradox (circa 600 BC) has been suggested as an example of the liar paradox, but they are not logically equivalent. The semi-mythical seer Epimenides, a Cretan, reportedly stated t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Use–mention Distinction
The use–mention distinction is a foundational concept of analytic philosophy, according to which it is necessary to make a distinction between a word (or phrase) and it.Devitt and Sterelny (1999) pp. 40–1 W.V. Quine (1940) p. 24 Many philosophical works have been "vitiated by a failure to distinguish use and mention". The distinction can sometimes be pedantic, especially in simple cases where it is obvious. The distinction between use and mention can be illustrated with the word ''cheese'': * ''Use'': Cheese is derived from milk. * ''Mention'': 'Cheese' is derived from (the Anglian variant of) the Old English word ''ċēse'' (). The first sentence is a statement about the substance called "cheese": it ''uses'' the word 'cheese' to refer to that substance. The second is a statement about the word 'cheese' as a signifier: it ''mentions'' the word without ''using'' it to refer to anything other than itself. Note the quotation marks. Grammar In written language, ''menti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metamagical Themas
''Metamagical Themas'' is an eclectic collection of articles that Douglas Hofstadter wrote for the popular science magazine ''Scientific American'' during the early 1980s. The anthology was published in 1985 by Basic Books. The volume is substantial in size and contains extensive notes concerning responses to the articles and other information relevant to their content. (One of the notes—page 65—suggested memetics for the study of memes.) Major themes include: self-reference in memes, language, art and logic; discussions of philosophical issues important in cognitive science/AI; analogies and what makes something similar to something else (specifically what makes, for example, an uppercase letter 'A' recognizable as such); and lengthy discussions of the work of Robert Axelrod on the prisoner's dilemma, as well as the idea of superrationality. The concept of superrationality, and its relevance to the Cold War, environmental issues and such, is accompanied by notes on exper ... [...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] |
Set (mathematics)
A set is the mathematical model for a collection of different things; a set contains '' elements'' or ''members'', which can be mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other sets. The set with no element is the empty set; a set with a single element is a singleton. A set may have a finite number of elements or be an infinite set. Two sets are equal if they have precisely the same elements. Sets are ubiquitous in modern mathematics. Indeed, set theory, more specifically Zermelo–Fraenkel set theory, has been the standard way to provide rigorous foundations for all branches of mathematics since the first half of the 20th century. History The concept of a set emerged in mathematics at the end of the 19th century. The German word for set, ''Menge'', was coined by Bernard Bolzano in his work ''Paradoxes of the Infinite''. Georg Cantor, one of the founders of set theory, gave the following defin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bertrand Russell
Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, artificial intelligence, cognitive science, computer science and various areas of analytic philosophy, especially philosophy of mathematics, philosophy of language, epistemology, and metaphysics.Stanford Encyclopedia of Philosophy"Bertrand Russell" 1 May 2003. He was one of the early 20th century's most prominent logicians, and a founder of analytic philosophy, along with his predecessor Gottlob Frege, his friend and colleague G. E. Moore and his student and protégé Ludwig Wittgenstein. Russell with Moore led the British "revolt against idealism". Together with his former teacher A. N. Whitehead, Russell wrote ''Principia Mathematica'', a milestone in the development of classical logic, and a major attempt to reduce the whole ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Contradiction
In traditional logic, a contradiction occurs when a proposition conflicts either with itself or established fact. It is often used as a tool to detect disingenuous beliefs and bias. Illustrating a general tendency in applied logic, Aristotle's law of noncontradiction states that "It is impossible that the same thing can at the same time both belong and not belong to the same object and in the same respect." In modern formal logic and type theory, the term is mainly used instead for a ''single'' proposition, often denoted by the falsum symbol \bot; a proposition is a contradiction if false can be derived from it, using the rules of the logic. It is a proposition that is unconditionally false (i.e., a self-contradictory proposition). This can be generalized to a collection of propositions, which is then said to "contain" a contradiction. History By creation of a paradox, Plato's '' Euthydemus'' dialogue demonstrates the need for the notion of ''contradiction''. In the ensuing ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tautology (logic)
In mathematical logic, a tautology (from el, ταυτολογία) is a formula or assertion that is true in every possible interpretation. An example is "x=y or x≠y". Similarly, "either the ball is green, or the ball is not green" is always true, regardless of the colour of the ball. The philosopher Ludwig Wittgenstein first applied the term to redundancies of propositional logic in 1921, borrowing from rhetoric, where a tautology is a repetitive statement. In logic, a formula is satisfiable if it is true under at least one interpretation, and thus a tautology is a formula whose negation is unsatisfiable. In other words, it cannot be false. It cannot be untrue. Unsatisfiable statements, both through negation and affirmation, are known formally as contradictions. A formula that is neither a tautology nor a contradiction is said to be Contingency (philosophy), logically contingent. Such a formula can be made either true or false based on the values assigned to its propositi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
