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Branching Quantifier
In logic a branching quantifier, also called a Henkin quantifier, finite partially ordered quantifier or even nonlinear quantifier, is a partial ordering :\langle Qx_1\dots Qx_n\rangle of quantifiers for ''Q'' ∈ . It is a special case of generalized quantifier. In classical logic, quantifier prefixes are linearly ordered such that the value of a variable ''ym'' bound by a quantifier ''Qm'' depends on the value of the variables : ''y''1, ..., ''y''''m''−1 bound by quantifiers : ''Qy''1, ..., ''Qy''''m''−1 preceding ''Qm''. In a logic with (finite) partially ordered quantification this is not in general the case. Branching quantification first appeared in a 1959 conference paper of Leon Henkin. Systems of partially ordered quantification are intermediate in strength between first-order logic and second-order logic. They are being used as a basis for Hintikka's and Gabriel Sandu's independence-friendly logic. Definition and properties The simplest Henkin quan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises in a topic-neutral way. When used as a countable noun, the term "a logic" refers to a logical formal system that articulates a proof system. Formal logic contrasts with informal logic, which is associated with informal fallacies, critical thinking, and argumentation theory. While there is no general agreement on how formal and informal logic are to be distinguished, one prominent approach associates their difference with whether the studied arguments are expressed in formal or informal languages. Logic plays a central role in multiple fields, such as philosophy, mathematics, computer science, and linguistics. Logic studies arguments, which consist of a set of premises together with a conclusion. Premises and conclusions are usua ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Herbert Enderton
Herbert Bruce Enderton (April 15, 1936 – October 20, 2010) was an American mathematician. He was a Professor Emeritus of Mathematics at UCLA and a former member of the faculties of Mathematics and of Logic and the Methodology of Science at the University of California, Berkeley. Enderton also contributed to recursion theory, the theory of definability, models of analysis, computational complexity, and the history of logic. He earned his Ph.D. at Harvard in 1962. He was a member of the American Mathematical Society from 1961 until his death. Personal life He lived in Santa Monica. He married his wife, Cathy, in 1961 and they had two sons; Eric and Bert. Later years From 1980 to 2002 he was coordinating editor of the reviews section of the Association for Symbolic Logic's Journal of Symbolic Logic. Death He died from leukemia in 2010. Selected publications * * * References External links Herbert B. Enderton home pageHerbert Enderton UCLA lectureson YouTube YouTu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mostowski Quantifier
Mostowski (feminine: Mostowska, plural: Mostowscy) is a surname. It may refer to: * Mostowski Palace ( pl, Pałac Mostowskich), an 18th-century palace in Warsaw * Andrzej Mostowski (1913 - 1975), a Polish mathematician ** Mostowski collapse lemma, in mathematical logic ** Ehrenfeucht–Mostowski theorem, in model theory ** Mostowski model In mathematical set theory, the Mostowski model is a model of set theory with atoms where the full axiom of choice fails, but every set can be linearly ordered. It was introduced by . The Mostowski model can be constructed as the permutation mode ... in set theory See also * Mostovsky {{surname, Mostowski, Mostowsky, Mostowska, etc. Polish-language surnames ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Independence-friendly Logic
Independence-friendly logic (IF logic; proposed by Jaakko Hintikka and in 1989) is an extension of classical first-order logic (FOL) by means of slashed quantifiers of the form (\exists v/V) and (\forall v/V), where V is a finite set of variables. The intended reading of (\exists v/V) is "there is a v which is functionally independent from the variables in V". IF logic allows one to express more general patterns of dependence between variables than those which are implicit in first-order logic. This greater level of generality leads to an actual increase in expressive power; the set of IF sentences can characterize the same classes of structures as existential second-order logic (\Sigma^1_1). For example, it can express branching quantifier sentences, such as the formula \exists c\forall x\exists y\forall z(\exists w/\)((x=z \leftrightarrow y=w) \land y \neq c) which expresses infinity in the empty signature; this cannot be done in FOL. Therefore, first-order logic cannot, in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dependence Logic
Dependence logic is a logical formalism, created by Jouko Väänänen, which adds ''dependence atoms'' to the language of first-order logic. A dependence atom is an expression of the form =\!\!(t_1 \ldots t_n), where t_1 \ldots t_n are terms, and corresponds to the statement that the value of t_n is functionally dependent on the values of t_1\ldots t_. Dependence logic is a logic of imperfect information, like branching quantifier logic or independence-friendly logic: in other words, its game theoretic semantics can be obtained from that of first-order logic by restricting the availability of information to the players, thus allowing for non-linearly ordered patterns of dependence and independence between variables. However, dependence logic differs from these logics in that it separates the notions of dependence and independence from the notion of quantification. Syntax The syntax of dependence logic is an extension of that of first-order logic. For a fixed signature σ = (' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Game Semantics
Game semantics (german: dialogische Logik, translated as ''dialogical logic'') is an approach to Formal semantics (logic), formal semantics that grounds the concepts of truth or Validity (logic), validity on game theory, game-theoretic concepts, such as the existence of a winning strategy for a player, somewhat resembling Socratic dialogues or medieval theory of Obligationes. History In the late 1950s Paul Lorenzen was the first to introduce a game semantics for logic, and it was further developed by Kuno Lorenz. At almost the same time as Lorenzen, Jaakko Hintikka developed a model-theoretical approach known in the literature as ''GTS'' (game-theoretical semantics). Since then, a number of different game semantics have been studied in logic. Shahid Rahman (Lille) and collaborators developed dialogical logic into a general framework for the study of logical and philosophical issues related to logical pluralism. Beginning 1994 this triggered a kind of renaissance with lasting conseq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bipartite Graph
In the mathematical field of graph theory, a bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint and independent sets U and V, that is every edge connects a vertex in U to one in V. Vertex sets U and V are usually called the ''parts'' of the graph. Equivalently, a bipartite graph is a graph that does not contain any odd-length cycles. The two sets U and V may be thought of as a coloring of the graph with two colors: if one colors all nodes in U blue, and all nodes in V red, each edge has endpoints of differing colors, as is required in the graph coloring problem.. In contrast, such a coloring is impossible in the case of a non-bipartite graph, such as a triangle: after one node is colored blue and another red, the third vertex of the triangle is connected to vertices of both colors, preventing it from being assigned either color. One often writes G=(U,V,E) to denote a bipartite graph whose partition has the parts U and V, with E denot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Quantifiers
In formal semantics, a generalized quantifier (GQ) is an expression that denotes a set of sets. This is the standard semantics assigned to quantified noun phrases. For example, the generalized quantifier ''every boy'' denotes the set of sets of which every boy is a member: \ This treatment of quantifiers has been essential in achieving a compositional semantics for sentences containing quantifiers. Type theory A version of type theory is often used to make the semantics of different kinds of expressions explicit. The standard construction defines the set of types recursively as follows: #''e'' and ''t'' are types. #If ''a'' and ''b'' are both types, then so is \langle a,b\rangle #Nothing is a type, except what can be constructed on the basis of lines 1 and 2 above. Given this definition, we have the simple types ''e'' and ''t'', but also a countable infinity of complex types, some of which include: \langle e,t\rangle;\qquad \langle t,t\rangle;\qquad \langle\langle e,t\rangle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jon Barwise
Kenneth Jon Barwise (; June 29, 1942 – March 5, 2000) was an American mathematician, philosopher and logician who proposed some fundamental revisions to the way that logic is understood and used. Education and career Born in Independence, Missouri to Kenneth T. and Evelyn Barwise, Jon was a precocious child. A pupil of Solomon Feferman at Stanford University, Barwise started his research in infinitary logic. After positions as assistant professor at Yale University and the University of Wisconsin, during which time his interests turned to natural language, he returned to Stanford in 1983 to direct the Center for the Study of Language and Information. He began teaching at Indiana University in 1990. He was elected a Fellow of the American Academy of Arts and Sciences in 1999. In his last year, Barwise was invited to give the 2000 Gödel Lecture; he died prior to the lecture. Philosophical and logical work Barwise contended that, by being explicit about the context in w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lindström Quantifier
In mathematical logic, a Lindström quantifier is a generalized polyadic quantifier. Lindström quantifiers generalize first-order quantifiers, such as the existential quantifier, the universal quantifier, and the counting quantifiers. They were introduced by Per Lindström in 1966. They were later studied for their applications in logic in computer science and database query languages. Generalization of first-order quantifiers In order to facilitate discussion, some notational conventions need explaining. The expression : \phi^=\ for ''A'' an ''L''-structure (or ''L''-model) in a language ''L'', ''φ'' an ''L''-formula, and \bar a tuple of elements of the domain dom(''A'') of ''A''. In other words, \phi^ denotes a ( monadic) property defined on dom(A). In general, where ''x'' is replaced by an ''n''-tuple \bar of free variables, \phi^ denotes an ''n''-ary relation defined on dom(''A''). Each quantifier Q_A is relativized to a structure, since each quantifier is viewe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alice Ter Meulen
Alice Geraldine Baltina ter Meulen (born 4 March 1952) is a Dutch linguist, logician, and philosopher of language whose research topics include genericity in linguistics, intensional logic, generalized quantifiers, discourse representation theory, and the linguistic representation of time. She is a professor emerita at the University of Geneva. Education and career Ter Meulen was born in Amsterdam on 4 March 1952. She studied philosophy and linguistics at the University of Amsterdam, earning a bachelor's degree in philosophy in 1972 and master's degrees in philosophy and linguistics in 1976, all three degrees cum laude. She was granted a Ph.D. in philosophy of language at Stanford University in 1980; her dissertation, ''Substances, quantities and individuals: A study in the formal semantics of mass terms'', was jointly supervised by mathematical logician Jon Barwise and philosopher Julius Moravcsik. After postdoctoral research at the Max Planck Institute for Psycholinguistics ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Existential Second-order Logic
In logic and mathematics, second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. Second-order logic is in turn extended by higher-order logic and type theory. First-order logic quantifies only variables that range over individuals (elements of the domain of discourse); second-order logic, in addition, also quantifies over relations. For example, the second-order sentence \forall P\,\forall x (Px \lor \neg Px) says that for every formula ''P'', and every individual ''x'', either ''Px'' is true or not(''Px'') is true (this is the law of excluded middle). Second-order logic also includes quantification over sets, functions, and other variables (see section below). Both first-order and second-order logic use the idea of a domain of discourse (often called simply the "domain" or the "universe"). The domain is a set over which individual elements may be quantified. Examples First-order logic can quantify over individuals, but no ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
