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Game Semantics
Game semantics (german: dialogische Logik, translated as ''dialogical logic'') is an approach to formal semantics that grounds the concepts of truth or validity on 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 consequences. This new philosophical impulse experienced a pa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dialogical Logic
Dialogical logic (also known as the logic of dialogues) was conceived as a pragmatic approach to the semantics of logic that resorts to concepts of game theory such as "winning a play" and that of "winning strategy". Since dialogical logic was the first approach to the semantics of logic using notions stemming from game theory, game theoretical semantics (GTS) and dialogical logic are often conflated under the term ''game semantics''. However, as discussed below, though GTS and dialogical logic are both rooted in a game-theoretical perspective, in fact, they have quite different philosophical and logical background. Nowadays it has been extended to a general framework for the study of meaning, knowledge, and inference constituted during interaction. The new developments include cooperative dialogues and dialogues deploying a fully interpreted language (''dialogues with content''). Origins and further developments The philosopher and mathematician Paul Lorenzen ( Erlangen-Nürnb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Logic
Linear logic is a substructural logic proposed by Jean-Yves Girard as a refinement of classical and intuitionistic logic, joining the dualities of the former with many of the constructive properties of the latter. Although the logic has also been studied for its own sake, more broadly, ideas from linear logic have been influential in fields such as programming languages, game semantics, and quantum physics (because linear logic can be seen as the logic of quantum information theory), as well as linguistics, particularly because of its emphasis on resource-boundedness, duality, and interaction. Linear logic lends itself to many different presentations, explanations, and intuitions. Proof-theoretically, it derives from an analysis of classical sequent calculus in which uses of (the structural rules) contraction and weakening are carefully controlled. Operationally, this means that logical deduction is no longer merely about an ever-expanding collection of persistent "truths", ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Negation
In logic, negation, also called the logical complement, is an operation that takes a proposition P to another proposition "not P", written \neg P, \mathord P or \overline. It is interpreted intuitively as being true when P is false, and false when P is true. Negation is thus a unary logical connective. It may be applied as an operation on notions, propositions, truth values, or semantic values more generally. In classical logic, negation is normally identified with the truth function that takes ''truth'' to ''falsity'' (and vice versa). In intuitionistic logic, according to the Brouwer–Heyting–Kolmogorov interpretation, the negation of a proposition P is the proposition whose proofs are the refutations of P. Definition ''Classical negation'' is an operation on one logical value, typically the value of a proposition, that produces a value of ''true'' when its operand is false, and a value of ''false'' when its operand is true. Thus if statement is true, then \neg P (pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Winning Strategy
Determinacy is a subfield of set theory, a branch of mathematics, that examines the conditions under which one or the other player of a game has a winning strategy, and the consequences of the existence of such strategies. Alternatively and similarly, "determinacy" is the property of a game whereby such a strategy exists. Determinacy was introduced by Gale and Stewart in 1950, under the name "determinateness". The games studied in set theory are usually Gale–Stewart games—two-player games of perfect information in which the players make an infinite sequence of moves and there are no draws. The field of game theory studies more general kinds of games, including games with draws such as tic-tac-toe, chess, or infinite chess, or games with imperfect information such as poker. Basic notions Games The first sort of game we shall consider is the two-player game of perfect information of length ω, in which the players play natural numbers. These games are often called Ga ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logical Conjunction
In logic, mathematics and linguistics, And (\wedge) is the truth-functional operator of logical conjunction; the ''and'' of a set of operands is true if and only if ''all'' of its operands are true. The logical connective that represents this operator is typically written as \wedge or . A \land B is true if and only if A is true and B is true, otherwise it is false. An operand of a conjunction is a conjunct. Beyond logic, the term "conjunction" also refers to similar concepts in other fields: * In natural language, the denotation of expressions such as English "and". * In programming languages, the short-circuit and control structure. * In set theory, intersection. * In lattice theory, logical conjunction ( greatest lower bound). * In predicate logic, universal quantification. Notation And is usually denoted by an infix operator: in mathematics and logic, it is denoted by \wedge, or ; in electronics, ; and in programming languages, &, &&, or and. In Jan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Disjunction
In logic, disjunction is a logical connective typically notated as \lor and read aloud as "or". For instance, the English language sentence "it is raining or it is snowing" can be represented in logic using the disjunctive formula R \lor S , assuming that R abbreviates "it is raining" and S abbreviates "it is snowing". In classical logic, disjunction is given a truth functional semantics according to which a formula \phi \lor \psi is true unless both \phi and \psi are false. Because this semantics allows a disjunctive formula to be true when both of its disjuncts are true, it is an ''inclusive'' interpretation of disjunction, in contrast with exclusive disjunction. Classical proof theoretical treatments are often given in terms of rules such as disjunction introduction and disjunction elimination. Disjunction has also been given numerous non-classical treatments, motivated by problems including Aristotle's sea battle argument, Heisenberg's uncertainty principle, as well t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Propositional Logic
Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations between propositions, including the construction of arguments based on them. Compound propositions are formed by connecting propositions by logical connectives. Propositions that contain no logical connectives are called atomic propositions. Unlike first-order logic, propositional logic does not deal with non-logical objects, predicates about them, or quantifiers. However, all the machinery of propositional logic is included in first-order logic and higher-order logics. In this sense, propositional logic is the foundation of first-order logic and higher-order logic. Explanation Logical connectives are found in natural languages. In English for example, some examples are "and" (conjunction), "or" (disjunction), "not" (negation) and "if" ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Giorgi Japaridze
Giorgi Japaridze (also spelled Giorgie Dzhaparidze) is a Georgian-American researcher in logic and theoretical computer science. He currently holds the title of Full Professor at the Computing Sciences Department of Villanova University. Japaridze is best known for his invention of computability logic, cirquent calculus, and Japaridze's polymodal logic. Research During 1985–1988 Japaridze elaborated the system GLP, known as Japaridze's polymodal logic. This is a system of modal logic with the "necessity" operators …, understood as a natural series of incrementally weak provability predicates for Peano arithmetic. In "The polymodal logic of provability" Japaridze proved the arithmetical completeness of this system, as well as its inherent incompleteness with respect to Kripke frames. GLP has been extensively studied by various authors during the subsequent three decades, especially after Lev Beklemishev, in 2004, pointed out its usefulness in understanding the proof theory of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wilfred Hodges
Wilfrid Augustine Hodges, FBA (born 27 May 1941) is a British mathematician and logician known for his work in model theory. Life Hodges attended New College, Oxford (1959–65), where he received degrees in both '' Literae Humaniores'' and (Christianic) Theology. In 1970 he was awarded a doctorate for a thesis in Logic. He lectured in both Philosophy and Mathematics at Bedford College, University of London. He has held visiting appointments in the department of philosophy at the University of California and in the department of mathematics at University of Colorado. Hodges was Professor of Mathematics at Queen Mary College, University of London from 1987 to 2006 and is the author of books on logic. Honors and awards Hodges was President of the British Logic Colloquium, of the European Association for Logic, Language and Information and of the Division of Logic, Methodology, and Philosophy of Science. In 2009 he was elected a Fellow of the British Academy. Writing style Hodge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Martin Hyland
(John) Martin Elliott Hyland is professor of mathematical logic at the University of Cambridge and a fellow of King's College, Cambridge. His interests include mathematical logic, category theory, and theoretical computer science. Education Hyland was educated at the University of Oxford where he was awarded a Doctor of Philosophy degree in 1975 for research supervised by Robin Gandy. Research and career Martin Hyland is best known for his work on category theory applied to logic (proof theory, recursion theory), theoretical computer science (lambda-calculus and semantics) and higher-dimensional algebra. In particular he is known for work on the effective topos (within topos theory) and on game semantics. His former doctoral students include Eugenia Cheng and Valeria de Paiva Valeria Correa Vaz de Paiva is a Brazilian mathematician, logician, and computer scientist. Her work includes research on logical approaches to computation, especially using category theory, knowledg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dov Gabbay
Dov M. Gabbay (; born October 23, 1945) is an Israeli logician. He is Augustus De Morgan Professor Emeritus of Logic at the Group of Logic, Language and Computation, Department of Computer Science, King's College London. Work Gabbay has authored over four hundred and fifty research papers and over thirty research monographs. He is editor of several international journals, and of many reference works and handbooks of logic, including the ''Handbook of Philosophical Logic'' (with Franz Guenthner), the ''Handbook of Logic in Computer Science]'' (with Samson Abramsky and T. S. E. Maibaum), and the ''Handbook of Logic in Artificial Intelligence and Logic Programming'' (with C.J. Hogger and J.A. Robinson). He is well-known for pioneering work on logic in computer science and artificial intelligence, especially the application of (executable) temporal logics in computer science, in particular formal verification, the logical foundations of non-monotonic reasoning and artificial intel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Andreas Blass
Andreas Raphael Blass (born October 27, 1947) is a mathematician, currently a professor at the University of Michigan. He works in mathematical logic, particularly set theory, and theoretical computer science. Blass graduated from the University of Detroit, where he was a Putnam Fellow in 1965, in 1966 with a B.S. in physics. He received his Ph.D. in 1970 from Harvard University, with a thesis on ''Orderings of Ultrafilters'' written under the supervision of Frank Wattenberg. Since 1970 he has been employed by the University of Michigan, first as a ''T.H. Hildebrandt Research Instructor'' (1970–72), then assistant professor (1972–76), associate professor (1976–84) and since 1984 he has been a full professor there. In 2014, he became a Fellow of the American Mathematical Society. Selected publications and results In 1984 Blass proved that the existence of a basis for every vector space is equivalent to the axiom of choice. He made important contributions in the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
