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ﾅ「kasiewicz Logic
In mathematics and philosophy, ﾅ「kasiewicz logic ( , ) is a non-classical, many-valued logic. It was originally defined in the early 20th century by Jan ﾅ「kasiewicz as a three-valued modal logic;ﾅ「kasiewicz J., 1920, O logice trﾃｳjwartoﾅ嫩iowej (in Polish). Ruch filozoficzny 5:170窶171. English translation: On three-valued logic, in L. Borkowski (ed.), ''Selected works by Jan ﾅ「kasiewicz'', North窶滴olland, Amsterdam, 1970, pp. 87窶88. it was later generalized to ''n''-valued (for all finite ''n'') as well as infinitely-many-valued ( 邃ｵ0-valued) variants, both propositional and first order.Hay, L.S., 1963Axiomatization of the infinite-valued predicate calculus ''Journal of Symbolic Logic'' 28:77窶86. The 邃ｵ0-valued version was published in 1930 by ﾅ「kasiewicz and Alfred Tarski; consequently it is sometimes called the ﾅ「kasiewiczTarski logic. citing ﾅ「kasiewicz, J., Tarski, A.Untersuchungen ﾃｼber den Aussagenkalkﾃｼl Comp. Rend. Soc. Sci. et Lettres Varsovie Cl. III ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Law Of Excluded Middle
In logic, the law of excluded middle or the principle of excluded middle states that for every proposition, either this proposition or its negation is true. It is one of the three laws of thought, along with the law of noncontradiction and the law of identity; however, no system of logic is built on just these laws, and none of these laws provides inference rules, such as modus ponens or De Morgan's laws. The law is also known as the law/principle of the excluded third, in Latin ''principium tertii exclusi''. Another Latin designation for this law is ''tertium non datur'' or "no third ossibilityis given". In classical logic, the law is a tautology. In contemporary logic the principle is distinguished from the semantical principle of bivalence, which states that every proposition is either true or false. The principle of bivalence always implies the law of excluded middle, while the converse is not always true. A commonly cited counterexample uses statements unprovable n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Agata Ciabattoni
Agata Ciabattoni is an Italian mathematical logician specializing in non-classical logic. She is a full professor at the Institute of Logic and Computation of the Faculty of Informatics at the Vienna University of Technology (TU Wien), and a co-chair of thVienna Center for Logic and Algorithms of TU Wien(VCLA). Education and career Ciabattoni is originally from Ripatransone. She studied computer science at the University of Bologna, and completed her Ph.D. in 2000 at the University of Milan. Her dissertation, ''Proof-theory in many-valued logics'', was supervised by Daniele Mundici. She moved to Vienna in 2000 with the support of an EU Marie Curie Fellowship, and In 2007, she earned her habilitation at TU Wien. She remains affiliated with TU Wien, as a professor in the faculty of informatics. She also serves as the Collegium Logicum lecture series chair for the Kurt Gﾃｶdel Society. Contributions One of Ciabattoni's projects at TU Wien involves using mathematical logic to form ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cut-elimination Theorem
The cut-elimination theorem (or Gentzen's ''Hauptsatz'') is the central result establishing the significance of the sequent calculus. It was originally proved by Gerhard Gentzen in part I of his landmark 1935 paper "Investigations in Logical Deduction" for the systems LJ and LK formalising intuitionistic and classical logic respectively. The cut-elimination theorem states that any judgement that possesses a proof in the sequent calculus making use of the cut rule also possesses a cut-free proof, that is, a proof that does not make use of the cut rule. The Natural Deduction version of cut-elimination, known as ''normalization theorem'', has been first proved for a variety of logics by Dag Prawitz in 1965 (a similar but less general proof was given the same year by Andrﾃｨs Raggio). The cut rule A sequent is a logical expression relating multiple formulas, in the form , which is to be read as "If all of hold, then at least one of must hold", or (as Gentzen glossed): "If (A_1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Studia Logica
''Studia Logica'' (full name: ''Studia Logica, An International Journal for Symbolic Logic'') is a scientific journal publishing papers employing formal tools from Mathematics and Logic. The scope of papers published in Studia Logica covers all scientific disciplines; the key criterion for published papers is not their topic but their method: they are required to contain significant and original results concerning formal systems and their properties. The journal offers papers on topics in general logic and on applications of logic to methodology of science, linguistics, philosophy, and other branches of knowledge. The journal is published by the Institute of Philosophy and Sociology of the Polish Academy of Sciences and Springer publications. History The name Studia Logica appeared for the first time in 1934, but only one volume (edited by Jan ﾅ「kasiewicz Jan ﾅ「kasiewicz (; 21 December 1878 窶 13 February 1956) was a Polish logician and philosopher who is best known for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Logic
Linear logic is a substructural logic proposed by French logician 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 pe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sequent Calculus
In mathematical logic, sequent calculus is a style of formal logical argumentation in which every line of a proof is a conditional tautology (called a sequent by Gerhard Gentzen) instead of an unconditional tautology. Each conditional tautology is inferred from other conditional tautologies on earlier lines in a formal argument according to rules and procedures of inference, giving a better approximation to the natural style of deduction used by mathematicians than David Hilbert's earlier style of formal logic, in which every line was an unconditional tautology. More subtle distinctions may exist; for example, propositions may implicitly depend upon non-logical axioms. In that case, sequents signify conditional theorems of a first-order theory rather than conditional tautologies. Sequent calculus is one of several extant styles of proof calculus for expressing line-by-line logical arguments. * Hilbert style. Every line is an unconditional tautology (or theorem). * Gentzen s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arnon Avron
Arnon Avron (; born 1952) is an Israeli mathematician and Professor at the School of Computer Science at Tel Aviv University. His research focuses on applications of mathematical logic to computer science and artificial intelligence. Biography Born in Tel Aviv in 1952, Arnon Avron studied mathematics at Tel Aviv University and the Hebrew University of Jerusalem, receiving a Ph.D. ''magna cum laude'' from Tel Aviv University in 1985. Between 1986 and 1988, he was a visitor at the University of Edinburgh's Laboratory for Foundations of Computer Science, where he began his association with computer science. In 1988 he became a senior faculty member of the Department of Computer Science (later School of Computer Science) of Tel Aviv University, chairing the School in 1996窶1998, and becoming a Full Professor in 1999. Research Avron's research interests include proof theory, automated reasoning, non-classical logics, foundations of mathematics. For example, using analytic geometry ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hypersequent
In mathematical logic, the hypersequent framework is an extension of the proof-theoretical framework of sequent calculi used in structural proof theory to provide analytic calculi for logics that are not captured in the sequent framework. A hypersequent is usually taken to be a finite multiset of ordinary sequents, written : \Gamma_1 \Rightarrow \Delta_1 \mid \cdots \mid \Gamma_n \Rightarrow \Delta_n The sequents making up a hypersequent are called components. The added expressivity of the hypersequent framework is provided by rules manipulating different components, such as the communication rule for the intermediate logic LC ( Gﾃｶdel窶泥ummett logic) : \frac or the modal splitting rule for the modal logic S5: : \frac Hypersequent calculi have been used to treat modal logics, intermediate logics, and substructural logics. Hypersequents usually have a formula interpretation, i.e., are interpreted by a formula in the object language, nearly always as some kind of disjunction ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Basic Fuzzy Logic
In mathematical logic, basic fuzzy logic (or shortly BL), the logic of the continuous t-norms, is one of the t-norm fuzzy logics. It belongs to the broader class of substructural logics, or logics of residuated lattices;Ono (2003). it extends the logic MTL of all left-continuous t-norms. Syntax Language The language of the propositional logic BL consists of countably many propositional variables and the following primitive logical connectives: * Implication \rightarrow ( binary) * Strong conjunction \otimes (binary). The sign & is a more traditional notation for strong conjunction in the literature on fuzzy logic, while the notation \otimes follows the tradition of substructural logics. * Bottom \bot (nullary 窶 a propositional constant); 0 or \overline are common alternative signs and zero a common alternative name for the propositional constant (as the constants bottom and zero of substructural logics coincide in MTL). The following are the most common defined logica ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monoidal T-norm Logic
In mathematical logic, monoidal t-norm based logic (or shortly MTL), the logic of left-continuous t-norms, is one of the t-norm fuzzy logics. It belongs to the broader class of substructural logics, or logics of residuated lattices;Ono (2003). it extends the logic of commutative bounded integral residuated lattices (known as Hﾃｶhle's monoidal logic, Ono's FLew, or intuitionistic logic without contraction) by the axiom of prelinearity. Motivation In fuzzy logic, rather than regarding statements as being either true or false, we associate each statement with a numerical ''confidence'' in that statement. By convention the confidences range over the unit interval ,1/math>, where the maximal confidence 1 corresponds to the classical concept of true and the minimal confidence 0 corresponds to the classical concept of false. T-norms are binary functions on the real unit interval , 1that in fuzzy logic are often used to represent a conjunction connective; if a,b \in ,1/math> ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modus Ponens
In propositional logic, (; MP), also known as (), implication elimination, or affirming the antecedent, is a deductive argument form and rule of inference. It can be summarized as "''P'' implies ''Q.'' ''P'' is true. Therefore, ''Q'' must also be true." ''Modus ponens'' is a mixed hypothetical syllogism and is closely related to another valid form of argument, '' modus tollens''. Both have apparently similar but invalid forms: affirming the consequent and denying the antecedent. Constructive dilemma is the disjunctive version of ''modus ponens''. The history of ''modus ponens'' goes back to antiquity. The first to explicitly describe the argument form ''modus ponens'' was Theophrastus. It, along with '' modus tollens'', is one of the standard patterns of inference that can be applied to derive chains of conclusions that lead to the desired goal. Explanation The form of a ''modus ponens'' argument is a mixed hypothetical syllogism, with two premises and a con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
