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MV-algebra
In abstract algebra, a branch of pure mathematics, an MV-algebra is an algebraic structure with a binary operation \oplus, a unary operation \neg, and the constant 0, satisfying certain axioms. MV-algebras are the algebraic semantics of ﾅ「kasiewicz logic; the letters MV refer to the ''many-valued'' logic of ﾅ「kasiewicz. MV-algebras coincide with the class of bounded commutative BCK algebras. Definitions An MV-algebra is an algebraic structure \langle A, \oplus, \lnot, 0\rangle, consisting of * a non-empty set A, * a binary operation \oplus on A, * a unary operation \lnot on A, and * a constant 0 denoting a fixed element of A, which satisfies the following identities: * (x \oplus y) \oplus z = x \oplus (y \oplus z), * x \oplus 0 = x, * x \oplus y = y \oplus x, * \lnot \lnot x = x, * x \oplus \lnot 0 = \lnot 0, and * \lnot ( \lnot x \oplus y)\oplus y = \lnot ( \lnot y \oplus x) \oplus x. By virtue of the first three axioms, \langle A, \oplus, 0 \rangle is a commutat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 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 23, 30窶5 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Effect Algebra
Effect algebras are partial algebras which abstract the (partial) algebraic properties of events that can be observed in quantum mechanics. Structures equivalent to effect algebras were introduced by three different research groups in theoretical physics or mathematics in the late 1980s and early 1990s. Since then, their mathematical properties and physical as well as computational significance have been studied by researchers in theoretical physics, mathematics and computer science. History In 1989, Giuntini and Greuling introduced structures for studying ''unsharp properties'', meaning those quantum events whose probability of occurring is strictly between zero and one (and is thus not an either-or event).Foulis, David J. "A Half-Century of Quantum Logic. What Have We Learned?" ''in'' Aerts, Diederik (ed.); Pykacz, JarosﾅＢw (ed.) ''Quantum Structures and the Nature of Reality.'' Springer, Dordrecht 1999. ISBN 978-94-017-2834-8. https://doi.org/10.1007/978-94-017-2834-8. In 19 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Semantics (mathematical Logic)
In mathematical logic, algebraic semantics is a formal semantics based on algebras studied as part of algebraic logic. For example, the modal logic S4 is characterized by the class of topological boolean algebras—that is, boolean algebras with an interior operator. Other modal logics are characterized by various other algebras with operators. The class of boolean algebras characterizes classical propositional logic, and the class of Heyting algebras propositional intuitionistic logic. MV-algebras are the algebraic semantics of ﾅ「kasiewicz logic. See also * Algebraic semantics (computer science) * Lindenbaum窶典arski algebra Further reading * (2nd published by ASL in 2009open accessat Project Euclid * * * Good introduction for readers with prior exposure to non-classical logics but without much background in order theory and/or universal algebra Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures thems ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fuzzy Logic
Fuzzy logic is a form of many-valued logic in which the truth value of variables may be any real number between 0 and 1. It is employed to handle the concept of partial truth, where the truth value may range between completely true and completely false. By contrast, in Boolean logic, the truth values of variables may only be the integer values 0 or 1. The term ''fuzzy logic'' was introduced with the 1965 proposal of fuzzy set theory by Iranian Azerbaijani mathematician Lotfi Zadeh. Fuzzy logic had, however, been studied since the 1920s, as infinite-valued logic窶馬otably by ﾅ「kasiewicz and Tarski. Fuzzy logic is based on the observation that people make decisions based on imprecise and non-numerical information. Fuzzy models or sets are mathematical means of representing vagueness and imprecise information (hence the term fuzzy). These models have the capability of recognising, representing, manipulating, interpreting, and using data and information that are vague and lack ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
BCK Algebra
In mathematics, BCI and BCK algebras are algebraic structures in universal algebra, which were introduced by Y. Imai, K. Isﾃｩki and S. Tanaka in 1966, that describe fragments of the propositional calculus involving implication known as BCI and BCK logics. Definition BCI algebra An algebra (in the sense of universal algebra) \left( X;\ast ,0\right) of type \left( 2,0\right) is called a ''BCI-algebra'' if, for any x,y,z\in X, it satisfies the following conditions. (Informally, we may read 0 as "truth" and x\ast y as "y implies x".) ; BCI-1: \left( \left( x\ast y\right) \ast \left( x\ast z\right) \right) \ast \left( z\ast y\right) =0 ; BCI-2: \left( x\ast \left( x\ast y\right) \right) \ast y=0 ; BCI-3: x\ast x=0 ; BCI-4: x\ast y=0 \land y\ast x=0\implies x=y ; BCI-5: x\ast 0=0 \implies x=0 BCK algebra A BCI-algebra \left( X;\ast ,0\right) is called a ''BCK-algebra'' if it satisfies the following condition: ; BCK-1: \forall x\in X: 0\ast x=0. A partial order can then be defined as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Many-valued Logic
Many-valued logic (also multi- or multiple-valued logic) refers to a propositional calculus in which there are more than two truth values. Traditionally, in Aristotle's logical calculus, there were only two possible values (i.e., "true" and "false") for any proposition. Classical two-valued logic may be extended to ''n''-valued logic for ''n'' greater than 2. Those most popular in the literature are three-valued (e.g., ﾅ「kasiewicz's and Kleene's, which accept the values "true", "false", and "unknown"), four-valued, nine-valued, the finite-valued (finitely-many valued) with more than three values, and the infinite-valued (infinitely-many-valued), such as fuzzy logic and probability logic. History It is wrong that the first known classical logician who did not fully accept the law of excluded middle was Aristotle (who, ironically, is also generally considered to be the first classical logician and the "father of wo-valuedlogic"). In fact, Aristotle did not contest the univer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
BCK Algebra
In mathematics, BCI and BCK algebras are algebraic structures in universal algebra, which were introduced by Y. Imai, K. Isﾃｩki and S. Tanaka in 1966, that describe fragments of the propositional calculus involving implication known as BCI and BCK logics. Definition BCI algebra An algebra (in the sense of universal algebra) \left( X;\ast ,0\right) of type \left( 2,0\right) is called a ''BCI-algebra'' if, for any x,y,z\in X, it satisfies the following conditions. (Informally, we may read 0 as "truth" and x\ast y as "y implies x".) ; BCI-1: \left( \left( x\ast y\right) \ast \left( x\ast z\right) \right) \ast \left( z\ast y\right) =0 ; BCI-2: \left( x\ast \left( x\ast y\right) \right) \ast y=0 ; BCI-3: x\ast x=0 ; BCI-4: x\ast y=0 \land y\ast x=0\implies x=y ; BCI-5: x\ast 0=0 \implies x=0 BCK algebra A BCI-algebra \left( X;\ast ,0\right) is called a ''BCK-algebra'' if it satisfies the following condition: ; BCK-1: \forall x\in X: 0\ast x=0. A partial order can then be defined as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Semantics (mathematical Logic)
In mathematical logic, algebraic semantics is a formal semantics based on algebras studied as part of algebraic logic. For example, the modal logic S4 is characterized by the class of topological boolean algebras—that is, boolean algebras with an interior operator. Other modal logics are characterized by various other algebras with operators. The class of boolean algebras characterizes classical propositional logic, and the class of Heyting algebras propositional intuitionistic logic. MV-algebras are the algebraic semantics of ﾅ「kasiewicz logic. See also * Algebraic semantics (computer science) * Lindenbaum窶典arski algebra Further reading * (2nd published by ASL in 2009open accessat Project Euclid * * * Good introduction for readers with prior exposure to non-classical logics but without much background in order theory and/or universal algebra Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures thems ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Many-valued Logic
Many-valued logic (also multi- or multiple-valued logic) refers to a propositional calculus in which there are more than two truth values. Traditionally, in Aristotle's logical calculus, there were only two possible values (i.e., "true" and "false") for any proposition. Classical two-valued logic may be extended to ''n''-valued logic for ''n'' greater than 2. Those most popular in the literature are three-valued (e.g., ﾅ「kasiewicz's and Kleene's, which accept the values "true", "false", and "unknown"), four-valued, nine-valued, the finite-valued (finitely-many valued) with more than three values, and the infinite-valued (infinitely-many-valued), such as fuzzy logic and probability logic. History It is wrong that the first known classical logician who did not fully accept the law of excluded middle was Aristotle (who, ironically, is also generally considered to be the first classical logician and the "father of wo-valuedlogic"). In fact, Aristotle did not contest the univer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Residuated Lattice
In abstract algebra, a residuated lattice is an algebraic structure that is simultaneously a lattice ''x'' 竕､ ''y'' and a monoid ''x''窶｢''y'' which admits operations ''x''\''z'' and ''z''/''y'', loosely analogous to division or implication, when ''x''窶｢''y'' is viewed as multiplication or conjunction, respectively. Called respectively right and left residuals, these operations coincide when the monoid is commutative. The general concept was introduced by Morgan Ward and Robert P. Dilworth in 1939. Examples, some of which existed prior to the general concept, include Boolean algebras, Heyting algebras, residuated Boolean algebras, relation algebras, and MV-algebras. Residuated semilattices omit the meet operation ∧, for example Kleene algebras and action algebras. Definition In mathematics, a residuated lattice is an algebraic structure L = (''L'', 竕､, 窶｢, I) such that : (i) (''L'', 竕､) is a lattice. : (ii) (''L'', 窶｢, I) is a monoid. :(iii) For all ''z'' there ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Categorical Equivalence
In category theory, a branch of abstract mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences from many areas of mathematics. Establishing an equivalence involves demonstrating strong similarities between the mathematical structures concerned. In some cases, these structures may appear to be unrelated at a superficial or intuitive level, making the notion fairly powerful: it creates the opportunity to "translate" theorems between different kinds of mathematical structures, knowing that the essential meaning of those theorems is preserved under the translation. If a category is equivalent to the opposite (or dual) of another category then one speaks of a duality of categories, and says that the two categories are dually equivalent. An equivalence of categories consists of a functor between the involved categories, which is required t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Petr Hﾃ｡jek
Petr Hﾃ｡jek (; 6 February 1940 窶 26 December 2016) was a Czech scientist in the area of mathematical logic and a professor of mathematics. Born in Prague, he worked at the Institute of Computer Science at the Academy of Sciences of the Czech Republic and as a lecturer at the Faculty of Mathematics and Physics at the Charles University in Prague and at the Faculty of Nuclear Sciences and Physical Engineering of the Czech Technical University in Prague. Academics Petr Hﾃ｡jek studied at the Faculty of Mathematics and Physics of the Charles University in Prague. Influenced by Petr Vopﾄ嬾ka, he specialized in set theory and arithmetic, and later also in logic and artificial intelligence. He contributed to establishing the mathematical fundamentals of fuzzy logic. Following the Velvet Revolution, he was appointed a senior lecturer (1993) and a professor (1997). From 1992 to 2000 he held the position of chairman of the Institute of Computer Science at the Academy of Sciences of the Czec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
