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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Leibniz operator ： ウィキペディア英語版 In abstract algebraic logic the Leibniz operator is a tool used to classify deductive systems, which have a precise technical definition, and capture a large number of logics. The Leibniz operator was introduced by Willem Blok and Don Pigozzi, two of the founders of the field, as a means to abstract the well-known Lindenbaum-Tarski process, that leads to the association of Boolean algebras to classical propositional calculus, and make it applicableto as wide a variety of sentential logics as possible. It is an operator that assigns to a given theory of a given sentential logic, perceived as a free algebrawith a consequence operation on its universe, thelargest congruence on the algebra that is compatible with the theory.==Formulation==In this article, we introduce the Leibniz operator in the special case of classicalpropositional calculus, then we abstract it to the general notion applied to an arbitrary sentential logic and, finally, we summarizesome of the most important consequences ofits use in the theory of abstract algebraic logic. Let :\mathcal=\langle,\vdash_,if \equiv_denotes the binary relation on the set of formulas of \mathcal, defined by:\phi\equiv_\psi if and only if \phi\leftrightarrow\psi\in T, where \leftrightarrow denotes the usualclassical propositional equivalence connective, then\equiv_ turns out to be a congruenceon the formula algebra. Furthermore, the quotient /\psi is equivalent to the condition :T\vdash_}\psi.Passing now to an arbitrary sentential logic :\mathcal=\langle,\vdash_, by:\phi\Omega(T)\psi if and only if, for every formula \alpha(x,\vec) containing a variable xand possibly other variables in the list \vec,and all formulas \vec forming a list of the same length as that of \vec, we have that:T\vdash_)if and only if T\vdash_).It turns out that this binary relation is a congruence relationon the formula algebra and, in fact, may alternatively be characterizedas the largest congruence on the formula algebra that is compatiblewith the theory T, in the sense thatif \phi\Omega(T)\psi and \phi\in T, then we must have also \psi\in T. It is this congruence thatplays the same role as the congruence used in thetraditional Lindenbaum-Tarski process described above in the context of an arbitrary sentential logic. It is not, however, the case that for arbitrary sentential logics the quotients of the free algebras bythese Leibniz congruences over different theories yield all algebrasin the class that forms the natural algebraic counterpart of thesentential logic. This phenomenon occurs only in the caseof "nice" logics and one of the main goals of Abstract Algebraic Logicis to make this vague notion of a logic being "nice", in thissense, mathematically precise. The Leibniz operator :\Omegais the operator that maps a theory T of a given logic to the Leibniz congruence :\Omega(T),that is associated with the theory. Thus, formally, :\Omega:\mathcal\rightarrow is a mapping from the collection :\mathcal of the theories of a sentential logic \mathcal to the collection :of all congruences on the formula algebra of the sentential logic.==Hierarchy==Abstract algebraic hierarchy redirects here -->The Leibniz operator and the study of various of its properties that may or may not be satisfied for particularsentential logics have given rise to what is now known asthe abstract algebraic hierarchy or Leibniz hierarchy ofsentential logics. Logics are classified in various steps of this hierarchy depending on how strong a tie exists between the logic and its algebraic counterpart.The properties of the Leibniz operator that help classifythe logics are monotonicity, injectivity, continuityand commutativity with inverse substitutions. For instance,protoalgebraic logics, forming the widest class in the hierarchy,i.e., the one that lies in the bottom of the hierarchyand contains all other classes, are characterized bythe monotonicity of the Leibniz operator on their theories.Other famous classes are formed by the equivalential logics,the weakly algebraizable logics, the algebraizable logicsetc. By now, there is a generalization of the Leibniz operator in the context of CategoricalAbstract Algebraic Logic, that makes it possibleto apply a wide variety of techniques that werepreviously applicable in the sentential logicframework to logics formalized as \pi-institutions.The \pi-institution framework is significantly widerin scope than the framework of sentential logicsbecause it allows incorporating multiple signaturesand quantifiers in the language and it provides a mechanism forhandling logics that are not syntactically-based. In abstract algebraic logic the Leibniz operator is a tool used to classify deductive systems, which have a precise technical definition, and capture a large number of logics. The Leibniz operator was introduced by Willem Blok and Don Pigozzi, two of the founders of the field, as a means to abstract the well-known Lindenbaum-Tarski process, that leads to the association of Boolean algebras to classical propositional calculus, and make it applicable to as wide a variety of sentential logics as possible. It is an operator that assigns to a given theory of a given sentential logic, perceived as a free algebra with a consequence operation on its universe, the largest congruence on the algebra that is compatible with the theory. ==Formulation== In this article, we introduce the Leibniz operator in the special case of classical propositional calculus, then we abstract it to the general notion applied to an arbitrary sentential logic and, finally, we summarize some of the most important consequences of its use in the theory of abstract algebraic logic. Let :$\backslash mathcal=\backslash langle,\backslash vdash\_$, if $\backslash equiv\_$ denotes the binary relation on the set of formulas of $\backslash mathcal$, defined by :$\backslash phi\backslash equiv\_\backslash psi$ if and only if $\backslash phi\backslash leftrightarrow\backslash psi\backslash in\; T,$ where $\backslash leftrightarrow$ denotes the usual classical propositional equivalence connective, then $\backslash equiv\_$ turns out to be a congruence on the formula algebra. Furthermore, the quotient $/\backslash psi$ is equivalent to the condition :$T\backslash vdash\_\}\backslash psi$. Passing now to an arbitrary sentential logic :$\backslash mathcal=\backslash langle,$ \vdash_, by :$\backslash phi\backslash Omega(T)\backslash psi$ if and only if, for every formula $\backslash alpha(x,\backslash vec)$ containing a variable $x$ and possibly other variables in the list $\backslash vec$, and all formulas $\backslash vec$ forming a list of the same length as that of $\backslash vec$, we have that :$T\backslash vdash\_)$ if and only if $T\backslash vdash\_)$. It turns out that this binary relation is a congruence relation on the formula algebra and, in fact, may alternatively be characterized as the largest congruence on the formula algebra that is compatible with the theory $T$, in the sense that if $\backslash phi\backslash Omega(T)\backslash psi$ and $\backslash phi\backslash in\; T$, then we must have also $\backslash psi\backslash in\; T$. It is this congruence that plays the same role as the congruence used in the traditional Lindenbaum-Tarski process described above in the context of an arbitrary sentential logic. It is not, however, the case that for arbitrary sentential logics the quotients of the free algebras by these Leibniz congruences over different theories yield all algebras in the class that forms the natural algebraic counterpart of the sentential logic. This phenomenon occurs only in the case of "nice" logics and one of the main goals of Abstract Algebraic Logic is to make this vague notion of a logic being "nice", in this sense, mathematically precise. The Leibniz operator :$\backslash Omega$ is the operator that maps a theory $T$ of a given logic to the Leibniz congruence :$\backslash Omega(T),$ that is associated with the theory. Thus, formally, :$\backslash Omega:\backslash mathcal$ \rightarrow is a mapping from the collection :$\backslash mathcal$ of the theories of a sentential logic $\backslash mathcal$ to the collection :$$ of all congruences on the formula algebra $$ of the sentential logic. ==Hierarchy== The Leibniz operator and the study of various of its properties that may or may not be satisfied for particular sentential logics have given rise to what is now known as the abstract algebraic hierarchy or Leibniz hierarchy of sentential logics. Logics are classified in various steps of this hierarchy depending on how strong a tie exists between the logic and its algebraic counterpart. The properties of the Leibniz operator that help classify the logics are monotonicity, injectivity, continuity and commutativity with inverse substitutions. For instance, protoalgebraic logics, forming the widest class in the hierarchy, i.e., the one that lies in the bottom of the hierarchy and contains all other classes, are characterized by the monotonicity of the Leibniz operator on their theories. Other famous classes are formed by the equivalential logics, the weakly algebraizable logics, the algebraizable logics etc. By now, there is a generalization of the Leibniz operator in the context of Categorical Abstract Algebraic Logic, that makes it possible to apply a wide variety of techniques that were previously applicable in the sentential logic framework to logics formalized as $\backslash pi$-institutions. The $\backslash pi$-institution framework is significantly wider in scope than the framework of sentential logics because it allows incorporating multiple signatures and quantifiers in the language and it provides a mechanism for handling logics that are not syntactically-based. 抄文引用元・出典: フリー百科事典『 The Leibniz operator and the study of various of its properties that may or may not be satisfied for particularsentential logics have given rise to what is now known asthe abstract algebraic hierarchy or Leibniz hierarchy ofsentential logics. Logics are classified in various steps of this hierarchy depending on how strong a tie exists between the logic and its algebraic counterpart.The properties of the Leibniz operator that help classifythe logics are monotonicity, injectivity, continuityand commutativity with inverse substitutions. For instance,protoalgebraic logics, forming the widest class in the hierarchy,i.e., the one that lies in the bottom of the hierarchyand contains all other classes, are characterized bythe monotonicity of the Leibniz operator on their theories.Other famous classes are formed by the equivalential logics,the weakly algebraizable logics, the algebraizable logicsetc. By now, there is a generalization of the Leibniz operator in the context of CategoricalAbstract Algebraic Logic, that makes it possibleto apply a wide variety of techniques that werepreviously applicable in the sentential logicframework to logics formalized as \pi-institutions.The \pi-institution framework is significantly widerin scope than the framework of sentential logicsbecause it allows incorporating multiple signaturesand quantifiers in the language and it provides a mechanism forhandling logics that are not syntactically-based.">ウィキペディア（Wikipedia）』 ■The Leibniz operator and the study of various of its properties that may or may not be satisfied for particularsentential logics have given rise to what is now known asthe abstract algebraic hierarchy or Leibniz hierarchy ofsentential logics. Logics are classified in various steps of this hierarchy depending on how strong a tie exists between the logic and its algebraic counterpart.The properties of the Leibniz operator that help classifythe logics are monotonicity, injectivity, continuityand commutativity with inverse substitutions. For instance,protoalgebraic logics, forming the widest class in the hierarchy,i.e., the one that lies in the bottom of the hierarchyand contains all other classes, are characterized bythe monotonicity of the Leibniz operator on their theories.Other famous classes are formed by the equivalential logics,the weakly algebraizable logics, the algebraizable logicsetc. By now, there is a generalization of the Leibniz operator in the context of CategoricalAbstract Algebraic Logic, that makes it possibleto apply a wide variety of techniques that werepreviously applicable in the sentential logicframework to logics formalized as \pi-institutions.The \pi-institution framework is significantly widerin scope than the framework of sentential logicsbecause it allows incorporating multiple signaturesand quantifiers in the language and it provides a mechanism forhandling logics that are not syntactically-based.">ウィキペディアで「In abstract algebraic logic the Leibniz operator is a tool used to classify deductive systems, which have a precise technical definition, and capture a large number of logics. The Leibniz operator was introduced by Willem Blok and Don Pigozzi, two of the founders of the field, as a means to abstract the well-known Lindenbaum-Tarski process, that leads to the association of Boolean algebras to classical propositional calculus, and make it applicableto as wide a variety of sentential logics as possible. It is an operator that assigns to a given theory of a given sentential logic, perceived as a free algebrawith a consequence operation on its universe, thelargest congruence on the algebra that is compatible with the theory.==Formulation==In this article, we introduce the Leibniz operator in the special case of classicalpropositional calculus, then we abstract it to the general notion applied to an arbitrary sentential logic and, finally, we summarizesome of the most important consequences ofits use in the theory of abstract algebraic logic. Let :\mathcal=\langle,\vdash_,if \equiv_denotes the binary relation on the set of formulas of \mathcal, defined by:\phi\equiv_\psi if and only if \phi\leftrightarrow\psi\in T, where \leftrightarrow denotes the usualclassical propositional equivalence connective, then\equiv_ turns out to be a congruenceon the formula algebra. Furthermore, the quotient /\psi is equivalent to the condition :T\vdash_}\psi.Passing now to an arbitrary sentential logic :\mathcal=\langle,\vdash_, by:\phi\Omega(T)\psi if and only if, for every formula \alpha(x,\vec) containing a variable xand possibly other variables in the list \vec,and all formulas \vec forming a list of the same length as that of \vec, we have that:T\vdash_)if and only if T\vdash_).It turns out that this binary relation is a congruence relationon the formula algebra and, in fact, may alternatively be characterizedas the largest congruence on the formula algebra that is compatiblewith the theory T, in the sense thatif \phi\Omega(T)\psi and \phi\in T, then we must have also \psi\in T. It is this congruence thatplays the same role as the congruence used in thetraditional Lindenbaum-Tarski process described above in the context of an arbitrary sentential logic. It is not, however, the case that for arbitrary sentential logics the quotients of the free algebras bythese Leibniz congruences over different theories yield all algebrasin the class that forms the natural algebraic counterpart of thesentential logic. This phenomenon occurs only in the caseof "nice" logics and one of the main goals of Abstract Algebraic Logicis to make this vague notion of a logic being "nice", in thissense, mathematically precise. The Leibniz operator :\Omegais the operator that maps a theory T of a given logic to the Leibniz congruence :\Omega(T),that is associated with the theory. Thus, formally, :\Omega:\mathcal\rightarrow is a mapping from the collection :\mathcal of the theories of a sentential logic \mathcal to the collection :of all congruences on the formula algebra of the sentential logic.==Hierarchy==Abstract algebraic hierarchy redirects here -->The Leibniz operator and the study of various of its properties that may or may not be satisfied for particularsentential logics have given rise to what is now known asthe abstract algebraic hierarchy or Leibniz hierarchy ofsentential logics. Logics are classified in various steps of this hierarchy depending on how strong a tie exists between the logic and its algebraic counterpart.The properties of the Leibniz operator that help classifythe logics are monotonicity, injectivity, continuityand commutativity with inverse substitutions. For instance,protoalgebraic logics, forming the widest class in the hierarchy,i.e., the one that lies in the bottom of the hierarchyand contains all other classes, are characterized bythe monotonicity of the Leibniz operator on their theories.Other famous classes are formed by the equivalential logics,the weakly algebraizable logics, the algebraizable logicsetc. By now, there is a generalization of the Leibniz operator in the context of CategoricalAbstract Algebraic Logic, that makes it possibleto apply a wide variety of techniques that werepreviously applicable in the sentential logicframework to logics formalized as \pi-institutions.The \pi-institution framework is significantly widerin scope than the framework of sentential logicsbecause it allows incorporating multiple signaturesand quantifiers in the language and it provides a mechanism forhandling logics that are not syntactically-based.」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. 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