In modal logic,topological semantics is an intuitive and natural special case of neighbourhood semantics.This paper stems from the observation that the satisfaction relation of topological semantics applies to subset ...In modal logic,topological semantics is an intuitive and natural special case of neighbourhood semantics.This paper stems from the observation that the satisfaction relation of topological semantics applies to subset spaces which are more general than topological spaces.The minimal modal logic which is strongly sound and complete with respect to the class of subset spaces is found.Soundness and completeness results of some famous modal logics(e.g.S4,S5 and Tr)with respect to various important classes of subset spaces(eg intersection structures and complete fields of sets)are also proved.In the meantime,some known results,e.g.the soundness and completeness of Tr with respect to the class of discrete topological spaces,are proved directly using some modifications of the method of canonical mode1,without a detour via neighbourhood semantics or relational semantics.展开更多
This article extends the foundational work of Wang and Wang on modal logic over lattices.Building upon their framework using polyadic modal logic with binary modalities<sup>and<inf>under standard Kripke se...This article extends the foundational work of Wang and Wang on modal logic over lattices.Building upon their framework using polyadic modal logic with binary modalities<sup>and<inf>under standard Kripke semantics to axiomatize lattice structures,we focus on the modal characterization of bounded lattices and their extensions relevant to logical systems.By introducing nullary modalities 1(maximum element)and 0(minimum element),we first establish a modal axiomatic system for bounded lattices.Subsequently,we provide pure formula characterizations of complementation and orthocomplementation relations in lattices,along with corresponding completeness results.As key applications,we present modal characterizations of fundamental logical algebraic structures:Boolean algebras,orthomodular lattices,and Heyting algebras.The last section develops novel axiomatization results for atomic lattices and atomless lattices.Throughout this work,all axiomatic systems are shown to be strongly complete via pureformula extensions,demonstrating how hybrid modal languages with nullary operators can uniformly capture boundary elements,complementation properties,and latticetheoretic operations central to both classical and nonclassical logics.展开更多
In the present paper,we give a systematic study of the discrete correspondence the-ory and topological correspondence theory of modal meet-implication logic and moda1 meet-semilattice logic,in the semantics provided i...In the present paper,we give a systematic study of the discrete correspondence the-ory and topological correspondence theory of modal meet-implication logic and moda1 meet-semilattice logic,in the semantics provided in[21].The special features of the present paper include the following three points:the first one is that the semantic structure used is based on a semilattice rather than an ordinary partial order,the second one is that the propositional vari-ables are interpreted as filters rather than upsets,and the nominals,which are the“first-order counterparts of propositional variables,are interpreted as principal filters rather than principal upsets;the third one is that in topological correspondence theory,the collection of admissi-ble valuations is not closed under taking disjunction,which makes the proof of the topological Ackermann 1emma different from existing settings.展开更多
The theory of （n） truth degrees of formulas is proposed in modal logic for the first time. A consistency theorem is obtained which says that the （n） truth degree of a modality-free formula equals the truth degree ...The theory of （n） truth degrees of formulas is proposed in modal logic for the first time. A consistency theorem is obtained which says that the （n） truth degree of a modality-free formula equals the truth degree of the formula in two-valued propositional logic. Variations of （n） truth degrees of formulas w.r.t. n in temporal logic is investigated. Moreover, the theory of （n） similarity degrees among modal formulas is proposed and the （n） modal logic metric space is derived therefrom which contains the classical logic metric space as a subspace. Finally, a kind of approximate reasoning theory is proposed in modal logic.展开更多
Global property is the necessary condition which must be satisfied by the provable formulas. It can help to find out some unprovable formula that does not satisfy some global property before proving it using formal au...Global property is the necessary condition which must be satisfied by the provable formulas. It can help to find out some unprovable formula that does not satisfy some global property before proving it using formal automated rea-soning systems, thus the efficiency of the whole system is improved. This paper presents some global properties of valid formulas in modal logic K. Such prop-erties are structure characters of formulas, so they are simple and easy to check.At the same time, some global properties of K unsatisfiable formula set are also given.展开更多
The modal lattice implication algebra(i.e.,M-lattice implication algebra) is introduced and its properties are investigated.The modal lattice-valued propositional logical system is introduced by considering the M-latt...The modal lattice implication algebra(i.e.,M-lattice implication algebra) is introduced and its properties are investigated.The modal lattice-valued propositional logical system is introduced by considering the M-lattice implication algebra as the valuation field,and the syntax and semantic of the logical system are discussed,respectively.展开更多
A semantic interpretation of a first order extension of Hennessy-Milner logic for value-passing processes, named HML(FO), is presented. The semantics is based on symbolic transition graphs with assignment. It is shown...A semantic interpretation of a first order extension of Hennessy-Milner logic for value-passing processes, named HML(FO), is presented. The semantics is based on symbolic transition graphs with assignment. It is shown that the satisfiability of the two-variable sub-logic HML(FO2) of HML(FO) is decidable, and the complexity discussed. Finally, a decision procedure for model checking the value-passing processes with respect to HML(FO2) is obtained.展开更多
Based on the intuitionistic first order predicate calculusH given by Thomason with the modal machinery of MIPC put forward by Prior this paper obtains the intuitionistic quantified modal logic system MIPC*, gives it a...Based on the intuitionistic first order predicate calculusH given by Thomason with the modal machinery of MIPC put forward by Prior this paper obtains the intuitionistic quantified modal logic system MIPC*, gives it a semantic interpretation and proves its strong (thus also weak) completeness theorem and soundness theorem with respect to that semantic. Since Zorn lemma plays a decisive role in our discussion, methodologically, it was even farther from the intuitionistic point of view than Thomason's result.展开更多
Modal logic characterization in a higher-order setting is usually not a trivial task because higher-order process-passing is quite different from first-order name-passing. We study the logical characterization of high...Modal logic characterization in a higher-order setting is usually not a trivial task because higher-order process-passing is quite different from first-order name-passing. We study the logical characterization of higherorder processes constrained by linearity. Linearity respects resource-sensitiveness and does not allow processes to duplicate themselves arbitrarily. We provide a modal logic that characterizes linear higher-order processes,particularly the bisimulation called local bisimulation over them. More importantly, the logic has modalities for higher-order actions downscaled to resembling first-order ones in Hennessy-Milner logic, based on a formulation exploiting the linearity of processes.展开更多
Despite half a century of fuzzy sets and fuzzy logic progress, as fuzzy sets address complex and uncertain information through the lens of human knowledge and subjectivity, more progress is needed in the semantics of ...Despite half a century of fuzzy sets and fuzzy logic progress, as fuzzy sets address complex and uncertain information through the lens of human knowledge and subjectivity, more progress is needed in the semantics of fuzzy sets and in exploring the multi-modal aspect of fuzzy logic due to the different cognitive, emotional and behavioral angles of assessing truth. We lay here the foundations of a postmodern fuzzy set and fuzzy logic theory addressing these issues by deconstructing fuzzy truth values and fuzzy set membership functions to re-capture the human knowledge and subjectivity structure in membership function evaluations. We formulate a fractal multi-modal logic of Kabbalah which integrates the cognitive, emotional and behavioral levels of humanistic systems into epistemic and modal, deontic and doxastic and dynamic multi-modal logic. This is done by creating a fractal multi-modal Kabbalah possible worlds semantic frame of Kripke model type. The Kabbalah possible worlds semantic frame integrates together both the multi-modal logic aspects and their Kripke possible worlds model. We will not focus here on modal operators and axiom sets. We constructively define a fractal multi-modal Kabbalistic L-fuzzy set as the central concept of the postmodern fuzzy set theory based on Kabbalah logic and semantics.展开更多
Business letter with friendly modality is an effective means to represent the best advantage of one's self and firm. The paper in vestigates modality in business letter according to Halliday's view on modality...Business letter with friendly modality is an effective means to represent the best advantage of one's self and firm. The paper in vestigates modality in business letter according to Halliday's view on modality. The modal devices mainly cover modal operator, modal ad junct, and modal cohesion. By means of binary approach modal logic is tentative divided into: perfective and imperfective; positive and neg ative; subjective and objective or explicit and implicit; high, median and low value. Modal devices or modal logic has formal and semantic level. Modality is regarded as a potential analyzing method to business text in this paper. The significance of the study is for understanding modal expression in business letter, meanwhile, this is a way to better understand modal logic, and to better use modal device in business let ter.展开更多
基金supported by the National Social Science Fund of China(No.20CZX048)。
文摘In modal logic,topological semantics is an intuitive and natural special case of neighbourhood semantics.This paper stems from the observation that the satisfaction relation of topological semantics applies to subset spaces which are more general than topological spaces.The minimal modal logic which is strongly sound and complete with respect to the class of subset spaces is found.Soundness and completeness results of some famous modal logics(e.g.S4,S5 and Tr)with respect to various important classes of subset spaces(eg intersection structures and complete fields of sets)are also proved.In the meantime,some known results,e.g.the soundness and completeness of Tr with respect to the class of discrete topological spaces,are proved directly using some modifications of the method of canonical mode1,without a detour via neighbourhood semantics or relational semantics.
基金supported by China Postdoctoral Science Foundation(2024M750225).
文摘This article extends the foundational work of Wang and Wang on modal logic over lattices.Building upon their framework using polyadic modal logic with binary modalities<sup>and<inf>under standard Kripke semantics to axiomatize lattice structures,we focus on the modal characterization of bounded lattices and their extensions relevant to logical systems.By introducing nullary modalities 1(maximum element)and 0(minimum element),we first establish a modal axiomatic system for bounded lattices.Subsequently,we provide pure formula characterizations of complementation and orthocomplementation relations in lattices,along with corresponding completeness results.As key applications,we present modal characterizations of fundamental logical algebraic structures:Boolean algebras,orthomodular lattices,and Heyting algebras.The last section develops novel axiomatization results for atomic lattices and atomless lattices.Throughout this work,all axiomatic systems are shown to be strongly complete via pureformula extensions,demonstrating how hybrid modal languages with nullary operators can uniformly capture boundary elements,complementation properties,and latticetheoretic operations central to both classical and nonclassical logics.
基金supported by the Chinese Ministry of Education of Humanities and Social Science Project(23YJC72040003)the Key Project of Chinese Ministry of Education(22JJD720021)supported by the Natural Science Foundation of Shandong Province,China(project number:ZR2023QF021)。
文摘In the present paper,we give a systematic study of the discrete correspondence the-ory and topological correspondence theory of modal meet-implication logic and moda1 meet-semilattice logic,in the semantics provided in[21].The special features of the present paper include the following three points:the first one is that the semantic structure used is based on a semilattice rather than an ordinary partial order,the second one is that the propositional vari-ables are interpreted as filters rather than upsets,and the nominals,which are the“first-order counterparts of propositional variables,are interpreted as principal filters rather than principal upsets;the third one is that in topological correspondence theory,the collection of admissi-ble valuations is not closed under taking disjunction,which makes the proof of the topological Ackermann 1emma different from existing settings.
基金Supported by the National Natural Science Foundation of China (Grant Nos. 10331010 and 10771129)the Foundation of 211 Constructionof Shaanxi Normal University
文摘The theory of （n） truth degrees of formulas is proposed in modal logic for the first time. A consistency theorem is obtained which says that the （n） truth degree of a modality-free formula equals the truth degree of the formula in two-valued propositional logic. Variations of （n） truth degrees of formulas w.r.t. n in temporal logic is investigated. Moreover, the theory of （n） similarity degrees among modal formulas is proposed and the （n） modal logic metric space is derived therefrom which contains the classical logic metric space as a subspace. Finally, a kind of approximate reasoning theory is proposed in modal logic.
文摘Global property is the necessary condition which must be satisfied by the provable formulas. It can help to find out some unprovable formula that does not satisfy some global property before proving it using formal automated rea-soning systems, thus the efficiency of the whole system is improved. This paper presents some global properties of valid formulas in modal logic K. Such prop-erties are structure characters of formulas, so they are simple and easy to check.At the same time, some global properties of K unsatisfiable formula set are also given.
基金the National Natural Science Foundation of China(No.61175055)the Scientific Research Fund of Sichuan Provincial Education Department(11ZB023)the Sichuan Key Technology Research and Development Program(No.2011FZ0051)
文摘The modal lattice implication algebra(i.e.,M-lattice implication algebra) is introduced and its properties are investigated.The modal lattice-valued propositional logical system is introduced by considering the M-lattice implication algebra as the valuation field,and the syntax and semantic of the logical system are discussed,respectively.
基金This work was partially supported by the National Natural Science Foundationof China (Grant No. 69833020) the National High Technology Development Program of China (Grant No. 2002AA144050)the National Grand Fundamental Research 973 Program of China
文摘A semantic interpretation of a first order extension of Hennessy-Milner logic for value-passing processes, named HML(FO), is presented. The semantics is based on symbolic transition graphs with assignment. It is shown that the satisfiability of the two-variable sub-logic HML(FO2) of HML(FO) is decidable, and the complexity discussed. Finally, a decision procedure for model checking the value-passing processes with respect to HML(FO2) is obtained.
文摘Based on the intuitionistic first order predicate calculusH given by Thomason with the modal machinery of MIPC put forward by Prior this paper obtains the intuitionistic quantified modal logic system MIPC*, gives it a semantic interpretation and proves its strong (thus also weak) completeness theorem and soundness theorem with respect to that semantic. Since Zorn lemma plays a decisive role in our discussion, methodologically, it was even farther from the intuitionistic point of view than Thomason's result.
基金the National Natural Science Foundation of China(Nos.61202023,61261130589 and61173048)the PACE Project(No.12IS02001)the Specialized Research Fund for the Doctoral Program of Higher Edueation of China(No.20120073120031)
文摘Modal logic characterization in a higher-order setting is usually not a trivial task because higher-order process-passing is quite different from first-order name-passing. We study the logical characterization of higherorder processes constrained by linearity. Linearity respects resource-sensitiveness and does not allow processes to duplicate themselves arbitrarily. We provide a modal logic that characterizes linear higher-order processes,particularly the bisimulation called local bisimulation over them. More importantly, the logic has modalities for higher-order actions downscaled to resembling first-order ones in Hennessy-Milner logic, based on a formulation exploiting the linearity of processes.
文摘Despite half a century of fuzzy sets and fuzzy logic progress, as fuzzy sets address complex and uncertain information through the lens of human knowledge and subjectivity, more progress is needed in the semantics of fuzzy sets and in exploring the multi-modal aspect of fuzzy logic due to the different cognitive, emotional and behavioral angles of assessing truth. We lay here the foundations of a postmodern fuzzy set and fuzzy logic theory addressing these issues by deconstructing fuzzy truth values and fuzzy set membership functions to re-capture the human knowledge and subjectivity structure in membership function evaluations. We formulate a fractal multi-modal logic of Kabbalah which integrates the cognitive, emotional and behavioral levels of humanistic systems into epistemic and modal, deontic and doxastic and dynamic multi-modal logic. This is done by creating a fractal multi-modal Kabbalah possible worlds semantic frame of Kripke model type. The Kabbalah possible worlds semantic frame integrates together both the multi-modal logic aspects and their Kripke possible worlds model. We will not focus here on modal operators and axiom sets. We constructively define a fractal multi-modal Kabbalistic L-fuzzy set as the central concept of the postmodern fuzzy set theory based on Kabbalah logic and semantics.
文摘Business letter with friendly modality is an effective means to represent the best advantage of one's self and firm. The paper in vestigates modality in business letter according to Halliday's view on modality. The modal devices mainly cover modal operator, modal ad junct, and modal cohesion. By means of binary approach modal logic is tentative divided into: perfective and imperfective; positive and neg ative; subjective and objective or explicit and implicit; high, median and low value. Modal devices or modal logic has formal and semantic level. Modality is regarded as a potential analyzing method to business text in this paper. The significance of the study is for understanding modal expression in business letter, meanwhile, this is a way to better understand modal logic, and to better use modal device in business let ter.
