26 results on '"KRIPKE semantics"'
Search Results
2. Dynamic Łukasiewicz logic and its application to immune system.
- Author
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Di Nola, Antonio, Grigolia, Revaz, Mitskevich, Nunu, and Vitale, Gaetano
- Subjects
- *
KRIPKE semantics , *IMMUNE system , *ALGEBRAIC logic , *LOGIC , *ALGEBRA - Abstract
It is introduced an immune dynamic n-valued Łukasiewicz logic I D Ł n on the base of n-valued Łukasiewicz logic Ł n and corresponding to it immune dynamic M V n -algebra ( I D L n -algebra), 1 < n < ω , which are algebraic counterparts of the logic, that in turn represent two-sorted algebras (M , R , ◊) that combine the varieties of M V n -algebras M = (M , ⊕ , ⊙ , ∼ , 0 , 1) and regular algebras R = (R , ∪ , ; , ∗) into a single finitely axiomatized variety resembling R-module with "scalar" multiplication ◊ . Kripke semantics is developed for immune dynamic Łukasiewicz logic I D Ł n with application in immune system. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
3. BELNAP–DUNN MODAL LOGICS: TRUTH CONSTANTS VS. TRUTH VALUES.
- Author
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ODINTSOV, SERGEI P. and SPERANSKI, STANISLAV O.
- Subjects
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MODAL logic , *KRIPKE semantics , *TRUTH , *ALGEBRAIC logic , *MANY-valued logic - Abstract
We shall be concerned with the modal logic BK—which is based on the Belnap–Dunn four-valued matrix, and can be viewed as being obtained from the least normal modal logic K by adding 'strong negation'. Though all four values 'truth', 'falsity', 'neither' and 'both' are employed in its Kripke semantics, only the first two are expressible as terms. We show that expanding the original language of BK to include constants for 'neither' or/and 'both' leads to quite unexpected results. To be more precise, adding one of these constants has the effect of eliminating the respective value at the level of BK-extensions. In particular, if one adds both of these, then the corresponding lattice of extensions turns out to be isomorphic to that of ordinary normal modal logics. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
4. Dynamic Łukasiewicz Logic and Dynamic MV-algebras.
- Author
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Di Nola, Antonio, Grigolia, Revaz, and Vitale, Gaetano
- Subjects
- *
KRIPKE semantics , *MATHEMATICAL logic , *ALGEBRAIC logic , *LOGIC , *ALGEBRA - Abstract
Following K. Segerberg [22] , D. Kozen [15] and V. Pratt [19] , who have been introduced dynamic propositional logic and dynamic algebras, dynamic propositional Łukasiewicz logic DP Ł (dynamic n -valued propositional Łukasiewicz logic D P Ł n) and dynamic MV -algebras (dynamic M V n -algebras) are introduced and theories of the logic DP Ł (D P Ł n) and dynamic MV -algebras (M V n -algebras) are developed. Dynamic MV -algebras (dynamic M V n -algebras) are algebraic counterparts of the logic DP Ł (D P Ł n), that in turn represent two-sorted algebras that combine the varieties of MV -algebras (M V n -algebras) (M , ⊕ , ⊙ , ∼ , 0 , 1) and regular algebras (R , ∪ , ; , ⁎) into a single finitely axiomatized variety (M , R , ◇) resembling R -module with "scalar" multiplication ◇. Kripke semantics is developed for dynamic propositional Łukasiewicz logic (dynamic n -valued propositional Łukasiewicz logic D P Ł n). [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
5. Neighborhood semantics for modal many-valued logics.
- Author
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Cintula, Petr and Noguera, Carles
- Subjects
- *
MANY-valued logic , *KRIPKE semantics , *UNARY algebras , *ALGEBRAIC logic , *OPERATOR theory , *FUZZY logic - Abstract
The majority of works on modal many-valued logics consider Kripke-style possible worlds frames as the principal semantics despite their well-known axiomatizability issues when considering non-Boolean accessibility relations. The present work explores a more general semantical picture, namely a many-valued version of the classical neighborhood semantics. We present it in two levels of generality. First, we work with modal languages containing only the two usual unary modalities, define neighborhood frames over algebras of the logic FL ew with operators, and show their relation with the usual Kripke semantics (this is actually the highest level of generality where one can give a straightforward definition of the Kripke-style semantics). Second, we define generalized neighborhood frames for arbitrary modal languages over a given class of algebras for an arbitrary protoalgebraic logic and, assuming certain additional conditions, axiomatize the logic of all such frames (which generalizes the completeness theorem of the classical modal logic E with respect to classical neighborhood frames). [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
6. DISTRIBUTED RELATION LOGIC.
- Author
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Allwein, Gerard, Harrison, William L., and Reynolds, Thomas
- Subjects
BOOLEAN algebra ,RELATION algebras ,RELEVANCE logic ,KRIPKE semantics ,ALGEBRAIC logic - Abstract
We extend the relational algebra of Chin and Tarski so that it is multisorted or, as we prefer, typed. Each type supports a local Boolean algebra outfitted with a converse operator. From Lyndon, we know that relation algebras cannot be represented as proper relation algebras where a proper relation algebra has binary relations as elements and the algebra is singly-typed. Here, the intensional conjunction, which was to represent relational composition in Chin and Tarski, spans three different local alge- bras, thus the term distributed in the title. Since we do not rely on proper relation algebras, we are free to re-express the algebras as typed. In doing so, we allow many different intensional conjunction operators. We construct a typed logic over these algebras, also known as heterogeneous algebras of Birkhoff and Lipson. The logic can be seen as a form of relevance logic with a classical negation connective where the Routley-Meyer star operator is reified as a converse connective in the logic. Relevance logic itself is not typed but our work shows how it can be made so. Some of the properties of classical relevance logic are weakened from Routley-Meyer's version which is too strong for a logic over relation algebras. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
7. BELNAP–DUNN MODAL LOGICS: TRUTH CONSTANTS VS. TRUTH VALUES
- Author
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Stanislav O. Speranski and Sergei P. Odintsov
- Subjects
Philosophy ,Matrix (mathematics) ,Pure mathematics ,Mathematics (miscellaneous) ,Modal ,Negation ,Logic ,Normal modal logic ,Truth value ,Modal logic ,Kripke semantics ,Algebraic logic ,Mathematics - Abstract
We shall be concerned with the modal logic BK—which is based on the Belnap–Dunn four-valued matrix, and can be viewed as being obtained from the least normal modal logic K by adding ‘strong negation’. Though all four values ‘truth’, ‘falsity’, ‘neither’ and ‘both’ are employed in its Kripke semantics, only the first two are expressible as terms. We show that expanding the original language of BK to include constants for ‘neither’ or/and ‘both’ leads to quite unexpected results. To be more precise, adding one of these constants has the effect of eliminating the respective value at the level of BK-extensions. In particular, if one adds both of these, then the corresponding lattice of extensions turns out to be isomorphic to that of ordinary normal modal logics.
- Published
- 2019
8. Non-normal modalities in variants of linear logic.
- Author
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Porello, D. and Troquard, N.
- Subjects
KRIPKE semantics ,MATHEMATICAL logic ,ALGEBRAIC logic ,RELATION algebras ,NUMERICAL analysis - Abstract
This article presents modal versions of resource-conscious logics. We concentrate on extensions of variants of linear logic with one minimal non-normal modality. In earlier work, where we investigated agency in multi-agent systems, we have shown that the results scale up to logics with multiple non-minimal modalities. Here, we start with the language of propositional intuitionistic linear logic without the additive disjunction, to which we add a modality. We provide an interpretation of this language on a class of Kripke resource models extended with a neighbourhood function: modal Kripke resource models. We propose a Hilbert-style axiomatisation and a Gentzen-style sequent calculus. We show that the proof theories are sound and complete with respect to the class of modal Kripke resource models. We show that the sequent calculus admits cut elimination and that proof-search is in PSPACE. We then show how to extend the results when non-commutative connectives are added to the language. Finally, we put the logical framework to use by instantiating it as logics of agency. In particular, we propose a logic to reason about the resource-sensitive use of artefacts and illustrate it with a variety of examples. [ABSTRACT FROM PUBLISHER]
- Published
- 2015
- Full Text
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9. Bi-modal Gödel logic over [0,1]-valued Kripke frames.
- Author
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Caicedo, Xavier and Rodríguez, Ricardo Oscar
- Subjects
KRIPKE semantics ,MATHEMATICAL logic ,SEMANTICS ,COMPARATIVE linguistics ,ALGEBRAIC logic - Abstract
We consider the Gödel bi-modal logic determined by fuzzy Kripke models where both the propositions and the accessibility relation are infinitely valued over the standard Gödel algebra [0,1], and prove strong completeness of the Fischer Servi intuitionistic modal logic IK plus the prelinearity axiom with respect to this semantics. We axiomatize also the bi-modal analogues of classical T, S4 and S5, obtained by restricting to models over frames satisfying the [0,1]-valued versions of the structural properties which characterize these logics. As an application of the completeness theorems we obtain a representation theorem for bi-modal Gödel algebras. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
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10. A NEW SEMANTIC FRAMEWORK FOR MODAL LOGIC.
- Author
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PUNČOCHÁŘ, VÍT
- Subjects
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SEMANTICS , *BOOLEAN algebra , *ALGEBRAIC logic , *ALGEBRAIC geometry , *GEOMETRY - Abstract
The article presents a new semantic framework for modal propositional language. The basic structures of the semantics are Boolean algebras with operators. However, the semantics is not algebraic but rather relational; in it, Boolean algebras with operators play a similar role as Kripke models in standard relational semantics, and the semantics is based on a relation between the elements of Boolean algebras enriched with operators and formulas from modal language. Some basic connections between the new semantic framework and standard algebraic and relational semantics are studied. [ABSTRACT FROM AUTHOR]
- Published
- 2014
11. An AC-complete model checking problem for intuitionistic logic.
- Author
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Mundhenk, Martin and Wei, Felix
- Subjects
INTUITIONISTIC mathematics ,HEYTING algebras ,ALGEBRAIC logic ,COMPUTATIONAL complexity ,KRIPKE semantics - Abstract
We show that the model checking problem for intuitionistic propositional logic with one variable is complete for logspace-uniform AC. For superintuitionistic logics with one variable, we obtain NC-completeness for the model checking problem. [ABSTRACT FROM AUTHOR]
- Published
- 2014
- Full Text
- View/download PDF
12. ON FLATTENING ELIMINATION RULES.
- Author
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OLKHOVIKOV, GRIGORY K. and SCHROEDER-HEISTER, PETER
- Subjects
- *
KRIPKE semantics , *COMPARATIVE linguistics , *MATHEMATICAL logic , *LANGUAGE & languages , *INFORMATION theory , *ALGEBRAIC logic - Abstract
In proof-theoretic semantics of intuitionistic logic it is well known that elimination rules can be generated from introduction rules in a uniform way. If introduction rules discharge assumptions, the corresponding elimination rule is a rule of higher level, which allows one to discharge rules occurring as assumptions. In some cases, these uniformly generated elimination rules can be equivalently replaced with elimination rules that only discharge formulas or do not discharge any assumption at all—they can be flattened in a terminology proposed by Read. We show by an example from propositional logic that not all introduction rules have flat elimination rules. We translate the general form of flat elimination rules into a formula of second-order propositional logic and demonstrate that our example is not equivalent to any such formula. The proof uses elementary techniques from propositional logic and Kripke semantics. [ABSTRACT FROM PUBLISHER]
- Published
- 2014
- Full Text
- View/download PDF
13. An algebraic look at filtrations in modal logic.
- Author
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Conradie, Willem, Morton, Wilmari, and van Alten, Clint J.
- Subjects
ALGEBRAIC logic ,MODAL logic ,KRIPKE semantics ,MODULAR arithmetic ,FINITE model theory ,DUALITY theory (Mathematics) - Abstract
Filtration constructions are among the oldest and best known methods for obtaining finite model properties for modal logics, and appear in the literature in both model-theoretic and algebraic versions. In this article we investigate definitions of algebraic filtrations by means of different types of binary relations on modal algebras, and the relationships between these. We generalize the notion of a model-theoretic filtration somewhat while simultaneously lifting it to the level of frames. We proceed to link algebraic filtrations with their model- or frame-theoretic counterparts by showing how our filtration notions interface neatly with the well-known duality theory of modal algebras and Kripke frames. We illustrate, by means of some examples, how this theory enables one to easily translate between algebraic and model-theoretic versions of some well-known filtrations. We obtain some order theoretic insights regarding the (usually model-theoretically specified) smallest and largest filtrations by considering their algebraic versions, thus demonstrating the utility of having ready access to both versions. [ABSTRACT FROM AUTHOR]
- Published
- 2013
- Full Text
- View/download PDF
14. Intuitionistic Trilattice Logics.
- Author
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WANSING, HEINRICH and KAMIDE, NORIHIRO
- Subjects
ALGEBRAIC logic ,SEMANTICS ,CALCULUS ,MATHEMATICS ,COMPLETENESS theorem ,MATHEMATICAL functions - Abstract
We take up a suggestion by Odintsov (2009, Studia Logica, 91, 407–428) and define intuitionistic variants of certain logics arising from the trilattice SIXTEEN3 introduced in Shramko and Wansing (2005, Journal of Philosophical Logic, 34, 121–153 and 2006, Journal of Logic, Language and Information, 15, 403–424). In a first step, a logic I16 is presented as a Gentzen-type sequent calculus for an intuitionistic version of Odintsov’s Hilbert-style axiom system LT (Kamide and Wansing, 2009, Review of Symbolic Logic, 2, 374–395; Odintsov, 2009, Studia Logica, 91, 407–428). The cut-elimination theorem for I16 is proved using an embedding of I16 into Gentzen’s LJ. The completeness theorem with respect to a Kripke-style semantics is also proved for I16. The framework of I16 is regarded as plausible and natural for the following reasons: (i) the properties of constructible falsity and paraconsistency with respect to some negation connectives hold for I16, and (ii) sequent calculi for Belnap and Dunn’s four-valued logic (Anderson et al., 1992, Entailment: The Logic of Relevance and Necessity; Belnap, 1977, A useful four-valued logic, In Modern uses of Multiple Valued Logic, pp. 5–37; Dunn, 1976, Philosophical Studies, 29, 149–168) and for Nelson’s constructive four-valued logic (Almukdad and Nelson, 1984, Journal of Symbolic Logic, 49, 231–233) are included as natural subsystems of I16. In a second step, a logic IT16 is introduced as a tableau calculus. The tableau system IT16 is an intuitionistic counterpart of Odintsov’s axiom system for truth entailment ⊨t in SIXTEEN3 and of the sequent calculus for ⊨t presented in Wansing (2010, Journal of Philosophical Logic, to appear). The tableau calculus is also shown to be sound and complete with respect to a Kripke-style semantics. A tableau calculus for falsity entailment can be obtained by suitably modifying the notion of provability. [ABSTRACT FROM PUBLISHER]
- Published
- 2010
- Full Text
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15. Basic Core Fuzzy Logics and Algebraic Routley–Meyer-Style Semantics.
- Author
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Yang, Eunsuk
- Subjects
- *
ALGEBRAIC logic , *SEMANTICS , *KRIPKE semantics , *LOGIC - Abstract
Recently, algebraic Routley–Meyer-style semantics was introduced for basic substructural logics. This paper extends it to fuzzy logics. First, we recall the basic substructural core fuzzy logic MIAL (Mianorm logic) and its axiomatic extensions, together with their algebraic semantics. Next, we introduce two kinds of ternary relational semantics, called here linear Urquhart-style and Fine-style Routley–Meyer semantics, for them as algebraic Routley–Meyer-style semantics. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
16. An Algebraic View of Super-Belnap Logics
- Author
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Hugo Albuquerque, Umberto Rivieccio, and Adam Přenosil
- Subjects
Logic ,Substructural logic ,010102 general mathematics ,Classical logic ,06 humanities and the arts ,Non-classical logic ,0603 philosophy, ethics and religion ,01 natural sciences ,Algebraic logic ,Algebra ,TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES ,History and Philosophy of Science ,Computer Science::Logic in Computer Science ,Monoidal t-norm logic ,060302 philosophy ,Kripke semantics ,0101 mathematics ,T-norm fuzzy logics ,Łukasiewicz logic ,Mathematics - Abstract
The Belnap–Dunn logic (also known as First Degree Entailment, or FDE) is a well-known and well-studied four-valued logic, but until recently little has been known about its extensions, i.e. stronger logics in the same language, called super-Belnap logics here. We give an overview of several results on these logics which have been proved in recent works by Přenosil and Rivieccio. We present Hilbert-style axiomatizations, describe reduced matrix models, and give a description of the lattice of super-Belnap logics and its connections with graph theory. We adopt the point of view of Abstract Algebraic Logic, exploring applications of the general theory of algebraization of logics to the super-Belnap family. In this respect we establish a number of new results, including a description of the algebraic counterparts, Leibniz filters, and strong versions of super-Belnap logics, as well as the classification of these logics within the Leibniz and Frege hierarchies.
- Published
- 2017
17. Combinators and structurally free logic.
- Author
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Dunn, JM and Meyer, RK
- Subjects
COMBINATORIAL number theory ,LOGIC ,KRIPKE semantics ,ALGEBRAIC logic ,CALCULUS - Abstract
A 'Kripke-style' semantics is given for combinatory logic using frames with a ternary accessibility relation, much as in the Tourley-Meyer semantics for relevance logic. We prove by algebraic means a completeness theorem for combinatory logic, by proving a representation theorem for 'combinatory posets.' A philosophical interpretation is given of the models, showing that an element of a combinatory poset can be understood simultaneously as a set of states and as a set of (untyped) actions on states. This double interpretation allows for one such element to be applied to another (including itself). Application turns out to be modeled the same way as 'fusion' in relevance logic.We also introduce 'dual combinators' that apply from the right. We then explore relationships to some well-known substructural logics, showing that each can be embedded into the structurally free. non-associative Lambek calculus, with the embedding taking a theorem ω to a statement of the form Γ ⊨ ω, where Γ is some fusion of the combinators (sometimes dual combinators as well) needed to justify the structural assumptions of the given substructural logic. This builds on earlier ideas from Belnap and Meyer about a Gentzen system wherein structural rules are replaced with rules for introducing combinators. We develop such a system and prove a cut theorem.Keywords: combinatory logic, substructural logic, Gentzen, gaggle, relevance logic, semantics, algebraic logic [ABSTRACT FROM PUBLISHER]
- Published
- 1997
- Full Text
- View/download PDF
18. Algebraic Kripke-Style Semantics for Relevance Logics
- Author
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Eunsuk Yang
- Subjects
Theoretical computer science ,Classical logic ,Algebraic logic ,Operational semantics ,Action semantics ,Algebra ,Philosophy ,Algebraic sentence ,TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES ,Algebraic semantics ,Well-founded semantics ,TheoryofComputation_LOGICSANDMEANINGSOFPROGRAMS ,Computer Science::Logic in Computer Science ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,Computer Science::Programming Languages ,Kripke semantics ,Mathematics - Abstract
This paper deals with one kind of Kripke-style semantics, which we shall call algebraic Kripke-style semantics, for relevance logics. We first recall the logic R of relevant implication and some closely related systems, their corresponding algebraic structures, and algebraic completeness results. We provide simpler algebraic completeness proofs. We then introduce various types of algebraic Kripke-style semantics for these systems and connect them with algebraic semantics.
- Published
- 2013
19. Possible Worlds in Use
- Author
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Andrzej Indrzejczak
- Subjects
Possible world ,Theoretical computer science ,History and Philosophy of Science ,Logic ,Well-founded semantics ,Impossible world ,Semantics of logic ,Modal logic ,Kripke semantics ,Computational linguistics ,Algebraic logic ,Mathematics ,Epistemology - Abstract
The paper is a brief survey of the most important semantic constructions founded on the concept of possible world. It is impossible to capture in one short paper the whole variety of the problems connected with manifold applications of possible worlds. Hence, after a brief explanation of some philosophical matters I take a look at possible worlds from rather technical standpoint of logic and focus on the applications in formal semantics. In particular, I would like to focus on the fruitful marriage of possible world semantics and algebra and its evolution leading to very general construction of Wojcicki called referential semantics and some of its refinements. The presentation is informal and sketchy; the main purpose is to put in one place a short, and readable I hope, description of the most important constructions and to point out the main sources of these solutions.
- Published
- 2011
20. Fuzzy Topology and Łukasiewicz Logics from the Viewpoint of Duality Theory
- Author
-
Yoshihiro Maruyama
- Subjects
compactness theorem ,Mathematics::General Mathematics ,Logic ,Duality (mathematics) ,fuzzy topology ,many-valued logic ,MV-algebra ,Stone duality ,Fuzzy logic ,Jónsson-Tarski duality ,Algebra ,algebraic logic ,Interior algebra ,History and Philosophy of Science ,Monoidal t-norm logic ,Computer Science::Logic in Computer Science ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,T-norm fuzzy logics ,Łukasiewicz logic ,Mathematics ,modal logic ,Kripke semantics - Abstract
This paper explores relationships between many-valued logic and fuzzy topology from the viewpoint of duality theory. We first show a fuzzy topological duality for the algebras of Łukasiewicz n-valued logic with truth constants, which generalizes Stone duality for Boolean algebras to the n-valued case via fuzzy topology. Then, based on this duality, we show a fuzzy topological duality for the algebras of modal Łukasiewicz n-valued logic with truth constants, which generalizes Jonsson-Tarski duality for modal algebras to the n-valued case via fuzzy topology. We emphasize that fuzzy topological spaces naturally arise as spectrums of algebras of many-valued logics.
- Published
- 2010
21. Fuzzy Topology and Łukasiewicz Logics from the Viewpoint of Duality Theory
- Author
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Maruyama, Yoshihiro
- Published
- 2010
- Full Text
- View/download PDF
22. Definability and Interpolation in Non-Classical Logics
- Author
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Larisa Maksimova
- Subjects
Logic ,Classical logic ,Non-classical logic ,Algebraic logic ,Algebra ,TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES ,History and Philosophy of Science ,Algebraic semantics ,Computer Science::Logic in Computer Science ,Monoidal t-norm logic ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,Kripke semantics ,T-norm fuzzy logics ,Łukasiewicz logic ,Mathematics - Abstract
Algebraic approach to study of classical and non-classical logical calculi was developed and systematically presented by Helena Rasiowa in [48], [47]. It is very fruitful in investigation of non-classical logics because it makes possible to study large families of logics in an uniform way. In such research one can replace logics with suitable classes of algebras and apply powerful machinery of universal algebra. In this paper we present an overview of results on interpolation and definability in modal and positive logics,and also in extensions of Johansson's minimal logic. All these logics are strongly complete under algebraic semantics. It allows to combine syntactic methods with studying varieties of algebras and to flnd algebraic equivalents for interpolation and related properties. Moreover, we give exhaustive solution to interpolation and some related problems for many families of propositional logics and calculi.
- Published
- 2006
23. Leibniz filters and the strong version of a protoalgebraic logic
- Author
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Ramon Jansana and Josep Font
- Subjects
Pure mathematics ,Logic ,Classical logic ,Non-classical logic ,Algebraic logic ,Algebra ,Philosophy ,Computer Science::Logic in Computer Science ,Monoidal t-norm logic ,Kripke semantics ,T-norm fuzzy logics ,Principle of bivalence ,Łukasiewicz logic ,Mathematics - Abstract
A filter of a sentential logic ? is Leibniz when it is the smallest one among all the ?-filters on the same algebra having the same Leibniz congruence. This paper studies these filters and the sentential logic ?+ defined by the class of all ?-matrices whose filter is Leibniz, which is called the strong version of ?, in the context of protoalgebraic logics with theorems. Topics studied include an enhanced Correspondence Theorem, characterizations of the weak algebraizability of ?+ and of the explicit definability of Leibniz filters, and several theorems of transfer of metalogical properties from ? to ?+. For finitely equivalential logics stronger results are obtained. Besides the general theory, the paper examines the examples of modal logics, quantum logics and Łukasiewicz's finitely-valued logics. One finds that in some cases the existence of a weak and a strong version of a logic corresponds to well-known situations in the literature, such as the local and the global consequences for normal modal logics; while in others these constructions give an independent interest to the study of other lesser-known logics, such as the lattice-based many-valued logics.
- Published
- 2001
24. An algebraic approach to intuitionistic modal logics in connection with intermediate predicate logics
- Author
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Nobu-Yuki Suzuki
- Subjects
Discrete mathematics ,Logic ,Normal modal logic ,Predicate (mathematical logic) ,Algebraic logic ,Algebra ,Mathematics::Logic ,TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES ,History and Philosophy of Science ,Algebraic semantics ,TheoryofComputation_LOGICSANDMEANINGSOFPROGRAMS ,Computer Science::Logic in Computer Science ,Monoidal t-norm logic ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,Accessibility relation ,Computer Science::Programming Languages ,Kripke semantics ,T-norm fuzzy logics ,Hardware_LOGICDESIGN ,Mathematics - Abstract
Modal counterparts of intermediate predicate logics will be studied by means of algebraic devise. Our main tool will be a construction of algebraic semantics for modal logics from algebraic frames for predicate logics. Uncountably many examples of modal counterparts of intermediate predicate logics will be given.
- Published
- 1989
25. Algebraic semantics and program logics: Algorithmic logic for program trees
- Author
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Patrice Enjalbert
- Subjects
Discrete mathematics ,Algebraic sentence ,Theoretical computer science ,Algebraic semantics ,Classical logic ,Computer Science::Programming Languages ,Kripke semantics ,Algorithmic logic ,Higher-order logic ,Algebraic logic ,First-order logic ,Mathematics - Abstract
The aim of this paper is to ground Algorithmic (or Dynamic) Logic on Algebraic semantics in the french acceptation of the term, i.e. theory in which the meaning of a program is a tree resulting from an infinite formal unfolding. We present an algorithmic system in which programs are program-trees and also an example of how it can be applied in order to design systems for programs. Another feature is the use of techniques of Lω1ω (the notion of Consistency Property) for proving completeness and Model Existence theorems.
- Published
- 1983
26. A unifying theorem for algebraic semantics and dynamic logics
- Author
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Hajnal Andréka, István Németi, Irène Guessarian, Laboratoire d'informatique Algorithmique : Fondements et Applications (LIAFA), Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), and Guessarian, Irene
- Subjects
Correctness ,Computer science ,Semantics (computer science) ,02 engineering and technology ,0102 computer and information sciences ,01 natural sciences ,Theoretical Computer Science ,Monoidal t-norm logic ,Computer Science::Logic in Computer Science ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,0202 electrical engineering, electronic engineering, information engineering ,0101 mathematics ,Algebra over a field ,Principle of bivalence ,Mathematics ,Discrete mathematics ,Mathematical logic ,Classical logic ,010102 general mathematics ,Ultraproduct ,16. Peace & justice ,Algebraic logic ,Computer Science Applications ,Algebra ,TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES ,Algebraic semantics ,Computational Theory and Mathematics ,010201 computation theory & mathematics ,Computer Science::Programming Languages ,020201 artificial intelligence & image processing ,Kripke semantics ,T-norm fuzzy logics ,Partially ordered set ,Information Systems - Abstract
A unified single proof is given which implies theorems in such diverse fields as continuous algebras of algebraic semantics, dynamic algebras of logics of programs, and program verification methods for total correctness. The proof concerns ultraproducts and diagonalization.
- Published
- 1987
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