13 results on '"KRIPKE semantics"'
Search Results
2. Dynamic Łukasiewicz logic and its application to immune system.
- Author
-
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
-
ODINTSOV, SERGEI P. and SPERANSKI, STANISLAV O.
- Subjects
- *
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
-
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
-
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
-
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
-
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
-
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
- View/download PDF
9. Bi-modal Gödel logic over [0,1]-valued Kripke frames.
- Author
-
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
- View/download PDF
10. A NEW SEMANTIC FRAMEWORK FOR MODAL LOGIC.
- Author
-
PUNČOCHÁŘ, VÍT
- Subjects
- *
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
-
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. Basic Core Fuzzy Logics and Algebraic Routley–Meyer-Style Semantics.
- Author
-
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
13. An Algebraic View of Super-Belnap Logics
- Author
-
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
Catalog
Discovery Service for Jio Institute Digital Library
For full access to our library's resources, please sign in.