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Journal articles on the topic 'Modal logic'

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Published: 4 June 2021

Last updated: 6 July 2024

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1

Schumann, Andrew. "Nāgārjunian-Yogācārian Modal Logic versus Aristotelian Modal Logic." Journal of Indian Philosophy 49, no. 3 (May 26, 2021): 467–98. http://dx.doi.org/10.1007/s10781-021-09470-5.

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AbstractThere are two different modal logics: the logic T assuming contingency and the logic K = assuming logical determinism. In the paper, I show that the Aristotelian treatise On Interpretation (Περί ερμηνείας, De Interpretatione) has introduced some modal-logical relationships which correspond to T. In this logic, it is supposed that there are contingent events. The Nāgārjunian treatise Īśvara-kartṛtva-nirākṛtiḥ-viṣṇoḥ-ekakartṛtva-nirākaraṇa has introduced some modal-logical relationships which correspond to K =. In this logic, it is supposed that there is a logical determinism: each event happens necessarily (siddha) or it does not happen necessarily (asiddha). The Nāgārjunian approach was inherited by the Yogācārins who developed, first, the doctrine of causality of all real entities (arthakriyātva) and, second, the doctrine of momentariness of all real entities (kṣaṇikavāda). Both doctrines were a philosophical ground of the Yogācārins for the logical determinism. Hence, Aristotle implicitly used the logic T in his modal reasoning. The Madhyamaka and Yogācāra schools implicitly used the logic K = in their modal reasoning.

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2

Rybakov, V. V. "Hereditarily structurally complete modal logics." Journal of Symbolic Logic 60, no. 1 (March 1995): 266–88. http://dx.doi.org/10.2307/2275521.

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AbstractWe consider structural completeness in modal logics. The main result is the necessary and sufficient condition for modal logics over K4 to be hereditarily structurally complete: a modal logic λ is hereditarily structurally complete iff λ is not included in any logic from the list of twenty special tabular logics. Hence there are exactly twenty maximal structurally incomplete modal logics above K4 and they are all tabular.

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3

Kooi, Barteld, and Allard Tamminga. "Three-valued Logics in Modal Logic." Studia Logica 101, no. 5 (August 21, 2012): 1061–72. http://dx.doi.org/10.1007/s11225-012-9420-0.

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4

BEZHANISHVILI, GURAM, NICK BEZHANISHVILI, and JULIA ILIN. "STABLE MODAL LOGICS." Review of Symbolic Logic 11, no. 3 (September 2018): 436–69. http://dx.doi.org/10.1017/s1755020317000375.

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AbstractStable logics are modal logics characterized by a class of frames closed under relation preserving images. These logics admit all filtrations. Since many basic modal systems such as K4 and S4 are not stable, we introduce the more general concept of an M-stable logic, where M is an arbitrary normal modal logic that admits some filtration. Of course, M can be chosen to be K4 or S4. We give several characterizations of M-stable logics. We prove that there are continuum many S4-stable logics and continuum many K4-stable logics between K4 and S4. We axiomatize K4-stable and S4-stable logics by means of stable formulas and discuss the connection between S4-stable logics and stable superintuitionistic logics. We conclude the article with many examples (and nonexamples) of stable, K4-stable, and S4-stable logics and provide their axiomatization in terms of stable rules and formulas.

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5

Demey, Lorenz. "Logic-Sensitivity of Aristotelian Diagrams in Non-Normal Modal Logics." Axioms 10, no. 3 (June 22, 2021): 128. http://dx.doi.org/10.3390/axioms10030128.

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Aristotelian diagrams, such as the square of opposition, are well-known in the context of normal modal logics (i.e., systems of modal logic which can be given a relational semantics in terms of Kripke models). This paper studies Aristotelian diagrams for non-normal systems of modal logic (based on neighborhood semantics, a topologically inspired generalization of relational semantics). In particular, we investigate the phenomenon of logic-sensitivity of Aristotelian diagrams. We distinguish between four different types of logic-sensitivity, viz. with respect to (i) Aristotelian families, (ii) logical equivalence of formulas, (iii) contingency of formulas, and (iv) Boolean subfamilies of a given Aristotelian family. We provide concrete examples of Aristotelian diagrams that illustrate these four types of logic-sensitivity in the realm of normal modal logic. Next, we discuss more subtle examples of Aristotelian diagrams, which are not sensitive with respect to normal modal logics, but which nevertheless turn out to be highly logic-sensitive once we turn to non-normal systems of modal logic.

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6

Venema, Yde, Alexander Chagrov, and Michael Zakharyaschev. "Modal Logic." Philosophical Review 109, no. 2 (April 2000): 286. http://dx.doi.org/10.2307/2693587.

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7

Kracht, Marcus, Patrick Blackburn, Maarten de Rijke, and Yde Venema. "Modal Logic." Bulletin of Symbolic Logic 8, no. 2 (June 2002): 299. http://dx.doi.org/10.2307/2693968.

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8

Venema, Y. "MODAL LOGIC." Philosophical Review 109, no. 2 (April 1, 2000): 286–89. http://dx.doi.org/10.1215/00318108-109-2-286.

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9

Spencer-Smith, Richard. "Modal logic." Artificial Intelligence Review 5, no. 1-2 (1991): 5–34. http://dx.doi.org/10.1007/bf00129533.

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10

Ewald, W. B. "Intuitionistic tense and modal logic." Journal of Symbolic Logic 51, no. 1 (March 1986): 166–79. http://dx.doi.org/10.2307/2273953.

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In this article we shall construct intuitionistic analogues to the main systems of classical tense logic. Since each classical modal logic can be gotten from some tense logic by one of the definitions(i) □ p ≡ p ∧ Gp ∧ Hp, ◇p ≡ p ∨ Fp ∨ Pp; or,(ii) □ p ≡ p ∧ Gp, ◇p = p ∨ Fp(see [5]), we shall find that our intuitionistic tense logics give us analogues to the classical modal logics as well.We shall not here discuss the philosophical issues raised by our logics. Readers interested in the intuitionistic view of time and modality should see [2] for a detailed discussion.In §2 we define the Kripke models for IKt, the intuitionistic analogue to Lemmon's system Kt. We then prove the completeness and decidability of this system (§§3–5). Finally, we extend our results to other sorts of tense logic and to modal logic.In the language of IKt, we have: sentence-letters p, q, r, etc.; the (intuitionistic) connectives ∧, ∨, →, ¬; and unary operators P (“it was the case”), F (it will be the case”), H (“it has always been the case”) and G (“it will always be the case”). Formulas are defined inductively: all sentence-letters are formulas; if X is a formula, so are ¬X, PX, FX, HX, and GX; if X and Y are formulas, so are X ∧ Y, X ∨ Y, and X → Y. We shall see that, in contrast to classical tense logic, F and P cannot be defined in terms of G and H.

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11

De Araujo Feitosa, Hércules, Mariana Matulovic, and Ana Claudia de J. Golzio. "A basic epistemic logic and its algebraic model." INTERMATHS 4, no. 2 (December 30, 2023): 28–37. http://dx.doi.org/10.22481/intermaths.v4i2.14133.

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In this paper we propose an algebraic model for a modal epistemic logic. Although it is known the existence of algebraic models for modal logics, considering that there are so many different modal logics, so it is not usual to give an algebraic model for each such system. The basic epistemic logic used in the paper is bimodal and we can show that the epistemic algebra introduced in the paper is an adequate model for it.

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12

Calcagno, Cristiano, Philippa Gardner, and Uri Zarfaty. "Context logic as modal logic." ACM SIGPLAN Notices 42, no. 1 (January 17, 2007): 123–34. http://dx.doi.org/10.1145/1190215.1190236.

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13

Demri, Stéphane, and Raul Fervari. "The power of modal separation logics." Journal of Logic and Computation 29, no. 8 (December 2019): 1139–84. http://dx.doi.org/10.1093/logcom/exz019.

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Abstract We introduce a modal separation logic MSL whose models are memory states from separation logic and the logical connectives include modal operators as well as separating conjunction and implication from separation logic. With such a combination of operators, some fragments of MSL can be seen as genuine modal logics whereas some others capture standard separation logics, leading to an original language to speak about memory states. We analyse the decidability status and the computational complexity of several fragments of MSL, obtaining surprising results by design of proof methods that take into account the modal and separation features of MSL. For example, the satisfiability problem for the fragment of MSL with $\Diamond $, the difference modality $\langle \neq \rangle $ and separating conjunction $\ast $ is shown Tower-complete whereas the restriction either to $\Diamond $ and $\ast $ or to $\langle \neq \rangle $ and $\ast $ is only NP-complete. We establish that the full logic MSL admits an undecidable satisfiability problem. Furthermore, we investigate variants of MSL with alternative semantics and we build bridges with interval temporal logics and with logics equipped with sabotage operators.

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Mruczek-Nasieniewska, Krystyna, and Marek Nasieniewski. "A Kotas-Style Characterisation of Minimal Discussive Logic." Axioms 8, no. 4 (October 1, 2019): 108. http://dx.doi.org/10.3390/axioms8040108.

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In this paper, we discuss a version of discussive logic determined by a certain variant of Jaśkowski’s original model of discussion. The obtained system can be treated as the minimal discussive logic. It is determined by frames with serial accessibility relation. As the smallest one, this logic can be treated as a basis which could be extended to richer discussive logics that are obtained by varying accessibility relation and resulting in a lattice of discussive logics. One has to remember that while formulating discussive logics there is no one-to-one determination of discussive logics by modal logics. For example, it is proved that Jaśkowski’s logic D 2 can be expressed by other than S 5 modal logics. In this paper we consider a deductive system for the sketchily described minimal logic. While formulating the deductive system, we apply a method of Kotas that was used to axiomatize D 2 . The obtained system determines a logic D 0 as a set of theses that is contained in D 2 . Moreover, any discussive logic that would be expressed by means of the provided model of discussion would contain D 0 , so it is the smallest discussive logic.

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15

Kupke, Clemens, Dirk Pattinson, and Lutz Schröder. "Coalgebraic Reasoning with Global Assumptions in Arithmetic Modal Logics." ACM Transactions on Computational Logic 23, no. 2 (April 30, 2022): 1–34. http://dx.doi.org/10.1145/3501300.

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We establish a generic upper bound ExpTime for reasoning with global assumptions (also known as TBoxes) in coalgebraic modal logics. Unlike earlier results of this kind, our bound does not require a tractable set of tableau rules for the instance logics, so that the result applies to wider classes of logics. Examples are Presburger modal logic, which extends graded modal logic with linear inequalities over numbers of successors, and probabilistic modal logic with polynomial inequalities over probabilities. We establish the theoretical upper bound using a type elimination algorithm. We also provide a global caching algorithm that potentially avoids building the entire exponential-sized space of candidate states, and thus offers a basis for practical reasoning. This algorithm still involves frequent fixpoint computations; we show how these can be handled efficiently in a concrete algorithm modelled on Liu and Smolka’s linear-time fixpoint algorithm. Finally, we show that the upper complexity bound is preserved under adding nominals to the logic, i.e., in coalgebraic hybrid logic.

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16

Balbiani, Philippe. "A Modal Semantics of Negation in Logic Programming." Fundamenta Informaticae 16, no. 3-4 (May 1, 1992): 231–62. http://dx.doi.org/10.3233/fi-1992-163-403.

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The beauty of modal logics and their interest lie in their ability to represent such different intensional concepts as knowledge, time, obligation, provability in arithmetic, … according to the properties satisfied by the accessibility relations of their Kripke models (transitivity, reflexivity, symmetry, well-foundedness, …). The purpose of this paper is to study the ability of modal logics to represent the concepts of provability and unprovability in logic programming. The use of modal logic to study the semantics of logic programming with negation is defended with the help of a modal completion formula. This formula is a modal translation of Clack’s formula. It gives soundness and completeness proofs for the negation as failure rule. It offers a formal characterization of unprovability in logic programs. It characterizes as well its stratified semantics.

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17

SASAKI, KATSUMI. "FORMULAS IN MODAL LOGIC S4." Review of Symbolic Logic 3, no. 4 (September 13, 2010): 600–627. http://dx.doi.org/10.1017/s1755020310000043.

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Here, we provide a detailed description of the mutual relation of formulas with finite propositional variables p1, …, pm in modal logic S4. Our description contains more information on S4 than those given in Shehtman (1978) and Moss (2007); however, Shehtman (1978) also treated Grzegorczyk logic and Moss (2007) treated many other normal modal logics. Specifically, we construct normal forms, which behave like the principal conjunctive normal forms in the classical propositional logic. The results include finite and effective methods to find a normal form equivalent to a given formula A by clarifying the behavior of connectives and giving a finite method to list all exact models.

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18

Milnikel, Robert. "Embedding Modal Nonmonotonic Logics into Default Logic." Studia Logica 75, no. 3 (December 2003): 377–82. http://dx.doi.org/10.1023/b:stud.0000009566.83940.4f.

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19

Rimatskiy, V. V. "Admissible Inference Rules and Semantic Property of Modal Logics." Bulletin of Irkutsk State University. Series Mathematics 37 (2021): 104–17. http://dx.doi.org/10.26516/1997-7670.2021.37.104.

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Firstly semantic property of nonstandart logics were described by formulas which are peculiar to studied a models in general, and do not take to consideration a variable conditions and a changing assumptions. Evidently the notion of inference rule generalizes the notion of formulas and brings us more flexibility and more expressive power to model human reasoning and computing. In 2000-2010 a few results on describing of explicit bases for admissible inference rules for nonstandard logics (S4, K4, H etc.) appeared. The key property of these logics was weak co-cover property. Beside the improvement of deductive power in logic, an admissible rule are able to describe some semantic property of given logic. We describe a semantic property of modal logics in term of admissibility of given set of inference rules. We prove that modal logic over logic 𝐺𝐿 enjoys weak co-cover property iff all given rules are admissible for logic.

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20

HOLLIDAY, WESLEY H., and TADEUSZ LITAK. "COMPLETE ADDITIVITY AND MODAL INCOMPLETENESS." Review of Symbolic Logic 12, no. 3 (July 4, 2019): 487–535. http://dx.doi.org/10.1017/s1755020317000259.

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AbstractIn this article, we tell a story about incompleteness in modal logic. The story weaves together an article of van Benthem (1979), “Syntactic aspects of modal incompleteness theorems,” and a longstanding open question: whether every normal modal logic can be characterized by a class ofcompletely additivemodal algebras, or as we call them,${\cal V}$-baos. Using a first-order reformulation of the property of complete additivity, we prove that the modal logic that starred in van Benthem’s article resolves the open question in the negative. In addition, for the case of bimodal logic, we show that there is a naturally occurring logic that is incomplete with respect to${\cal V}$-baos, namely the provability logic$GLB$(Japaridze, 1988; Boolos, 1993). We also show that even logics that are unsound with respect to such algebras do not have to be more complex than the classical propositional calculus. On the other hand, we observe that it is undecidable whether a syntactically defined logic is${\cal V}$-complete. After these results, we generalize the Blok Dichotomy (Blok, 1978) to degrees of${\cal V}$-incompleteness. In the end, we return to van Benthem’s theme of syntactic aspects of modal incompleteness.

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21

Lin, Zhe, and Minghui Ma. "Gentzen sequent calculi for some intuitionistic modal logics." Logic Journal of the IGPL 27, no. 4 (May 30, 2019): 596–623. http://dx.doi.org/10.1093/jigpal/jzz020.

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Abstract Intuitionistic modal logics are extensions of intuitionistic propositional logic with modal axioms. We treat with two modal languages ${\mathscr{L}}_\Diamond $ and $\mathscr{L}_{\Diamond ,\Box }$ which extend the intuitionistic propositional language with $\Diamond $ and $\Diamond ,\Box $, respectively. Gentzen sequent calculi are established for several intuitionistic modal logics. In particular, we introduce a Gentzen sequent calculus for the well-known intuitionistic modal logic $\textsf{MIPC}$. These sequent calculi admit cut elimination and subformula property. They are decidable.

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22

Parisi, Andrew. "Second-Order Modal Logic." Bulletin of Symbolic Logic 27, no. 4 (December 2021): 530–31. http://dx.doi.org/10.1017/bsl.2020.45.

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AbstractThe dissertation introduces new sequent-calculi for free first- and second-order logic, and a hyper-sequent calculus for modal logics K, D, T, B, S4, and S5; to attain the calculi for the stronger modal logics, only external structural rules need to be added to the calculus for K, while operational and internal structural rules remain the same. Completeness and cut-elimination are proved for all calculi presented.Philosophically, the dissertation develops an inferentialist, or proof-theoretic, theory of meaning. It takes as a starting point that the sense of a sentence is determined by the rules governing its use. In particular, there are two features of the use of a sentence that jointly determine its sense, the conditions under which it is coherent to assert that sentence and the conditions under which it is coherent to deny that sentence. The dissertation develops a theory of quantification as marking coherent ways a language can be expanded and modality as the means by which we can reflect on the norms governing the assertion and denial conditions of our language. If the view of quantification that is argued for is correct, then there is no tension between second-order quantification and nominalism. In particular, the ontological commitments one can incur through the use of a quantifier depend wholly on the ontological commitments one can incur through the use of atomic sentences. The dissertation concludes by applying the developed theory of meaning to the metaphysical issue of necessitism and contingentism. Two objections to a logic of contingentism are raised and addressed. The resulting logic is shown to meet all the requirement that the dissertation lays out for a theory of meaning for quantifiers and modal operators.Abstract prepared by Andrew ParisiE-mail: andrew.p.parisi@gmail.comURL: https://opencommons.uconn.edu/dissertations/1480/

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23

Li, Dazhu. "Losing connection: the modal logic of definable link deletion." Journal of Logic and Computation 30, no. 3 (April 2020): 715–43. http://dx.doi.org/10.1093/logcom/exz036.

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Abstract In this article, we start with a two-player game that models communication under adverse circumstances in everyday life and study it from the perspective of a modal logic of graphs, where links can be deleted locally according to definitions available to the adversarial player. We first introduce a new language, semantics and some typical validities. We then formulate a new type of first-order translation for this modal logic and prove its correctness. Then, a novel notion of bisimulation is proposed that leads to a characterization theorem for the logic as a fragment of first-order logic, and a further investigation is made of its expressive power against hybrid modal languages. Next, we discuss how to axiomatize this logic of link deletion, using dynamic-epistemic logics as a contrast. Finally, we show that our new modal logic lacks both the tree model property and the finite model property and that its satisfiability problem is undecidable.

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DROBYSHEVICH, SERGEY, and HEINRICH WANSING. "PROOF SYSTEMS FOR VARIOUS FDE-BASED MODAL LOGICS." Review of Symbolic Logic 13, no. 4 (June 17, 2019): 720–47. http://dx.doi.org/10.1017/s1755020319000261.

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AbstractWe present novel proof systems for various FDE-based modal logics. Among the systems considered are a number of Belnapian modal logics introduced in Odintsov & Wansing (2010) and Odintsov & Wansing (2017), as well as the modal logic KN4 with strong implication introduced in Goble (2006). In particular, we provide a Hilbert-style axiom system for the logic $BK^{\square - } $ and characterize the logic BK as an axiomatic extension of the system $BK^{FS} $. For KN4 we provide both an FDE-style axiom system and a decidable sequent calculus for which a contraction elimination and a cut elimination result are shown.

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25

Kontopoulos, Efstratios, Nick Bassiliades, Guido Governatori, and Grigoris Antoniou. "A Modal Defeasible Reasoner of Deontic Logic for the Semantic Web." International Journal on Semantic Web and Information Systems 7, no. 1 (January 2011): 18–43. http://dx.doi.org/10.4018/jswis.2011010102.

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Defeasible logic is a non-monotonic formalism that deals with incomplete and conflicting information, whereas modal logic deals with the concepts of necessity and possibility. These types of logics play a significant role in the emerging Semantic Web, which enriches the available Web information with meaning, leading to better cooperation between end-users and applications. Defeasible and modal logics, in general, and, particularly, deontic logic provide means for modeling agent communities, where each agent is characterized by its cognitive profile and normative system, as well as policies, which define privacy requirements, access permissions, and individual rights. Toward this direction, this article discusses the extension of DR-DEVICE, a Semantic Web-aware defeasible reasoner, with a mechanism for expressing modal logic operators, while testing the implementation via deontic logic operators, concerned with obligations, permissions, and related concepts. The motivation behind this work is to develop a practical defeasible reasoner for the Semantic Web that takes advantage of the expressive power offered by modal logics, accompanied by the flexibility to define diverse agent behaviours. A further incentive is to study the various motivational notions of deontic logic and discuss the cognitive state of agents, as well as the interactions among them.

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Hirsch, R., I. Hodkinson, and A. Kurucz. "On modal logics betweenK × K × KandS5 × S5 × S5." Journal of Symbolic Logic 67, no. 1 (March 2002): 221–34. http://dx.doi.org/10.2178/jsl/1190150040.

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AbstractWe prove that everyn-modal logic betweenKnandS5nis undecidable, whenever n ≥ 3. We also show that each of these logics is non-finitely axiomatizable, lacks the product finite model property, and there is no algorithm deciding whether a finite frame validates the logic. These results answer several questions of Gabbay and Shehtman. The proofs combine the modal logic technique of Yankov–Fine frame formulas with algebraic logic results of Halmos, Johnson and Monk, and give a reduction of the (undecidable) representation problem of finite relation algebras.

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Goguadze, George, Carla Piazza, and Yde Venema. "Simulating polyadic modal logics by monadic ones." Journal of Symbolic Logic 68, no. 2 (June 2003): 419–62. http://dx.doi.org/10.2178/jsl/1052669058.

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AbstractWe define an interpretation of modal languages with polyadic operators in modal languages that use monadic operators (diamonds) only. We also define a simulation operator which associates a logic Λsim in the diamond language with each logic Λ in the language with polyadic modal connectives. We prove that this simulation operator transfers several useful properties of modal logics, such as finite/recursive axiomatizability, frame completeness and the finite model property, canonicity and first-order definability.

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Rybakov, Mikhail, and Dmitry Shkatov. "Recursive enumerability and elementary frame definability in predicate modal logic." Journal of Logic and Computation 30, no. 2 (December 20, 2019): 549–60. http://dx.doi.org/10.1093/logcom/exz028.

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Abstract We investigate the relationship between recursive enumerability and elementary frame definability in first-order predicate modal logic. On one hand, it is well known that every first-order predicate modal logic complete with respect to an elementary class of Kripke frames, i.e. a class of frames definable by a classical first-order formula, is recursively enumerable. On the other, numerous examples are known of predicate modal logics, based on ‘natural’ propositional modal logics with essentially second-order Kripke semantics, that are either not recursively enumerable or Kripke incomplete. This raises the question of whether every Kripke complete, recursively enumerable predicate modal logic can be characterized by an elementary class of Kripke frames. We answer this question in the negative, by constructing a normal predicate modal logic which is Kripke complete, recursively enumerable, but not complete with respect to an elementary class of frames. We also present an example of a normal predicate modal logic that is recursively enumerable, Kripke complete, and not complete with respect to an elementary class of rooted frames, but is complete with respect to an elementary class of frames that are not rooted.

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Shapirovsky, Ilya. "Satisfiability Problems on Sums of Kripke Frames." ACM Transactions on Computational Logic 23, no. 3 (July 31, 2022): 1–25. http://dx.doi.org/10.1145/3508068.

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We consider the operation of sum on Kripke frames, where a family of frames-summands is indexed by elements of another frame. In many cases, the modal logic of sums inherits the finite model property and decidability from the modal logic of summands [Babenyshev and Rybakov 2010 ; Shapirovsky 2018 ]. In this paper we show that, under a general condition, the satisfiability problem on sums is polynomial space Turing reducible to the satisfiability problem on summands. In particular, for many modal logics decidability in PSpace is an immediate corollary from the semantic characterization of the logic.

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Nocic, Vladimir, and Jasmina Nocic. "Modal logic and logic of fiction." Theoria, Beograd 56, no. 4 (2013): 47–62. http://dx.doi.org/10.2298/theo1304047n.

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This paper analyzes the views of representative theoreticians of possible worlds semantics and possible worlds theory in an attempt to ascertain the degree and manner of interdisciplinary borrowing through focusing on possible worlds and individuals in those worlds. The paper first clarifies the general perceptions of possible worlds, perceptions in the field of modal restrictions, transworld identity, and identity over time, as presented in the works of Saul Kripke, David Lewis, and Nicholas Rescher, the representative semanticists of possible worlds, and then ascertains the degree and manner of their adaptations in the theory proposed by Ljubomir Dolezel within literary theory. The conclusion is that the cooperation between the two disciplines stands on fertile ground but that it is necessary to perform more systematic adaptations due to different subjects of research and different objectives.

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Hodkinson, Ian, and Louis Paternault. "Axiomatizing hybrid logic using modal logic." Journal of Applied Logic 8, no. 4 (December 2010): 386–96. http://dx.doi.org/10.1016/j.jal.2010.08.005.

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Tulenheimo, Tero. "Hybrid Logic Meets IF Modal Logic." Journal of Logic, Language and Information 18, no. 4 (April 28, 2009): 559–91. http://dx.doi.org/10.1007/s10849-009-9092-y.

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Allo, Patrick. "Adaptive Logic as a Modal Logic." Studia Logica 101, no. 5 (October 11, 2012): 933–58. http://dx.doi.org/10.1007/s11225-012-9403-1.

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MURAI, Tetsuya, and Satoru FUKAMI. "Modal Logic (1)." Journal of Japan Society for Fuzzy Theory and Systems 7, no. 1 (1995): 3–18. http://dx.doi.org/10.3156/jfuzzy.7.1_3.

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MURAI, Tetsuya, and Satoru FUKAMI. "Modal Logic (2)." Journal of Japan Society for Fuzzy Theory and Systems 7, no. 2 (1995): 222–38. http://dx.doi.org/10.3156/jfuzzy.7.2_222.

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Degnan, Michael J. "Aristotle’s Modal Logic." Ancient Philosophy 20, no. 1 (2000): 215–22. http://dx.doi.org/10.5840/ancientphil200020118.

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37

Venema, Yde. "Cylindric modal logic." Journal of Symbolic Logic 60, no. 2 (June 1995): 591–623. http://dx.doi.org/10.2307/2275853.

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AbstractTreating the existential quantification ∃νi as a diamond ♢i and the identity νi = νj as a constant δij, we study restricted versions of first order logic as if they were modal formalisms. This approach is closely related to algebraic logic, as the Kripke frames of our system have the type of the atom structures of cylindric algebras; the full cylindric set algebras are the complex algebras of the intended multidimensional frames called cubes.The main contribution of the paper is a characterization of these cube frames for the finite-dimensional case and, as a consequence of the special form of this characterization, a completeness theorem for this class. These results lead to finite, though unorthodox, derivation systems for several related formalisms, e.g. for the valid n-variable first order formulas, for type-free valid formulas and for the equational theory of representable cylindric algebras. The result for type-free valid formulas indicates a positive solution to Problem 4.16 of Henkin, Monk and Tarski [16].

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38

Kracht, Marcus. "Technical Modal Logic." Philosophy Compass 6, no. 5 (May 2011): 350–59. http://dx.doi.org/10.1111/j.1747-9991.2011.00396.x.

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Bozzelli, Laura, Hans van Ditmarsch, Tim French, James Hales, and Sophie Pinchinat. "Refinement modal logic." Information and Computation 239 (December 2014): 303–39. http://dx.doi.org/10.1016/j.ic.2014.07.013.

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40

Dunn, J. Michael. "Positive modal logic." Studia Logica 55, no. 2 (1995): 301–17. http://dx.doi.org/10.1007/bf01061239.

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41

Kamide, Norihiro, and Yaroslav Shramko. "Modal Multilattice Logic." Logica Universalis 11, no. 3 (July 20, 2017): 317–43. http://dx.doi.org/10.1007/s11787-017-0172-5.

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42

Ostertag, Gary. "Russell’s Modal Logic?" Russell: The Journal of Bertrand Russell Studies 20, no. 2 (December 2000): 165–72. http://dx.doi.org/10.1353/rss.2000.0005.

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43

HARRISON-TRAINOR, MATTHEW. "FIRST-ORDER POSSIBILITY MODELS AND FINITARY COMPLETENESS PROOFS." Review of Symbolic Logic 12, no. 4 (September 2, 2019): 637–62. http://dx.doi.org/10.1017/s1755020319000418.

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AbstractThis article builds on Humberstone’s idea of defining models of propositional modal logic where total possible worlds are replaced by partial possibilities. We follow a suggestion of Humberstone by introducing possibility models for quantified modal logic. We show that a simple quantified modal logic is sound and complete for our semantics. Although Holliday showed that for many propositional modal logics, it is possible to give a completeness proof using a canonical model construction where every possibility consists of finitely many formulas, we show that this is impossible to do in the first-order case. However, one can still construct a canonical model where every possibility consists of a computable set of formulas and thus still of finitely much information.

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GYENIS, ZALÁN. "STANDARD BAYES LOGIC IS NOT FINITELY AXIOMATIZABLE." Review of Symbolic Logic 13, no. 2 (March 22, 2019): 326–37. http://dx.doi.org/10.1017/s1755020319000157.

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AbstractIn the article [2] a hierarchy of modal logics has been defined to capture the logical features of Bayesian belief revision. Elements in that hierarchy were distinguished by the cardinality of the set of elementary propositions. By linking the modal logics in the hierarchy to the modal logics of Medvedev frames it has been shown that the modal logic of Bayesian belief revision determined by probabilities on a finite set of elementary propositions is not finitely axiomatizable. However, the infinite case remained open. In this article we prove that the modal logic of Bayesian belief revision determined by standard Borel spaces (these cover probability spaces that occur in most of the applications) is also not finitely axiomatizable.

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BEZHANISHVILI, GURAM, DAVID GABELAIA, and JOEL LUCERO-BRYAN. "MODAL LOGICS OF METRIC SPACES." Review of Symbolic Logic 8, no. 1 (December 18, 2014): 178–91. http://dx.doi.org/10.1017/s1755020314000446.

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AbstractIt is a classic result (McKinsey & Tarski, 1944; Rasiowa & Sikorski, 1963) that if we interpret modal diamond as topological closure, then the modal logic of any dense-in-itself metric space is the well-known modal system S4. In this paper, as a natural follow-up, we study the modal logic of an arbitrary metric space. Our main result establishes that modal logics arising from metric spaces form the following chain which is order-isomorphic (with respect to the ⊃ relation) to the ordinal ω + 3:$S4.Gr{z_1} \supset S4.Gr{z_2} \supset S4.Gr{z_3} \supset \cdots \,S4.Grz \supset S4.1 \supset S4.$It follows that the modal logic of an arbitrary metric space is finitely axiomatizable, has the finite model property, and hence is decidable.

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Hella, Lauri, Antti Kuusisto, Arne Meier, and Jonni Virtema. "Model checking and validity in propositional and modal inclusion logics." Journal of Logic and Computation 29, no. 5 (April 10, 2019): 605–30. http://dx.doi.org/10.1093/logcom/exz008.

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Abstract Propositional and modal inclusion logic are formalisms that belong to the family of logics based on team semantics. This article investigates the model checking and validity problems of these logics. We identify complexity bounds for both problems, covering both lax and strict team semantics. By doing so, we come close to finalizing the programme that aims to completely classify the complexities of the basic reasoning problems for modal and propositional dependence, independence and inclusion logics.

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Gabelaia, D., A. Kurucz, F. Wolter, and M. Zakharyaschev. "Products of ‘transitive” modal logics." Journal of Symbolic Logic 70, no. 3 (September 2005): 993–1021. http://dx.doi.org/10.2178/jsl/1122038925.

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AbstractWe solve a major open problem concerning algorithmic properties of products of ‘transitive’ modal logics by showing that products and commutators of such standard logics as K4, S4, S4.1, K4.3, GL, or Grz are undecidable and do not have the finite model property. More generally, we prove that no Kripke complete extension of the commutator [K4, K4] with product frames of arbitrary finite or infinite depth (with respect to both accessibility relations) can be decidable. In particular, if l1 and l2 are classes of transitive frames such that their depth cannot be bounded by any fixed n < ω, then the logic of the class {5ℑ1 × ℑ2 ∣ ℑ1 ∈ l1, ℑ2, ∈ l2} is undecidable. (On the contrary, the product of, say, K4 and the logic of all transitive Kripke frames of depth ≤ n, for some fixed n < ω, is decidable.) The complexity of these undecidable logics ranges from r.e. to co-r.e. and Π11-complete. As a consequence, we give the first known examples of Kripke incomplete commutators of Kripke complete logics.

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Kracht, Marcus, and Frank Wolter. "Normal monomodal logics can simulate all others." Journal of Symbolic Logic 64, no. 1 (March 1999): 99–138. http://dx.doi.org/10.2307/2586754.

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AbstractThis paper shows that non-normal modal logics can be simulated by certain polymodal normal logics and that polymodal normal logics can be simulated by monomodal (normal) logics. Many properties of logics are shown to be reflected and preserved by such simulations. As a consequence many old and new results in modal logic can be derived in a straightforward way, sheding new light on the power of normal monomodal logic.

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Marek, V. Wiktor, and Miroslaw Truszczynski. "More on Modal Aspects of Default Logic1." Fundamenta Informaticae 17, no. 1-2 (July 1, 1992): 99–116. http://dx.doi.org/10.3233/fi-1992-171-207.

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Investigations of default logic have been so far mostly concerned with the notion of an extension of a default theory. It turns out, however, that default logic is much richer. Namely, there are other natural classes of objects that might be associated with default reasoning. We study two such classes of objects with emphasis on their relations with modal nonmonotonic formalisms. First, we introduce the concept of a weak extension and study its properties. It has long been suspected that there are close connections between default and autoepistemic logics. The notion of weak extension allows us to precisely describe the relationship between these two formalisms. In particular, we show that default logic with weak extensions is essentially equivalent to autoepistemic logic, that is, nonmonotonic logic KD45. In the paper we also study the notion of a set of formulas closed under a default theory. These objects are shown to correspond to stable theories and to modal logic S5. In particular, we show that skeptical reasoning with sets closed under default theories is closely related with provability in S5. As an application of our results we determine the complexity of reasoning with weak extensions and sets closed under default theories.

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PRIOR, A. N. "Modal Logic and the Logic of Applicability." Theoria 34, no. 3 (February 11, 2008): 183–202. http://dx.doi.org/10.1111/j.1755-2567.1968.tb00350.x.

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