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  1. Simplified Kripke Semantics for K45-Like Gödel Modal Logics and Its Axiomatic Extensions.Ricardo Oscar Rodriguez, Olim Frits Tuyt, Francesc Esteva & Lluís Godo - 2022 - Studia Logica 110 (4):1081-1114.
    In this paper we provide a simplified, possibilistic semantics for the logics K45, i.e. a many-valued counterpart of the classical modal logic K45 over the [0, 1]-valued Gödel fuzzy logic \. More precisely, we characterize K45 as the set of valid formulae of the class of possibilistic Gödel frames \, where W is a non-empty set of worlds and \ is a possibility distribution on W. We provide decidability results as well. Moreover, we show that all the results also apply (...)
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  • Continuous Propositional Modal Logic.Stefano Baratella - 2018 - Journal of Applied Non-Classical Logics 28 (4):297-312.
    We introduce a propositional many-valued modal logic which is an extension of the Continuous Propositional Logic to a modal system. Otherwise said, we extend the minimal modal logic to a Continuous Logic system. After introducing semantics, axioms and deduction rules, we establish some preliminary results. Then we prove the equivalence between consistency and satisfiability. As straightforward consequences, we get compactness, an approximated completeness theorem, in the vein of Continuous Logic, and a Pavelka-style completeness theorem.
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  • Modal Logics with Belnapian Truth Values.Serge P. Odintsov & Heinrich Wansing - 2010 - Journal of Applied Non-Classical Logics 20 (3):279-304.
    Various four- and three-valued modal propositional logics are studied. The basic systems are modal extensions BK and BS4 of Belnap and Dunn's four-valued logic of firstdegree entailment. Three-valued extensions of BK and BS4 are considered as well. These logics are introduced semantically by means of relational models with two distinct evaluation relations, one for verification and the other for falsification. Axiom systems are defined and shown to be sound and complete with respect to the relational semantics and with respect to (...)
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  • A Note on Strong Axiomatization of Gödel Justification Logic.Nicholas Pischke - 2020 - Studia Logica 108 (4):687-724.
    Justification logics are special kinds of modal logics which provide a framework for reasoning about epistemic justifications. For this, they extend classical boolean propositional logic by a family of necessity-style modal operators “t : ”, indexed over t by a corresponding set of justification terms, which thus explicitly encode the justification for the necessity assertion in the syntax. With these operators, one can therefore not only reason about modal effects on propositions but also about dynamics inside the justifications themselves. We (...)
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  • Expressivity in Chain-Based Modal Logics.Michel Marti & George Metcalfe - 2018 - Archive for Mathematical Logic 57 (3-4):361-380.
    We investigate the expressivity of many-valued modal logics based on an algebraic structure with a complete linearly ordered lattice reduct. Necessary and sufficient algebraic conditions for admitting a suitable Hennessy–Milner property are established for classes of image-finite and modally saturated models. Full characterizations are obtained for many-valued modal logics based on complete BL-chains that are finite or have the real unit interval [0, 1] as a lattice reduct, including Łukasiewicz, Gödel, and product modal logics.
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  • A Four-Valued Dynamic Epistemic Logic.Yuri David Santos - 2020 - Journal of Logic, Language and Information 29 (4):451-489.
    Epistemic logic is usually employed to model two aspects of a situation: the factual and the epistemic aspects. Truth, however, is not always attainable, and in many cases we are forced to reason only with whatever information is available to us. In this paper, we will explore a four-valued epistemic logic designed to deal with these situations, where agents have only knowledge about the available information, which can be incomplete or conflicting, but not explicitly about facts. This layer of available (...)
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  • A Logical Modeling of Severe Ignorance.Stefano Bonzio, Vincenzo Fano & Pierluigi Graziani - unknown
    In the logical context, ignorance is traditionally defined recurring to epistemic logic $S_4$ \cite{Hintikka1962}. In particular, an agent ignores a formula $\varphi$ when s/he does not know neither $\varphi$ nor its negation $\neg\varphi$: $\neg\K\varphi\land\neg\K\neg\varphi$. In other words, ignorance is essentially interpreted as ``lack of knowledge''. \textcolor{red}{This received view has - as we point out - some problems, in particular we will highlight how it does not allow to express a type of content-theoretic ignorance, i.e. an ignorance of $\varphi$ that stems (...)
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  • Modal and Intuitionistic Variants of Extended Belnap–Dunn Logic with Classical Negation.Norihiro Kamide - 2021 - Journal of Logic, Language and Information 30 (3):491-531.
    In this study, we introduce Gentzen-type sequent calculi BDm and BDi for a modal extension and an intuitionistic modification, respectively, of De and Omori’s extended Belnap–Dunn logic BD+ with classical negation. We prove theorems for syntactically and semantically embedding BDm and BDi into Gentzen-type sequent calculi S4 and LJ for normal modal logic and intuitionistic logic, respectively. The cut-elimination, decidability, and completeness theorems for BDm and BDi are obtained using these embedding theorems. Moreover, we prove the Glivenko theorem for embedding (...)
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  • Standard Gödel Modal Logics.Xavier Caicedo & Ricardo O. Rodriguez - 2010 - Studia Logica 94 (2):189-214.
    We prove strong completeness of the □-version and the ◊-version of a Gödel modal logic based on Kripke models where propositions at each world and the accessibility relation are both infinitely valued in the standard Gödel algebra [0,1]. Some asymmetries are revealed: validity in the first logic is reducible to the class of frames having two-valued accessibility relation and this logic does not enjoy the finite model property, while validity in the second logic requires truly fuzzy accessibility relations and this (...)
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  • Biological Modalities.Maximilian Huber - unknown
    Biological modalities play an important explanatory role in biological practice. However, biological modalities lack truth conditions and the inferential relationship between biological and other modalities is unclear. This thesis addresses these problems, first, by improving upon Daniel Dennett's Library of Mendel. Second, a family of modal logics is introduced. In the simplest model, states are interpreted as codons, the binary relation is interpreted as single substitution mutation and the valuation induces a partition of blocks of codons that code for some (...)
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  • Belnap–Dunn Modal Logic with Value Operators.Yuanlei Lin & Minghui Ma - 2020 - Studia Logica 109 (4):759-789.
    The language of Belnap–Dunn modal logic \ expands the language of Belnap–Dunn four-valued logic with the modal operator \. We introduce the polarity semantics for \ and its two expansions \ and \ with value operators. The local finitary consequence relation \ in the language \ with respect to the class of all frames is axiomatized by a sequent system \ where \. We prove by using translations between sequents and formulas that these languages under the polarity semantics have the (...)
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  • How True It Is = Who Says It’s True.Melvin Fitting - 2009 - Studia Logica 91 (3):335 - 366.
    This is a largely expository paper in which the following simple idea is pursued. Take the truth value of a formula to be the set of agents that accept the formula as true. This means we work with an arbitrary (finite) Boolean algebra as the truth value space. When this is properly formalized, complete modal tableau systems exist, and there are natural versions of bisimulations that behave well from an algebraic point of view. There remain significant problems concerning the proper (...)
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  • Many-Valued Non-Monotonic Modal Logics.Melvin Fitting - unknown
    Among non-monotonic systems of reasoning, non-monotonic modal logics, and autoepistemic logic in particular, have had considerable success. The presence of explicit modal operators allows flexibility in the embedding of other approaches. Also several theoretical results of interest have been established concerning these logics. In this paper we introduce non-monotonic modal logics based on many-valued logics, rather than on classical logic. This extends earlier work of ours on many-valued modal logics. Intended applications are to situations involving several reasoners, not just one (...)
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  • On Ignorance and Contradiction Considered as Truth-Values.Didier Dubois - 2008 - Logic Journal of the IGPL 16 (2):195-216.
    A critical view of the alleged significance of Belnap four-valued logic for reasoning under inconsistent and incomplete information is provided. The difficulty lies in the confusion between truth-values and information states, when reasoning about Boolean propositions. So our critique is along the lines of previous debates on the relevance of many-valued logics and especially of the extension of the Boolean truth-tables to more than two values as a tool for reasoning about uncertainty. The critique also questions the significance of partial (...)
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  • A Completeness Theorem for Continuous Predicate Modal Logic.Stefano Baratella - 2019 - Archive for Mathematical Logic 58 (1-2):183-201.
    We study a modal extension of the Continuous First-Order Logic of Ben Yaacov and Pedersen :168–190, 2010). We provide a set of axioms for such an extension. Deduction rules are just Modus Ponens and Necessitation. We prove that our system is sound with respect to a Kripke semantics and, building on Ben Yaacov and Pedersen, that it satisfies a number of properties similar to those of first-order predicate logic. Then, by means of a canonical model construction, we get that every (...)
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  • Extended Quantum Logic.Kenji Tokuo - 2003 - Journal of Philosophical Logic 32 (5):549-563.
    The concept of quantum logic is extended so that it covers a more general set of propositions that involve non-trivial probabilities. This structure is shown to be embedded into a multi-modal framework, which has desirable logical properties such as an axiomatization, the finite model property and decidability.
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  • Axiomatization of Crisp Gödel Modal Logic.Ricardo Oscar Rodriguez & Amanda Vidal - 2020 - Studia Logica 109 (2):367-395.
    In this paper we consider the modal logic with both \ and \ arising from Kripke models with a crisp accessibility and whose propositions are valued over the standard Gödel algebra \. We provide an axiomatic system extending the one from Caicedo and Rodriguez :37–55, 2015) for models with a valued accessibility with Dunn axiom from positive modal logics, and show it is strongly complete with respect to the intended semantics. The axiomatizations of the most usual frame restrictions are given (...)
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  • A Duality for the Algebras of a Łukasiewicz N + 1-Valued Modal System.Bruno Teheux - 2007 - Studia Logica 87 (1):13-36.
    In this paper, we develop a duality for the varieties of a Łukasiewicz n + 1-valued modal System. This duality is an extension of Stone duality for modal algebras. Some logical consequences (such as completeness results, correspondence theory...) are then derived and we propose some ideas for future research.
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  • Implicit, Explicit and Speculative Knowledge.Hans van Ditmarsch, Tim French, Fernando R. Velázquez-Quesada & Yì N. Wáng - 2018 - Artificial Intelligence 256:35-67.
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  • Logical Characterizations of Regular Equivalence in Weighted Social Networks.Tuan-Fang Fan & Churn-Jung Liau - 2014 - Artificial Intelligence 214:66-88.
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  • Many-Valued Hybrid Logic.Jens Ulrik Hansen, Thomas Bolander & Torben Braüner - 2008 - In Carlos Areces & Robert Goldblatt (eds.), Advances in Modal Logic, Volume 7. CSLI Publications. pp. 111-132.
    In this paper we define a family of many-valued semantics for hybrid logic, where each semantics is based on a finite Heyting algebra of truth-values. We provide sound and complete tableau systems for these semantics. Moreover, we show how the tableau systems can be made terminating and thereby give rise to decision procedures for the logics in question. Our many-valued hybrid logics turn out to be "intermediate" logics between intuitionistic hybrid logic and classical hybrid logic in a specific sense explained (...)
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  • On a Multilattice Analogue of a Hypersequent S5 Calculus.Oleg Grigoriev & Yaroslav Petrukhin - forthcoming - Logic and Logical Philosophy:1.
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  • Modal Extension of Ideal Paraconsistent Four-Valued Logic and its Subsystem.Norihiro Kamide & Yoni Zohar - 2020 - Annals of Pure and Applied Logic 171 (10):102830.
    This study aims to introduce a modal extension M4CC of Arieli, Avron, and Zamansky's ideal paraconsistent four-valued logic 4CC as a Gentzen-type sequent calculus and prove the Kripke-completeness and cut-elimination theorems for M4CC. The logic M4CC is also shown to be decidable and embeddable into the normal modal logic S4. Furthermore, a subsystem of M4CC, which has some characteristic properties that do not hold for M4CC, is introduced and the Kripke-completeness and cut-elimination theorems for this subsystem are proved. This subsystem (...)
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  • Bilattice Logic of Epistemic Actions and Knowledge.Zeinab Bakhtiari, Hans van Ditmarsch & Umberto Rivieccio - 2020 - Annals of Pure and Applied Logic 171 (6):102790.
    Baltag, Moss, and Solecki proposed an expansion of classical modal logic, called logic of epistemic actions and knowledge (EAK), in which one can reason about knowledge and change of knowledge. Kurz and Palmigiano showed how duality theory provides a flexible framework for modeling such epistemic changes, allowing one to develop dynamic epistemic logics on a weaker propositional basis than classical logic (for example an intuitionistic basis). In this paper we show how the techniques of Kurz and Palmigiano can be further (...)
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  • A Hennessy-Milner Property for Many-Valued Modal Logics.Michel Marti & George Metcalfe - 2014 - In Rajeev Goré, Barteld Kooi & Agi Kurucz (eds.), Advances in Modal Logic, Volume 10. CSLI Publications. pp. 407-420.
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  • Many-Valued Modal Logics II.Melvin Fitting - unknown
    Suppose there are several experts, with some dominating others (expert A dominates expert B if B says something is true whenever A says it is). Suppose, further, that each of the experts has his or her own view of what is possible — in other words each of the experts has their own Kripke model in mind (subject, of course, to the dominance relation that may hold between experts). How will they assign truth values to sentences in a common modal (...)
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  • The Family of Stable Models.Melvin Fitting - unknown
    The family of all stable models for a logic program has a surprisingly simple overall structure, once two naturally occurring orderings are made explicit. In a so-called knowledge ordering based on degree of definedness, every logic program P has a smallest stable model, sk P — it is the well-founded model. There is also a dual largest stable model, S k P, which has not been considered before. There is another ordering based on degree of truth. Taking the meet and (...)
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  • Tableaus for Many-Valued Modal Logic.Melvin Fitting - 1995 - Studia Logica 55 (1):63 - 87.
    We continue a series of papers on a family of many-valued modal logics, a family whose Kripke semantics involves many-valued accessibility relations. Earlier papers in the series presented a motivation in terms of a multiple-expert semantics. They also proved completeness of sequent calculus formulations for the logics, formulations using a cut rule in an essential way. In this paper a novel cut-free tableau formulation is presented, and its completeness is proved.
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  • Fuzzy Topology and Łukasiewicz Logics From the Viewpoint of Duality Theory.Yoshihiro Maruyama - 2010 - Studia Logica 94 (2):245-269.
    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 Jónsson-Tarski duality for modal algebras to the (...)
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  • Pavelka-Style Fuzzy Justification Logics.Meghdad Ghari - 2016 - Logic Journal of the IGPL 24 (5):743-773.
    Justification logics provide a framework for reasoning about justifications and evidence. In this article, we study a fuzzy variant of justification logics in which an agent’s justification for a belief has certainty degree between 0 and 1. We replace the classical base of justification logics with Hájek’s rational Pavelka logic. We introduce fuzzy possible world semantics with crisp accessibility relation and also single world models for our logics. We establish soundness and graded-style completeness for both kinds of semantics. We also (...)
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  • Axiomatizing a Real-Valued Modal Logic.Denisa Diaconescu, George Metcalfe & Laura Schnüriger - 2016 - In Lev Beklemishev, Stéphane Demri & András Máté (eds.), Advances in Modal Logic, Volume 11. CSLI Publications. pp. 236-251.
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  • Bisimulations and Boolean Vectors.Melvin Fitting - 2003 - In Philippe Balbiani, Nobu-Yuki Suzuki, Frank Wolter & Michael Zakharyaschev (eds.), Advances in Modal Logic, Volume 4. CSLI Publications. pp. 97-125.
    A modal accessibility relation is just a transition relation, and so can be represented by a {0, 1} valued transition matrix. Starting from this observation, I first show that the machinery of matrices, over Boolean algebras more general than the two-valued one, is appropriate for investigating multi-modal semantics. Then I show that bisimulations have a rather elegant theory, when expressed in terms of transformations on Boolean vector spaces. The resulting theory is a curious hybrid, fitting between conventional modal semantics and (...)
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  • Fibred Semantics and the Weaving of Logics Part 1: Modal and Intuitionistic Logics.D. M. Gabbay - 1996 - Journal of Symbolic Logic 61 (4):1057-1120.
    This is Part 1 of a paper on fibred semantics and combination of logics. It aims to present a methodology for combining arbitrary logical systems L i , i ∈ I, to form a new system L I . The methodology `fibres' the semantics K i of L i into a semantics for L I , and `weaves' the proof theory (axiomatics) of L i into a proof system of L I . There are various ways of doing this, we (...)
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  • Modal Twist-Structures Over Residuated Lattices.H. Ono & U. Rivieccio - 2014 - Logic Journal of the IGPL 22 (3):440-457.
  • A Family of Gödel Hybrid Logics.Didier Galmiche & Yakoub Salhi - 2010 - Journal of Applied Logic 8 (4):371-385.
  • Propositional Dynamic Logic for Searching Games with Errors.Bruno Teheux - 2014 - Journal of Applied Logic 12 (4):377-394.
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  • Subformula Property in Many-Valued Modal Logics.Mitio Takano - 1994 - Journal of Symbolic Logic 59 (4):1263-1273.
  • How True It Is = Who Says It’s True.Melvin Fitting - 2009 - Studia Logica 91 (3):335-366.
    This is a largely expository paper in which the following simple idea is pursued. Take the truth value of a formula to be the set of agents that accept the formula as true. This means we work with an arbitrary Boolean algebra as the truth value space. When this is properly formalized, complete modal tableau systems exist, and there are natural versions of bisimulations that behave well from an algebraic point of view. There remain significant problems concerning the proper formalization, (...)
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