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Modal Logic

Oxford, England: Oxford University Press (1997)

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  1. Substructural Logics: A Primer.Francesco Paoli - 2002 - Dordrecht, Netherland: Springer.
    The aim of the present book is to give a comprehensive account of the ‘state of the art’ of substructural logics, focusing both on their proof theory and on their semantics (both algebraic and relational. It is for graduate students in either philosophy, mathematics, theoretical computer science or theoretical linguistics as well as specialists and researchers.
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  • Leo Esakia on Duality in Modal and Intuitionistic Logics.Guram Bezhanishvili (ed.) - 2014 - Dordrecht, Netherland: Springer.
    This volume is dedicated to Leo Esakia's contributions to the theory of modal and intuitionistic systems. Consisting of 10 chapters, written by leading experts, this volume discusses Esakia’s original contributions and consequent developments that have helped to shape duality theory for modal and intuitionistic logics and to utilize it to obtain some major results in the area. Beginning with a chapter which explores Esakia duality for S4-algebras, the volume goes on to explore Esakia duality for Heyting algebras and its generalizations (...)
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  • Philosophical Investigation Series: Selected Texts on Logic / Série Investigação Filosófica: Textos Selecionados de Lógica.Danilo Fraga Dantas & Rodrigo Cid - 2020 - Pelotas - Princesa, Pelotas - RS, Brasil: UFPEL's Publisher / Editora da UFPEL.
    Este livro marca o início da Série Investigação Filosófica. Uma série de livros de traduções de textos de plataformas internacionalmente reconhecidas, que possa servir tanto como material didático para os professores das diferentes subáreas e níveis da Filosofia quanto como material de estudo para o desenvolvimento pesquisas relevantes na área. Nós, professores, sabemos o quão difícil é encontrar bons materiais em português para indicarmos. E há uma certa deficiência na graduação brasileira de filosofia, principalmente em localizações menos favorecidas, com relação (...)
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  • The Intensional Side of Algebraic-Topological Representation Theorems.Sara Negri - 2017 - Synthese 198 (Suppl 5):1121-1143.
    Stone representation theorems are a central ingredient in the metatheory of philosophical logics and are used to establish modal embedding results in a general but indirect and non-constructive way. Their use in logical embeddings will be reviewed and it will be shown how they can be circumvented in favour of direct and constructive arguments through the methods of analytic proof theory, and how the intensional part of the representation results can be recovered from the syntactic proof of those embeddings. Analytic (...)
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  • Tarskian Consequence Relations Bilaterally: Some Familiar Notions.Sergey Drobyshevich - 2019 - Synthese 198 (S22):5213-5240.
    This paper is dedicated to developing a formalism that takes rejection seriously. Bilateral notation of signed formulas with force indicators is adopted to define signed consequences which can be viewed as the bilateral counterpart of Tarskian consequence relations. Its relation to some other bilateral approaches is discussed. It is shown how David Nelson’s logic N4 can be characterized bilaterally and the corresponding completeness result is proved. Further, bilateral variants of three familiar notions are considered and investigated: that of a fragment, (...)
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  • Contra-Classical Logics.Lloyd Humberstone - 2000 - Australasian Journal of Philosophy 78 (4):438 – 474.
    Only propositional logics are at issue here. Such a logic is contra-classical in a superficial sense if it is not a sublogic of classical logic, and in a deeper sense, if there is no way of translating its connectives, the result of which translation gives a sublogic of classical logic. After some motivating examples, we investigate the incidence of contra-classicality (in the deeper sense) in various logical frameworks. In Sections 3 and 4 we will encounter, originally as an example of (...)
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  • Some Results on Modal Axiomatization and Definability for Topological Spaces.Guram Bezhanishvili, Leo Esakia & David Gabelaia - 2005 - Studia Logica 81 (3):325-355.
    We consider two topological interpretations of the modal diamond—as the closure operator (C-semantics) and as the derived set operator (d-semantics). We call the logics arising from these interpretations C-logics and d-logics, respectively. We axiomatize a number of subclasses of the class of nodec spaces with respect to both semantics, and characterize exactly which of these classes are modally definable. It is demonstrated that the d-semantics is more expressive than the C-semantics. In particular, we show that the d-logics of the six (...)
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  • Admissible Bases Via Stable Canonical Rules.Nick Bezhanishvili, David Gabelaia, Silvio Ghilardi & Mamuka Jibladze - 2016 - Studia Logica 104 (2):317-341.
    We establish the dichotomy property for stable canonical multi-conclusion rules for IPC, K4, and S4. This yields an alternative proof of existence of explicit bases of admissible rules for these logics.
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  • Modal Languages for Topology: Expressivity and Definability.Balder ten Cate, David Gabelaia & Dmitry Sustretov - 2009 - Annals of Pure and Applied Logic 159 (1-2):146-170.
    In this paper we study the expressive power and definability for modal languages interpreted on topological spaces. We provide topological analogues of the van Benthem characterization theorem and the Goldblatt–Thomason definability theorem in terms of the well-established first-order topological language.
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  • Prior’s OIC Nonconservativity Example Revisited.Lloyd Humberstone - 2014 - Journal of Applied Non-Classical Logics 24 (3):209-235.
    In his 1964 note, ‘Two Additions to Positive Implication’, A. N. Prior showed that standard axioms governing conjunction yield a nonconservative extension of the pure implicational intermediate logic OIC of R. A. Bull. Here, after reviewing the situation with the aid of an adapted form of the Kripke semantics for intuitionistic and intermediate logics, we proceed to illuminate this example by transposing it to the setting of modal logic, and then relate it to the propositional logic of what have been (...)
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  • The Range of Modal Logic: An Essay in Memory of George Gargov.Johan van Benthem - 1999 - Journal of Applied Non-Classical Logics 9 (2):407-442.
    ABSTRACT George Gargov was an active pioneer in the ‘Sofia School’ of modal logicians. Starting in the 1970s, he and his colleagues expanded the scope of the subject by introducing new modal expressive power, of various innovative kinds. The aim of this paper is to show some general patterns behind such extensions, and review some very general results that we know by now, 20 years later. We concentrate on simulation invariance, decidability, and correspondence. What seems clear is that ‘modal logic’ (...)
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  • On the Finite Model Property of Intuitionistic Modal Logics Over MIPC.Takahito Aoto & Hiroyuki Shirasu - 1999 - Mathematical Logic Quarterly 45 (4):435-448.
    MIPC is a well-known intuitionistic modal logic of Prior and Bull . It is shown that every normal intuitionistic modal logic L over MIPC has the finite model property whenever L is Kripke-complete and universal.
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  • On Finite Model Property for Admissible Rules.Vladimir V. Rybakov, Vladimir R. Kiyatkin & Tahsin Oner - 1999 - Mathematical Logic Quarterly 45 (4):505-520.
    Our investigation is concerned with the finite model property with respect to admissible rules. We establish general sufficient conditions for absence of fmp w. r. t. admissibility which are applicable to modal logics containing K4: Theorem 3.1 says that no logic λ containing K4 with the co-cover property and of width > 2 has fmp w. r. t. admissibility. Surprisingly many, if not to say all, important modal logics of width > 2 are within the scope of this theorem–K4 itself, (...)
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  • A Note on the Complexity of S4.2.Aggeliki Chalki, Costas D. Koutras & Yorgos Zikos - 2021 - Journal of Applied Non-Classical Logics 31 (2):108-129.
    S4.2 is the modal logic of directed partial pre-orders and/or the modal logic of reflexive and transitive relational frames with a final cluster. It holds a distinguished position in philosophical...
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  • Béziau's Translation Paradox.Lloyd Humberstone - 2005 - Theoria 71 (2):138-181.
    Jean-Yves Béziau (‘Classical Negation can be Expressed by One of its Halves’, Logic Journal of the IGPL 7 (1999), 145–151) has given an especially clear example of a phenomenon he considers a sufficiently puzzling to call the ‘paradox of translation’: the existence of pairs of logics, one logic being strictly weaker than another and yet such that the stronger logic can be embedded within it under a faithful translation. We elaborate on Béziau’s example, which concerns classical negation, as well as (...)
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  • A Topological Approach to Full Belief.Alexandru Baltag, Nick Bezhanishvili, Aybüke Özgün & Sonja Smets - 2019 - Journal of Philosophical Logic 48 (2):205-244.
    Stalnaker, 169–199 2006) introduced a combined epistemic-doxastic logic that can formally express a strong concept of belief, a concept of belief as ‘subjective certainty’. In this paper, we provide a topological semantics for belief, in particular, for Stalnaker’s notion of belief defined as ‘epistemic possibility of knowledge’, in terms of the closure of the interior operator on extremally disconnected spaces. This semantics extends the standard topological interpretation of knowledge with a new topological semantics for belief. We prove that the belief (...)
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  • On the Modal Logic of the Non-Orthogonality Relation Between Quantum States.Shengyang Zhong - 2018 - Journal of Logic, Language and Information 27 (2):157-173.
    It is well known that the non-orthogonality relation between the states of a quantum system is reflexive and symmetric, and the modal logic \ is sound and complete with respect to the class of sets each equipped with a reflexive and symmetric binary relation. In this paper, we consider two properties of the non-orthogonality relation: Separation and Superposition. We find sound and complete modal axiomatizations for the classes of sets each equipped with a reflexive and symmetric relation that satisfies each (...)
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  • The Power of Belnap: Sequent Systems for SIXTEEN ₃. [REVIEW]Heinrich Wansing - 2010 - Journal of Philosophical Logic 39 (4):369 - 393.
    The trilattice SIXTEEN₃ is a natural generalization of the wellknown bilattice FOUR₂. Cut-free, sound and complete sequent calculi for truth entailment and falsity entailment in SIXTEEN₃, are presented.
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  • Zolin and Pizzi: Defining Necessity From Noncontingency.Lloyd Humberstone - 2013 - Erkenntnis 78 (6):1275-1302.
    The point of the present paper is to draw attention to some interesting similarities, as well as differences, between the approaches to the logic of noncontingency of Evgeni Zolin and of Claudio Pizzi. Though neither of them refers to the work of the other, each is concerned with the definability of a (normally behaving, though not in general truth-implying) notion of necessity in terms of noncontingency, standard boolean connectives and additional but non-modal expressive resources. The notion of definability involved is (...)
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  • Possible Worlds in Use.Andrzej Indrzejczak - 2011 - Studia Logica 99 (1-3):229-248.
    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 (...)
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  • Inquisitive Logic.Ivano Ciardelli & Floris Roelofsen - 2011 - Journal of Philosophical Logic 40 (1):55-94.
    This paper investigates a generalized version of inquisitive semantics. A complete axiomatization of the associated logic is established, the connection with intuitionistic logic and several intermediate logics is explored, and the generalized version of inquisitive semantics is argued to have certain advantages over the system that was originally proposed by Groenendijk (2009) and Mascarenhas (2009).
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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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  • On Axiomatizing Shramko-Wansing’s Logic.Sergei P. Odintsov - 2009 - Studia Logica 91 (3):407 - 428.
    This work treats the problem of axiomatizing the truth and falsity consequence relations, $ \vDash _t $ and $ \vDash _f $, determined via truth and falsity orderings on the trilattice SIXTEEN₃. The approach is based on a representation of SIXTEEN₃ as a twist-structure over the two-element Boolean algebra.
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  • Three Dual Ontologies.Chris Brink & Ingrid Rewitzky - 2002 - Journal of Philosophical Logic 31 (6):543-568.
    In this paper we give an example of intertranslatability between an ontology of individuals (nominalism), an ontology of properties (realism), and an ontology of facts (factualism). We demonstrate that these three ontologies are dual to each other, meaning that each ontology can be translated into, and recaptured from, each of the others. The aiin of the enterprise is to raise the possibility that, at least in some settings, there may be no need for considerations of ontological primacy. Whether the world (...)
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  • Classifying material implications over minimal logic.Hannes Diener & Maarten McKubre-Jordens - 2020 - Archive for Mathematical Logic 59 (7-8):905-924.
    The so-called paradoxes of material implication have motivated the development of many non-classical logics over the years, such as relevance logics, paraconsistent logics, fuzzy logics and so on. In this note, we investigate some of these paradoxes and classify them, over minimal logic. We provide proofs of equivalence and semantic models separating the paradoxes where appropriate. A number of equivalent groups arise, all of which collapse with unrestricted use of double negation elimination. Interestingly, the principle ex falso quodlibet, and several (...)
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  • Counterfactual Logic and the Necessity of Mathematics.Samuel Elgin - manuscript
    This paper is concerned with counterfactual logic and its implications for the modal status of mathematical claims. It is most directly a response to an ambitious program by Yli-Vakkuri and Hawthorne (2018), who seek to establish that mathematics is committed to its own necessity. I claim that their argument fails to establish this result for two reasons. First, their assumptions force our hand on a controversial debate within counterfactual logic. In particular, they license counterfactual strengthening— the inference from ‘If A (...)
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  • Transitive Logics of Finite Width with Respect to Proper-Successor-Equivalence.Ming Xu - 2021 - Studia Logica 109 (6):1177-1200.
    This paper presents a generalization of Fine’s completeness theorem for transitive logics of finite width, and proves the Kripke completeness of transitive logics of finite “suc-eq-width”. The frame condition for each finite suc-eq-width axiom requires, in rooted transitive frames, a finite upper bound of cardinality for antichains of points with different proper successors. The paper also presents a generalization of Rybakov’s completeness theorem for transitive logics of prefinite width, and proves the Kripke completeness of transitive logics of prefinite “suc-eq-width”. The (...)
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  • Natural Factors of the Medvedev Lattice Capturing IPC.Rutger Kuyper - 2014 - Archive for Mathematical Logic 53 (7-8):865-879.
    Skvortsova showed that there is a factor of the Medvedev lattice which captures intuitionistic propositional logic. However, her factor is unnatural in the sense that it is constructed in an ad hoc manner. We present a more natural example of such a factor. We also show that the theory of every non-trivial factor of the Medvedev lattice is contained in Jankov’s logic, the deductive closure of IPC plus the weak law of the excluded middle ¬p∨¬¬p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} (...)
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  • Labelled Tree Sequents, Tree Hypersequents and Nested Sequents.Rajeev Goré & Revantha Ramanayake - 2012 - In Thomas Bolander, Torben Braüner, Silvio Ghilardi & Lawrence Moss (eds.), Advances in Modal Logic, Volume 9. CSLI Publications. pp. 279-299.
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  • Does the Deduction Theorem Fail for Modal Logic?Raul Hakli & Sara Negri - 2012 - Synthese 187 (3):849-867.
    Various sources in the literature claim that the deduction theorem does not hold for normal modal or epistemic logic, whereas others present versions of the deduction theorem for several normal modal systems. It is shown here that the apparent problem arises from an objectionable notion of derivability from assumptions in an axiomatic system. When a traditional Hilbert-type system of axiomatic logic is generalized into a system for derivations from assumptions, the necessitation rule has to be modified in a way that (...)
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  • Label-Free Natural Deduction Systems for Intuitionistic and Classical Modal Logics.Didier Galmiche & Yakoub Salhi - 2010 - Journal of Applied Non-Classical Logics 20 (4):373-421.
    In this paper we study natural deduction for the intuitionistic and classical modal logics obtained from the combinations of the axioms T, B, 4 and 5. In this context we introduce a new multi-contextual structure, called T-sequent, that allows to design simple labelfree natural deduction systems for these logics. After proving that they are sound and complete we show that they satisfy the normalization property and consequently the subformula property in the intuitionistic case.
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  • Knowledge Means ‘All’, Belief Means ‘Most’.Dimitris Askounis, Costas D. Koutras & Yorgos Zikos - 2016 - Journal of Applied Non-Classical Logics 26 (3):173-192.
    We introduce a bimodal epistemic logic intended to capture knowledge as truth in all epistemically alternative states and belief as a generalised ‘majority’ quantifier, interpreted as truth in most of the epistemically alternative states. This doxastic interpretation is of interest in knowledge-representation applications and it also holds an independent philosophical and technical appeal. The logic comprises an epistemic modal operator, a doxastic modal operator of consistent and complete belief and ‘bridge’ axioms which relate knowledge to belief. To capture the notion (...)
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  • A Strange Remark Attributed to Gödel.Lloyd Humberstone - 2003 - History and Philosophy of Logic 24 (1):39-44.
    We assemble material from the literature on matrix methodology for sentential logic?without claiming to present any new logical results?in order to show that Gödel once made (or at least, is quoted as having made) an uncharacteristically ill-considered remark in this area.
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  • Uniform Lyndon interpolation property in propositional modal logics.Taishi Kurahashi - 2020 - Archive for Mathematical Logic 59 (5-6):659-678.
    We introduce and investigate the notion of uniform Lyndon interpolation property which is a strengthening of both uniform interpolation property and Lyndon interpolation property. We prove several propositional modal logics including \, \, \ and \ enjoy ULIP. Our proofs are modifications of Visser’s proofs of uniform interpolation property using layered bisimulations Gödel’96, logical foundations of mathematics, computer science and physics—Kurt Gödel’s legacy, Springer, Berlin, 1996). Also we give a new upper bound on the complexity of uniform interpolants for \ (...)
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  • Proof Analysis in Intermediate Logics.Roy Dyckhoff & Sara Negri - 2012 - Archive for Mathematical Logic 51 (1-2):71-92.
    Using labelled formulae, a cut-free sequent calculus for intuitionistic propositional logic is presented, together with an easy cut-admissibility proof; both extend to cover, in a uniform fashion, all intermediate logics characterised by frames satisfying conditions expressible by one or more geometric implications. Each of these logics is embedded by the Gödel–McKinsey–Tarski translation into an extension of S4. Faithfulness of the embedding is proved in a simple and general way by constructive proof-theoretic methods, without appeal to semantics other than in the (...)
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  • Some Metacomplete Relevant Modal Logics.Takahiro Seki - 2013 - Studia Logica 101 (5):1115-1141.
    A logic is called metacomplete if formulas that are true in a certain preferred interpretation of that logic are theorems in its metalogic. In the area of relevant logics, metacompleteness is used to prove primeness, consistency, the admissibility of γ and so on. This paper discusses metacompleteness and its applications to a wider class of modal logics based on contractionless relevant logics and their neighbours using Slaney’s metavaluational technique.
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  • Possible-Worlds Semantics for Modal Notions Conceived as Predicates.Volker Halbach, Hannes Leitgeb & Philip Welch - 2003 - Journal of Philosophical Logic 32 (2):179-223.
    If □ is conceived as an operator, i.e., an expression that gives applied to a formula another formula, the expressive power of the language is severely restricted when compared to a language where □ is conceived as a predicate, i.e., an expression that yields a formula if it is applied to a term. This consideration favours the predicate approach. The predicate view, however, is threatened mainly by two problems: Some obvious predicate systems are inconsistent, and possible-worlds semantics for predicates of (...)
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  • Algorithmic Correspondence and Completeness in Modal Logic. IV. Semantic Extensions of SQEMA.Willem Conradie & Valentin Goranko - 2008 - Journal of Applied Non-Classical Logics 18 (2-3):175-211.
    In a previous work we introduced the algorithm \SQEMA\ for computing first-order equivalents and proving canonicity of modal formulae, and thus established a very general correspondence and canonical completeness result. \SQEMA\ is based on transformation rules, the most important of which employs a modal version of a result by Ackermann that enables elimination of an existentially quantified predicate variable in a formula, provided a certain negative polarity condition on that variable is satisfied. In this paper we develop several extensions of (...)
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  • BK-Lattices. Algebraic Semantics for Belnapian Modal Logics.Sergei P. Odintsov & E. I. Latkin - 2012 - Studia Logica 100 (1-2):319-338.
    Earlier algebraic semantics for Belnapian modal logics were defined in terms of twist-structures over modal algebras. In this paper we introduce the class of BK -lattices, show that this class coincides with the abstract closure of the class of twist-structures, and it forms a variety. We prove that the lattice of subvarieties of the variety of BK -lattices is dually isomorphic to the lattice of extensions of Belnapian modal logic BK . Finally, we describe invariants determining a twist-structure over a (...)
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  • Not Every Splitting Heyting or Interior Algebra is Finitely Presentable.Alex Citkin - 2012 - Studia Logica 100 (1-2):115-135.
    We give an example of a variety of Heyting algebras and of a splitting algebra in this variety that is not finitely presentable. Moreover, we show that the corresponding splitting pair cannot be defined by any finitely presentable algebra. Also, using the Gödel-McKinsey-Tarski translation and the Blok-Esakia theorem, we construct a variety of Grzegorczyk algebras with similar properties.
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  • The Modal Logic of Gödel Sentences.Hirohiko Kushida - 2010 - Journal of Philosophical Logic 39 (5):577 - 590.
    The modal logic of Gödel sentences, termed as GS, is introduced to analyze the logical properties of 'true but unprovable' sentences in formal arithmetic. The logic GS is, in a sense, dual to Grzegorczyk's Logic, where modality can be interpreted as 'true and provable'. As we show, GS and Grzegorczyk's Logic are, in fact, mutually embeddable. We prove Kripke completeness and arithmetical completeness for GS. GS is also an extended system of the logic of 'Essence and Accident' proposed by Marcos (...)
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  • The Power of a Propositional Constant.Robert Goldblatt & Tomasz Kowalski - 2012 - Journal of Philosophical Logic (1):1-20.
    Monomodal logic has exactly two maximally normal logics, which are also the only quasi-normal logics that are Post complete, and they are complete for validity in Kripke frames. Here we show that addition of a propositional constant to monomodal logic allows the construction of continuum many maximally normal logics that are not valid in any Kripke frame, or even in any complete modal algebra. We also construct continuum many quasi-normal Post complete logics that are not normal. The set of extensions (...)
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  • First-Order Logic in the Medvedev Lattice.Rutger Kuyper - 2015 - Studia Logica 103 (6):1185-1224.
    Kolmogorov introduced an informal calculus of problems in an attempt to provide a classical semantics for intuitionistic logic. This was later formalised by Medvedev and Muchnik as what has come to be called the Medvedev and Muchnik lattices. However, they only formalised this for propositional logic, while Kolmogorov also discussed the universal quantifier. We extend the work of Medvedev to first-order logic, using the notion of a first-order hyperdoctrine from categorical logic, to a structure which we will call the hyperdoctrine (...)
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  • Hereditarily Structurally Complete Superintuitionistic Deductive Systems.Alex Citkin - 2018 - Studia Logica 106 (4):827-856.
    Propositional logic is understood as a set of theorems defined by a deductive system: a set of axioms and a set of rules. Superintuitionistic logic is a logic extending intuitionistic propositional logic \. A rule is admissible for a logic if any substitution that makes each premise a theorem, makes the conclusion a theorem too. A deductive system \ is structurally complete if any rule admissible for the logic defined by \ is derivable in \. It is known that any (...)
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  • Interconnection of the Lattices of Extensions of Four Logics.Alexei Y. Muravitsky - 2017 - Logica Universalis 11 (2):253-281.
    We show that the lattices of the normal extensions of four well-known logics—propositional intuitionistic logic \, Grzegorczyk logic \, modalized Heyting calculus \ and \—can be joined in a commutative diagram. One connection of this diagram is an isomorphism between the lattices of the normal extensions of \ and \; we show some preservation properties of this isomorphism. Two other connections are join semilattice epimorphims of the lattice of the normal extensions of \ onto that of \ and of the (...)
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  • Intermediate Logics and the de Jongh Property.Dick de Jongh, Rineke Verbrugge & Albert Visser - 2011 - Archive for Mathematical Logic 50 (1-2):197-213.
    We prove that all extensions of Heyting Arithmetic with a logic that has the finite frame property possess the de Jongh property.
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  • First-Order Expressivity for S5-Models: Modal Vs. Two-Sorted Languages.Holger Sturm & Frank Wolter - 2001 - Journal of Philosophical Logic 30 (6):571-591.
    Standard models for model predicate logic consist of a Kripke frame whose worlds come equipped with relational structures. Both modal and two-sorted predicate logic are natural languages for speaking about such models. In this paper we compare their expressivity. We determine a fragment of the two-sorted language for which the modal language is expressively complete on S5-models. Decidable criteria for modal definability are presented.
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  • A Tableau Calculus for Propositional Intuitionistic Logic with a Refined Treatment of Nested Implications.Mauro Ferrari, Camillo Fiorentini & Guido Fiorino - 2009 - Journal of Applied Non-Classical Logics 19 (2):149-166.
    Since 1993, when Hudelmaier developed an O-space decision procedure for propositional Intuitionistic Logic, a lot of work has been done to improve the efficiency of the related proof-search algorithms. In this paper a tableau calculus using the signs T, F and F c with a new set of rules to treat signed formulas of the kind T → C) is provided. The main feature of the calculus is the reduction of both the non-determinism in proof-search and the width of proofs (...)
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  • Fatal Heyting Algebras and Forcing Persistent Sentences.Leo Esakia & Benedikt Löwe - 2012 - Studia Logica 100 (1-2):163-173.
    Hamkins and Löwe proved that the modal logic of forcing is S4.2 . In this paper, we consider its modal companion, the intermediate logic KC and relate it to the fatal Heyting algebra H ZFC of forcing persistent sentences. This Heyting algebra is equationally generic for the class of fatal Heyting algebras. Motivated by these results, we further analyse the class of fatal Heyting algebras.
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  • A Catalog ofWeak Many-Valued Modal Axioms and Their Corresponding Frame Classes.Costas D. Koutras - 2003 - Journal of Applied Non-Classical Logics 13 (1):47-71.
    In this paper we provide frame definability results for weak versions of classical modal axioms that can be expressed in Fitting's many-valued modal languages. These languages were introduced by M. Fitting in the early '90s and are built on Heyting algebras which serve as the space of truth values. The possible-worlds frames interpreting these languages are directed graphs whose edges are labelled with an element of the underlying Heyting algebra, providing us a form of many-valued accessibility relation. Weak axioms of (...)
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