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An essay in classical modal logic

Uppsala,: Filosofiska föreningen och Filosofiska institutionen vid Uppsala universitet (1971)

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  1. On the Logic of Belief and Propositional Quantification.Yifeng Ding - 2021 - Journal of Philosophical Logic 50 (5):1143-1198.
    We consider extending the modal logic KD45, commonly taken as the baseline system for belief, with propositional quantifiers that can be used to formalize natural language sentences such as “everything I believe is true” or “there is something that I neither believe nor disbelieve.” Our main results are axiomatizations of the logics with propositional quantifiers of natural classes of complete Boolean algebras with an operator validating KD45. Among them is the class of complete, atomic, and completely multiplicative BAOs validating KD45. (...)
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  • Modal and Hyperintensional Cognitivism and Modal and Hyperintensional Expressivism.David Elohim - manuscript
    This paper aims to provide a mathematically tractable background against which to model both modal and hyperintensional cognitivism and modal and hyperintensional expressivism. I argue that epistemic modal algebras, endowed with a hyperintensional, topic-sensitive epistemic two-dimensional truthmaker semantics, comprise a materially adequate fragment of the language of thought. I demonstrate, then, how modal expressivism can be regimented by modal coalgebraic automata, to which the above epistemic modal algebras are categorically dual. I examine five methods for modeling the dynamics of conceptual (...)
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  • The Moral Law and The Good in Temporal Modal Logic with Propositional Quantifiers.Daniel Rönnedal - 2020 - Australasian Journal of Logic 17 (1):22-69.
    The Moral Law is fulfilled iff everything that ought to be the case is the case, and The Good is realised in a possible world w at a time t iff w is deontically accessible from w at t. In this paper, I will introduce a set of temporal modal deontic systems with propositional quantifiers that can be used to prove some interesting theorems about The Moral Law and The Good. First, I will describe a set of systems without any (...)
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  • Hyperintensional logics for everyone.Igor Sedlár - 2019 - Synthese 198 (2):933-956.
    We introduce a general representation of unary hyperintensional modalities and study various hyperintensional modal logics based on the representation. It is shown that the major approaches to hyperintensionality known from the literature, that is state-based, syntactic and structuralist approaches, all correspond to special cases of the general framework. Completeness results pertaining to our hyperintensional modal logics are established.
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  • Standard State Space Models of Unawareness.Peter Fritz & Harvey Lederman - 2015 - Theoretical Aspects of Rationality and Knowledge 15.
    The impossibility theorem of Dekel, Lipman and Rustichini has been thought to demonstrate that standard state-space models cannot be used to represent unawareness. We first show that Dekel, Lipman and Rustichini do not establish this claim. We then distinguish three notions of awareness, and argue that although one of them may not be adequately modeled using standard state spaces, there is no reason to think that standard state spaces cannot provide models of the other two notions. In fact, standard space (...)
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  • On neighbourhood product of some Horn axiomatizable logics.Andrey Kudinov - 2018 - Logic Journal of the IGPL 26 (3):316-338.
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  • Characterization of Classes of Frames in Modal Language.Kazimierz Trzęsicki - 2012 - Studies in Logic, Grammar and Rhetoric 27 (40).
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  • Definability in the class of all -frames – computability and complexity.D. T. Georgiev - 2017 - Journal of Applied Non-Classical Logics 27 (1-2):1-26.
    In the basic modal language and in the basic modal language with the added universal modality, first-order definability of all formulas over the class of all frames is shown. Also, it is shown that the problems of modal definability of first-order sentences over the class of all frames in the languages and are both PSPACE-complete.
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  • Hyperintensional Foundations of Mathematical Platonism.David Elohim - manuscript
    This paper aims to provide hyperintensional foundations for mathematical platonism. I examine Hale and Wright's (2009) objections to the merits and need, in the defense of mathematical platonism and its epistemology, of the thesis of Necessitism. In response to Hale and Wright's objections to the role of epistemic and metaphysical modalities in providing justification for both the truth of abstraction principles and the success of mathematical predicate reference, I examine the Necessitist commitments of the abundant conception of properties endorsed by (...)
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  • (1 other version)Forms of Luminosity: Epistemic Modality and Hyperintensionality in Mathematics.David Elohim - 2017 - Dissertation, Arché, University of St Andrews
    This book concerns the foundations of epistemic modality and hyperintensionality and their applications to the philosophy of mathematics. David Elohim examines the nature of epistemic modality, when the modal operator is interpreted as concerning both apriority and conceivability, as well as states of knowledge and belief. The book demonstrates how epistemic modality and hyperintensionality relate to the computational theory of mind; metaphysical modality and hyperintensionality; the types of mathematical modality and hyperintensionality; to the epistemic status of large cardinal axioms, undecidable (...)
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  • Post Completeness in Congruential Modal Logics.Peter Fritz - 2016 - In Lev Beklemishev, Stéphane Demri & András Máté (eds.), Advances in Modal Logic, Volume 11. CSLI Publications. pp. 288-301.
    Well-known results due to David Makinson show that there are exactly two Post complete normal modal logics, that in both of them, the modal operator is truth-functional, and that every consistent normal modal logic can be extended to at least one of them. Lloyd Humberstone has recently shown that a natural analog of this result in congruential modal logics fails, by showing that not every congruential modal logic can be extended to one in which the modal operator is truth-functional. As (...)
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  • A perspective on modal sequent logic.Stephen Blamey & Lloyd Humberstone - 1991 - Publications of the Research Institute for Mathematical Sciences 27 (5):763-782.
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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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  • Temporal Reference in Linear Tense Logic.M. J. Cresswell - 2010 - Journal of Philosophical Logic 39 (2):173-200.
    The paper introduces a first-order theory in the language of predicate tense logic which contains a single simple axiom. It is shewn that this theory enables times to be referred to and sentences involving ‘now’ and ‘then’ to be formalised. The paper then compares this way of increasing the expressive capacity of predicate tense logic with other mechanisms, and indicates how to generalise the results to other modal and tense systems.
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  • On the Axiomatisation of Elgesem's Logic of Agency and Ability.Guido Governatori & Antonino Rotolo - 2005 - Journal of Philosophical Logic 34 (4):403-431.
    In this paper we show that the Hilbert system of agency and ability presented by Dag Elgesem is incomplete with respect to the intended semantics. We argue that completeness result may be easily regained. Finally, we shortly discuss some issues related to the philosophical intuition behind his approach. This is done by examining Elgesem's modal logic of agency and ability using semantics with different flavours.
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  • Combinations of Stit with Ought and Know.Ming Xu - 2015 - Journal of Philosophical Logic 44 (6):851-877.
    This paper presents a short survey of recent developments in stit theories, with an emphasis on combinations of stit and deontic logic, and those of stit and epistemic logic.
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  • (2 other versions)La connaissance commune en logique modale.Luc Lismont - 1993 - Mathematical Logic Quarterly 39 (1):115-130.
    The problem of Common Knowledge will be considered in two classes of models: a class K.* of Kripke models and a class S of Scott models. Two modal logic systems will be defined. Those systems, KC and MC, include an axiomatisation of Common Knowledge. We prove determination of each system by the corresponding class of models. MSC: 03B45, 68T25.
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  • Around provability logic.Leo Esakia - 2010 - Annals of Pure and Applied Logic 161 (2):174-184.
    We present some results on algebraic and modal analysis of polynomial distortions of the standard provability predicate in Peano Arithmetic PA, and investigate three provability-like modal systems related to the Gödel–Löb modal system GL. We also present a short review of relational and topological semantics for these systems, and describe the dual category of algebraic models of our main modal system.
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  • (1 other version)V čom sa nemôžete mýliť?Igor Sedlár - 2011 - Organon F: Medzinárodný Časopis Pre Analytickú Filozofiu 18 (3):351-362.
    The paper sketches an analysis of the notion of a self-fulfilling belief in terms of doxastic modal logic. We point out a connection between self-fulfilling beliefs and Moore’s paradox. Then we look at self-fulfilling beliefs in the context of neighborhood semantics. We argue that the analysis of several interesting self-fulfilling beliefs has to make essential use of propositional quantification.
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  • On the interpolation property of some intuitionistic modal logics.C. Luppi - 1996 - Archive for Mathematical Logic 35 (3):173-189.
    LetL be one of the intuitionistic modal logics considered in [7] (or one of its extensions) and letM L be the “algebraic semantics” ofL. In this paper we will extend toL the equivalence, proved in the classical case (see [6]), among he weak Craig interpolation theorem, the Robinson theorem and the amalgamation property of varietyM L. We will also prove the equivalence between the Craig interpolation theorem and the super-amalgamation property of varietyM L. Then we obtain the Craig interpolation theorem (...)
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  • (1 other version)A Descending Chain of Incomplete Extensions of Implicational S 5.Dolph Ulrich - 1985 - Mathematical Logic Quarterly 31 (13):201-208.
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  • Krister Segerberg on Logic of Actions.Robert Trypuz (ed.) - 2013 - Dordrecht, Netherland: Springer Verlag.
    Belief revision from the point of view of doxastic logic. Logic Journal of the IGPL, 3(4), 535–553. Segerberg, K. (1995). Conditional action. In G. Crocco, L. Fariñas, & A. Herzig (Eds.), Conditionals: From philosophy to computer science, Studies ...
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  • Identity, Leibniz’s Law and Non-Transitive Reasoning.Pablo Cobreros, Paul Egré, David Ripley & Robert van Rooij - 2013 - Metaphysica 14 (2):253-264.
    Arguments based on Leibniz's Law seem to show that there is no room for either indefinite or contingent identity. The arguments seem to prove too much, but their conclusion is hard to resist if we want to keep Leibniz's Law. We present a novel approach to this issue, based on an appropriate modification of the notion of logical consequence.
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  • Relational Semantics of the Lambek Calculus Extended with Classical Propositional Logic.Michael Kaminski & Nissim Francez - 2014 - Studia Logica 102 (3):479-497.
    We show that the relational semantics of the Lambek calculus, both nonassociative and associative, is also sound and complete for its extension with classical propositional logic. Then, using filtrations, we obtain the finite model property for the nonassociative Lambek calculus extended with classical propositional logic.
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  • What is the correct logic of necessity, actuality and apriority?Peter Fritz - 2014 - Review of Symbolic Logic 7 (3):385-414.
    This paper is concerned with a propositional modal logic with operators for necessity, actuality and apriority. The logic is characterized by a class of relational structures defined according to ideas of epistemic two-dimensional semantics, and can therefore be seen as formalizing the relations between necessity, actuality and apriority according to epistemic two-dimensional semantics. We can ask whether this logic is correct, in the sense that its theorems are all and only the informally valid formulas. This paper gives outlines of two (...)
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  • A logic for epistemic two-dimensional semantics.Peter Fritz - 2013 - Synthese 190 (10):1753-1770.
    Epistemic two-dimensional semantics is a theory in the philosophy of language that provides an account of meaning which is sensitive to the distinction between necessity and apriority. While this theory is usually presented in an informal manner, I take some steps in formalizing it in this paper. To do so, I define a semantics for a propositional modal logic with operators for the modalities of necessity, actuality, and apriority that captures the relevant ideas of epistemic two-dimensional semantics. I also describe (...)
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  • Modal logic: A semantic perspective.Patrick Blackburn & Johan van Benthem - 1988 - Ethics 98:501-517.
    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 BASIC MODAL LOGIC . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.
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  • Arithmetical necessity, provability and intuitionistic logic.Rob Goldblatt - 1978 - Theoria 44 (1):38-46.
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  • On systems of modal logic with provability interpretations.George Boolos - 1980 - Theoria 46 (1):7-18.
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  • Conditional obligation and positive permission for agents in time.Mark A. Brown - 2000 - Nordic Journal of Philosophical Logic 5 (2):83-111.
    This paper investigates the semantic treatment of conditional obligation, explicit permission (often called positive permission), and prohibition based on models with agents and branched time. In such models branches (rather than moments) are taken as basic, and the branching provides a way to represent the indeterminism which is normally presupposed by talk of free will, responsibility, action and ability. Careful treatment of the relation between ability and responsibility avoids many common problems with accounts of conditional obligation. Recognition of the generality (...)
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  • Natural deduction rules for a logic of vagueness.J. A. Burgess & I. L. Humberstone - 1987 - Erkenntnis 27 (2):197-229.
    Extant semantic theories for languages containing vague expressions violate intuition by delivering the same verdict on two principles of classical propositional logic: the law of noncontradiction and the law of excluded middle. Supervaluational treatments render both valid; many-Valued treatments, Neither. The core of this paper presents a natural deduction system, Sound and complete with respect to a 'mixed' semantics which validates the law of noncontradiction but not the law of excluded middle.
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  • Canonicity for intensional logics without iterative axioms.Timothy J. Surendonk - 1997 - Journal of Philosophical Logic 26 (4):391-409.
    David Lewis proved in 1974 that all logics without iterative axioms are weakly complete. In this paper we extend Lewis's ideas and provide a proof that such logics are canonical and so strongly complete. This paper also discusses the differences between relational and neighborhood frame semantics and poses a number of open questions about the latter.
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  • On superintuitionistic logics as fragments of proof logic extensions.A. V. Kuznetsov & A. Yu Muravitsky - 1986 - Studia Logica 45 (1):77 - 99.
    Coming fromI andCl, i.e. from intuitionistic and classical propositional calculi with the substitution rule postulated, and using the sign to add a new connective there have been considered here: Grzegorozyk's logicGrz, the proof logicG and the proof-intuitionistic logicI set up correspondingly by the calculiFor any calculus we denote by the set of all formulae of the calculus and by the lattice of all logics that are the extensions of the logic of the calculus, i.e. sets of formulae containing the axioms (...)
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  • Singulary extensional connectives: A closer look. [REVIEW]I. L. Humberstone - 1997 - Journal of Philosophical Logic 26 (3):341-356.
    The totality of extensional 1-ary connectives distinguishable in a logical framework allowing sequents with multiple or empty (alongside singleton) succedents form a lattice under a natural partial ordering relating one connective to another if all the inferential properties of the former are possessed by the latter. Here we give a complete description of that lattice; its Hasse diagram appears as Figure 1 in §2. Simple syntactic descriptions of the lattice elements are provided in §3; §§4 and 5 give some additional (...)
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  • Completeness and decidability results for some propositional modal logics containing “actually” operators.Dominic Gregory - 2001 - Journal of Philosophical Logic 30 (1):57-78.
    The addition of "actually" operators to modal languages allows us to capture important inferential behaviours which cannot be adequately captured in logics formulated in simpler languages. Previous work on modal logics containing "actually" operators has concentrated entirely upon extensions of KT5 and has employed a particular modeltheoretic treatment of them. This paper proves completeness and decidability results for a range of normal and nonnormal but quasi-normal propositional modal logics containing "actually" operators, the weakest of which are conservative extensions of K, (...)
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  • Refutation systems in modal logic.Valentin Goranko - 1994 - Studia Logica 53 (2):299 - 324.
    Complete deductive systems are constructed for the non-valid (refutable) formulae and sequents of some propositional modal logics. Thus, complete syntactic characterizations in the sense of Lukasiewicz are established for these logics and, in particular, purely syntactic decision procedures for them are obtained. The paper also contains some historical remarks and a general discussion on refutation systems.
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  • Cut-free sequent and tableau systems for propositional diodorean modal logics.Rajeev Goré - 1994 - Studia Logica 53 (3):433 - 457.
    We present sound, (weakly) complete and cut-free tableau systems for the propositional normal modal logicsS4.3, S4.3.1 andS4.14. When the modality is given a temporal interpretation, these logics respectively model time as a linear dense sequence of points; as a linear discrete sequence of points; and as a branching tree where each branch is a linear discrete sequence of points.Although cut-free, the last two systems do not possess the subformula property. But for any given finite set of formulaeX the superformulae involved (...)
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  • Neighborhoods for entailment.Lou Goble - 2003 - Journal of Philosophical Logic 32 (5):483-529.
    This paper presents a neighborhood semantics for logics of entailment. It begins with a minimal system Min that expresses the most fundamental assumptions about the entailment relation, and continues by examining various extensions that reflect further assumptions that might be made about entailment. This leads first to the logic B that is the basic relevant logic, and then to more powerful systems. All of these logics are proved to be sound and strongly complete. With B the neighborhood semantics meets the (...)
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  • Modal logic with names.George Gargov & Valentin Goranko - 1993 - Journal of Philosophical Logic 22 (6):607 - 636.
    We investigate an enrichment of the propositional modal language L with a "universal" modality ■ having semantics x ⊧ ■φ iff ∀y(y ⊧ φ), and a countable set of "names" - a special kind of propositional variables ranging over singleton sets of worlds. The obtained language ℒ $_{c}$ proves to have a great expressive power. It is equivalent with respect to modal definability to another enrichment ℒ(⍯) of ℒ, where ⍯ is an additional modality with the semantics x ⊧ ⍯φ (...)
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  • A completeness proof for a logic with an alternative necessity operator.Stéphane Demri - 1997 - Studia Logica 58 (1):99-112.
    We show the completeness of a Hilbert-style system LK defined by M. Valiev involving the knowledge operator K dedicated to the reasoning with incomplete information. The completeness proof uses a variant of Makinson's canonical model construction. Furthermore we prove that the theoremhood problem for LK is co-NP-complete, using techniques similar to those used to prove that the satisfiability problem for propositional S5 is NP-complete.
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  • Basic conditional logic.Brian F. Chellas - 1975 - Journal of Philosophical Logic 4 (2):133 - 153.
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  • Modal companions of intermediate propositional logics.Alexander Chagrov & Michael Zakharyashchev - 1992 - Studia Logica 51 (1):49 - 82.
    This paper is a survey of results concerning embeddings of intuitionistic propositional logic and its extensions into various classical modal systems.
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  • Variable-Sharing as Relevance.Shawn Standefer - forthcoming - In Andrew Tedder, Shawn Standefer & Igor Sedlar (eds.), New Directions in Relevant Logic. Springer.
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  • Model Theoretical Aspects of Weakly Aggregative Modal Logic.Jixin Liu, Yifeng Ding & Yanjing Wang - 2022 - Journal of Logic, Language and Information 31 (2):261-286.
    Weakly Aggregative Modal Logic ) is a collection of disguised polyadic modal logics with n-ary modalities whose arguments are all the same. \ has interesting applications on epistemic logic, deontic logic, and the logic of belief. In this paper, we study some basic model theoretical aspects of \. Specifically, we first give a van Benthem–Rosen characterization theorem of \ based on an intuitive notion of bisimulation. Then, in contrast to many well known normal or non-normal modal logics, we show that (...)
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  • Logic, Reasoning, and Rationality.Erik Weber, Joke Meheus & Dietlinde Wouters (eds.) - 2014 - Dordrecht, Netherland: Springer.
    This book contains a selection of the papers presented at the Logic, Reasoning and Rationality 2010 conference in Ghent. The conference aimed at stimulating the use of formal frameworks to explicate concrete cases of human reasoning, and conversely, to challenge scholars in formal studies by presenting them with interesting new cases of actual reasoning. According to the members of the Wiener Kreis, there was a strong connection between logic, reasoning, and rationality and that human reasoning is rational in so far (...)
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  • Closed without boundaries.Elia Zardini - 2020 - Synthese 199 (Suppl 3):641-679.
    The paper critically discusses two prominent arguments against closure principles for knowledge. The first one is the “argument from aggregation”, claiming that closure under conjunction has the consequence that, if one individually knows i premises, one also knows their i-fold conjunction—yet, every one of the premises might exhibit interesting positive epistemic properties while the i-fold conjunction might fail to do so. The second one is the “argument from concatenation”, claiming that closure under entailment has the consequence that, if one knows (...)
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  • Modal logic with non-deterministic semantics: Part I—Propositional case.Marcelo E. Coniglio, Fariñas Del Cerro Luis & Marques Peron Newton - 2020 - Logic Journal of the IGPL 28 (3):281-315.
    Dugundji proved in 1940 that most parts of standard modal systems cannot be characterized by a single finite deterministic matrix. In the eighties, Ivlev proposed a semantics of four-valued non-deterministic matrices, in order to characterize a hierarchy of weak modal logics without the necessitation rule. In a previous paper, we extended some systems of Ivlev’s hierarchy, also proposing weaker six-valued systems in which the axiom was replaced by the deontic axiom. In this paper, we propose even weaker systems, by eliminating (...)
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  • The polytopologies of transfinite provability logic.David Fernández-Duque - 2014 - Archive for Mathematical Logic 53 (3-4):385-431.
    Provability logics are modal or polymodal systems designed for modeling the behavior of Gödel’s provability predicate and its natural extensions. If Λ is any ordinal, the Gödel-Löb calculus GLPΛ contains one modality [λ] for each λ < Λ, representing provability predicates of increasing strength. GLPω has no non-trivial Kripke frames, but it is sound and complete for its topological semantics, as was shown by Icard for the variable-free fragment and more recently by Beklemishev and Gabelaia for the full logic. In (...)
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  • Proving unprovability in some normal modal logics.Valentin Goranko - 1991 - Bulletin of the Section of Logic 20 (1):23-29.
    This note considers deductive systems for the operator a of unprovability in some particular propositional normal modal logics. We give thus complete syntactic characterization of these logics in the sense of Lukasiewicz: for every formula  either `  or a  (but not both) is derivable. In particular, purely syntactic decision procedure is provided for the logics under considerations.
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  • (1 other version)A Descending Chain of Classical Logics for Which Necessitation Implies Regularity.Roy A. Benton - 1979 - Mathematical Logic Quarterly 25 (19‐24):289-291.
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