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About this topic
Summary Modal logic's premise, which has been disputed, is that "it is possible that" and other related natural language words and phrases express logical concepts, as do the words "and", "or", and "not". Modal logic is then the study of this set of related concepts. In its modern form, this work was initiated axiomatically by C.I.Lewis and continued model-theoretically by Saul Kripke and others.
Key works C. I. Lewis's axiomatic approach was set out in Symbolic Logic (1932), co-authored with C.H. Langford (Lewis 1932). Kripke's model-theoretic work began with Kripke 1963.
Introductions Blackburn et al 2007; Nino & Freund Ma 2008; Fitting unknown; Cresswell & Hughes 1996 (highly recommended); Lemmon 1977
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  1. A Note on Comparing Probabilistic and Modal Logics of Conditionals.Ernest W. Adams - 1977 - Theoria 43 (3):186-194.
  2. Polynomial Ring Calculus for Modal Logics: A New Semantics and Proof Method for Modalities.Juan C. Agudelo & Walter Carnielli - 2011 - Review of Symbolic Logic 4 (1):150-170.
    A new (sound and complete) proof style adequate for modal logics is defined from the polynomial ring calculus (PRC). The new semantics not only expresses truth conditions of modal formulas by means of polynomials, but also permits to perform deductions through polynomial handling. This paper also investigates relationships among the PRC here defined, the algebraic semantics for modal logics, equational logics, the Dijkstra–Scholten equational-proof style, and rewriting systems. The method proposed is throughly exemplified for S5, and can be easily extended (...)
  3. Modal Logics for Reasoning About Infinite Unions and Intersections of Binary Relations.Natasha Alechina, Philippe Balbiani & Dmitry Shkatov - 2012 - Journal of Applied Non-Classical Logics 22 (4):275 - 294.
    (2012). Modal logics for reasoning about infinite unions and intersections of binary relations. Journal of Applied Non-Classical Logics: Vol. 22, No. 4, pp. 275-294. doi: 10.1080/11663081.2012.705960.
  4. A General Method for Proving Decidability of Intuitionistic Modal Logics.Natasha Alechina & Dmitry Shkatov - 2006 - Journal of Applied Logic 4 (3):219-230.
  5. Adaptive Logic as a Modal Logic.Patrick Allo - 2013 - Studia Logica 101 (5):933-958.
    Modal logics have in the past been used as a unifying framework for the minimality semantics used in defeasible inference, conditional logic, and belief revision. The main aim of the present paper is to add adaptive logics, a general framework for a wide range of defeasible reasoning forms developed by Diderik Batens and his co-workers, to the growing list of formalisms that can be studied with the tools and methods of contemporary modal logic. By characterising the class of abnormality models, (...)
  6. A Uniform Tableau Method for Intuitionistic Modal Logics I.Giambattista Amati & Fiora Pirri - 1994 - Studia Logica 53 (1):29 - 60.
    We present tableau systems and sequent calculi for the intuitionistic analogues IK, ID, IT, IKB, IKDB, IB, IK4, IKD4, IS4, IKB4, IK5, IKD5, IK45, IKD45 and IS5 of the normal classical modal logics. We provide soundness and completeness theorems with respect to the models of intuitionistic logic enriched by a modal accessibility relation, as proposed by G. Fischer Servi. We then show the disjunction property for IK, ID, IT, IKB, IKDB, IB, IK4, IKD4, IS4, IKB4, IK5, IK45 and IS5. We (...)
  7. Cut-Free Tableau Calculi for Some Propositional Normal Modal Logics.Martin Amerbauer - 1996 - Studia Logica 57 (2-3):359 - 372.
    We give sound and complete tableau and sequent calculi for the prepositional normal modal logics S4.04, K4B and G 0(these logics are the smallest normal modal logics containing K and the schemata A A, A A and A ( A); A A and AA; A A and ((A A) A) A resp.) with the following properties: the calculi for S4.04 and G 0are cut-free and have the interpolation property, the calculus for K4B contains a restricted version of the cut-rule, the (...)
  8. Correction to a Paper on Modal Logic.Alan Ross Anderson - 1955 - Journal of Symbolic Logic 20 (2):150.
  9. Improved Decision Procedures for Lewis's Calculus S4 and Von Wright's Calculus M.Alan Ross Anderson - 1954 - Journal of Symbolic Logic 19 (3):201-214.
  10. Representability in Second-Order Propositional Poly-Modal Logic.G. Aldo Antonelli & Richmond H. Thomason - 2002 - Journal of Symbolic Logic 67 (3):1039-1054.
    A propositional system of modal logic is second-order if it contains quantifiers ∀p and ∃p, which, in the standard interpretation, are construed as ranging over sets of possible worlds (propositions). Most second-order systems of modal logic are highly intractable; for instance, when augmented with propositional quantifiers, K, B, T, K4 and S4 all become effectively equivalent to full second-order logic. An exception is S5, which, being interpretable in monadic second-order logic, is decidable.
  11. New Perspectives on Games and Interactions.Krzysztof R. Apt & Robert van Rooij (eds.) - 2008 - Amsterdam University Press.
    This volume is a collection of papers presented at the colloquium, and it testifies to the growing importance of game theory as a tool that can capture concepts ...
  12. Modal Logic with Subjunctive Conditionals and Dispositional Predicates.Lennart Åqvist - 1973 - Journal of Philosophical Logic 2 (1):1 - 76.
  13. Review: D. Makinson, On Some Completeness Theorems in Modal Logic. [REVIEW]Lennart Aqvist - 1970 - Journal of Symbolic Logic 35 (1):135-136.
  14. Results Concerning Some Modal Systems That Contain S.Lennart Åqvist - 1964 - Journal of Symbolic Logic 29 (2):79-87.
  15. Discovering Knowability: A Semantic Analysis.Sergei Artemov & Tudor Protopopescu - 2013 - Synthese 190 (16):3349-3376.
    In this paper, we provide a semantic analysis of the well-known knowability paradox stemming from the Church–Fitch observation that the meaningful knowability principle /all truths are knowable/, when expressed as a bi-modal principle F --> K♢F, yields an unacceptable omniscience property /all truths are known/. We offer an alternative semantic proof of this fact independent of the Church–Fitch argument. This shows that the knowability paradox is not intrinsically related to the Church–Fitch proof, nor to the Moore sentence upon which it (...)
  16. Topology and Modality: The Topological Interpretation of First-Order Modal Logic: Topology and Modality.Steve Awodey - 2008 - Review of Symbolic Logic 1 (2):146-166.
    As McKinsey and Tarski showed, the Stone representation theorem for Boolean algebras extends to algebras with operators to give topological semantics for propositional modal logic, in which the “necessity” operation is modeled by taking the interior of an arbitrary subset of a topological space. In this article, the topological interpretation is extended in a natural way to arbitrary theories of full first-order logic. The resulting system of S4 first-order modal logic is complete with respect to such topological semantics.
  17. Local Realizability Toposes and a Modal Logic for Computability.Steve Awodey, Lars Birkedal & Dana Scott - unknown
    This work is a step toward the development of a logic for types and computation that includes not only the usual spaces of mathematics and constructions, but also spaces from logic and domain theory. Using realizability, we investigate a configuration of three toposes that we regard as describing a notion of relative computability. Attention is focussed on a certain local map of toposes, which we first study axiomatically, and then by deriving a modal calculus as its internal logic. The resulting (...)
  18. Modal Operators and the Formal Dual of Birkhoff's Completeness Theorem.Steve Awodey & Jess Hughes - unknown
    Steve Awodey and Jesse Hughes. Modal Operators and the Formal Dual of Birkhoff's Completeness Theorem.
  19. Connecting Many-Sorted Theories.Franz Baader & Silvio Ghilardi - 2007 - Journal of Symbolic Logic 72 (2):535 - 583.
    Basically, the connection of two many-sorted theories is obtained by taking their disjoint union, and then connecting the two parts through connection functions that must behave like homomorphisms on the shared signature. We determine conditions under which decidability of the validity of universal formulae in the component theories transfers to their connection. In addition, we consider variants of the basic connection scheme. Our results can be seen as a generalization of the so-called E-connection approach for combining modal logics to an (...)
  20. Review: Isabel C. Hungerland, Contextual Implication. [REVIEW]John Bacon - 1970 - Journal of Symbolic Logic 35 (3):458-458.
  21. Bi-Simulating in Bi-Intuitionistic Logic.Guillermo Badia - 2016 - Studia Logica 104 (5):1037-1050.
    Bi-intuitionistic logic is the result of adding the dual of intuitionistic implication to intuitionistic logic. In this note, we characterize the expressive power of this logic by showing that the first order formulas equivalent to translations of bi-intuitionistic propositional formulas are exactly those preserved under bi-intuitionistic directed bisimulations. The proof technique is originally due to Lindstrom and, in contrast to the most common proofs of this kind of result, it does not use the machinery of neither saturated models nor elementary (...)
  22. Probabilization of Logics: Completeness and Decidability. [REVIEW]Pedro Baltazar - 2013 - Logica Universalis 7 (4):403-440.
    The probabilization of a logic system consists of enriching the language (the formulas) and the semantics (the models) with probabilistic features. Such an operation is said to be exogenous if the enrichment is done on top, without internal changes to the structure, and is called endogenous otherwise. These two different enrichments can be applied simultaneously to the language and semantics of a same logic. We address the problem of studying the transference of metaproperties, such as completeness and decidability, to the (...)
  23. A Functional Calculus of First Order Based on Strict Implication.Ruth C. Barcan - 1946 - Journal of Symbolic Logic 11 (1):1-16.
  24. The Deduction Theorem in a Functional Calculus of First Order Based on Strict Implication.Ruth C. Barcan - 1946 - Journal of Symbolic Logic 11 (4):115-118.
  25. On Some Models of Modal Logics.R. J. Baxter - 1973 - Notre Dame Journal of Formal Logic 14 (1):121-122.
  26. Lemmon E. J.. Quantifiers and Modal Operators. Proceedings of the Aristotelian Society, Vol. 58 , Pp. 245–268.Arnould Bayart - 1966 - Journal of Symbolic Logic 31 (2):275-276.
  27. Review: E. J. Lemmon, Quantifiers and Modal Operators. [REVIEW]Arnould Bayart - 1966 - Journal of Symbolic Logic 31 (2):275-276.
  28. Review: Tang Tsao-Chen, Algebraic Postulates and a Geometric Interpretation for the Lewis Calculus of Strict Implication. [REVIEW]Charles A. Baylis - 1939 - Journal of Symbolic Logic 4 (1):27-27.
  29. Tsao-Chen Tang. Algebraic Postulates and a Geometric Interpretation for the Lewis Calculus of Strict Implication. Bulletin of the American Mathematical Society, Vol. 44 , Pp. 737–744. [REVIEW]Charles A. Baylis - 1939 - Journal of Symbolic Logic 4 (1):27.
  30. Lewis C. I.. Emch's Calculus and Strict Implication.Charles A. Baylis - 1937 - Journal of Symbolic Logic 2 (1):46.
  31. Review: C. I. Lewis, Emch's Calculus and Strict Implication. [REVIEW]Charles A. Baylis - 1937 - Journal of Symbolic Logic 2 (1):46-46.
  32. Review: Roger W. Holmes, Classical and Relational Logic. [REVIEW]Charles A. Baylis - 1936 - Journal of Symbolic Logic 1 (2):69-69.
  33. Emch's Calculus and Strict Implication.Charles A. Baylis & C. I. Lewis - 1937 - Journal of Symbolic Logic 2 (1):46.
  34. Logic: The Basics (2nd Edition).Jc Beall & Shay A. Logan - 2017 - Routledge.
    Logic: the Basics is an accessible introduction to the core philosophy topic of standard logic. Focussing on traditional Classical Logic the book deals with topics such as mathematical preliminaries, propositional logic, monadic quantified logic, polyadic quantified logic, and English and standard ‘symbolic transitions’. With exercises and sample answers throughout this thoroughly revised new edition not only comprehensively covers the core topics at introductory level but also gives the reader an idea of how they can take their knowledge further and the (...)
  35. Free-Variable Tableaux for Propositional Modal Logics.Bernhard Beckert & Rajeev GorÉ - 2001 - Studia Logica 69 (1):59-96.
    Free-variable semantic tableaux are a well-established technique for first-order theorem proving where free variables act as a meta-linguistic device for tracking the eigenvariables used during proof search. We present the theoretical foundations to extend this technique to propositional modal logics, including non-trivial rigorous proofs of soundness and completeness, and also present various techniques that improve the efficiency of the basic naive method for such tableaux.
  36. Hypersequent Calculi for S5: The Methods of Cut Elimination.Kaja Bednarska & Andrzej Indrzejczak - 2015 - Logic and Logical Philosophy 24 (3):277–311.
  37. One Connection Between Standard Invariance Conditions on Modal Formulas and Generalized Quantifiers.Ben Shalom Dorit - 2003 - Journal of Logic, Language and Information 12 (1):47-52.
    The language of standard propositional modal logic has one operator (? or ?), that can be thought of as being determined by the quantifiers ? or ?, respectively: for example, a formula of the form ?F is true at a point s just in case all the immediate successors of s verify F.This paper uses a propositional modal language with one operator determined by a generalized quantifier to discuss a simple connection between standard invariance conditions on modal formulas and generalized (...)
  38. The Quantified Argument Calculus.Hanoch Ben-Yami - 2014 - Review of Symbolic Logic 7 (1):120-146.
    I develop a formal logic in which quantified arguments occur in argument positions of predicates. This logic also incorporates negative predication, anaphora and converse relation terms, namely, additional syntactic features of natural language. In these and additional respects, it represents the logic of natural language more adequately than does any version of Frege’s Predicate Calculus. I first introduce the system’s main ideas and familiarize it by means of translations of natural language sentences. I then develop a formal system built on (...)
  39. Why Rigidity?Hanoch Ben-Yami - 2014 - In J. Berg (ed.), Naming, Necessity and More: Explorations in the Philosophical Work of Saul Kripke. Palgrave. pp. 3-21.
    In Naming and Necessity Kripke argues 'intuitively' that names are rigid. Unlike Kripke, Ben-Yami first introduces and justifies the Principle of the Independence of Reference (PIR), according to which the reference of a name is independent of what is said in the rest of the sentence containing it. Ben-Yami then derives rigidity, or something close to it, from the PIR. Additional aspects of the use of names and other expressions in modal contexts, explained by the PIR but not by the (...)
  40. A New Modal Language with the Λ Operator.Ermanno Bencivenga & Peter W. Woodruff - 1981 - Studia Logica 40 (4):383 - 389.
    A system of modal logic with the operator is proposed, and proved complete. In contrast with a previous one by Stalnaker and Thomason, this system does not require two categories of singular terms.
  41. Modal Deduction in Second-Order Logic and Set Theory: II.Benthem Johan Van, D'Agostino Giovanna, Montanari Angelo & Policriti Alberto - 1998 - Studia Logica 60 (3):387 - 420.
    In this paper, we generalize the set-theoretic translation method for polymodal logic introduced in [11] to extended modal logics. Instead of devising an ad-hoc translation for each logic, we develop a general framework within which a number of extended modal logics can be dealt with. We first extend the basic set-theoretic translation method to weak monadic second-order logic through a suitable change in the underlying set theory that connects up in interesting ways with constructibility; then, we show how to tailor (...)
  42. Dynamic Logics of Evidence-Based Beliefs.J. Benthem & E. Pacuit - 2011 - Studia Logica 99 (1-3):61-92.
    This paper adds evidence structure to standard models of belief, in the form of families of sets of worlds. We show how these more fine-grained models support natural actions of “evidence management”, ranging from update with external new information to internal rearrangement. We show how this perspective leads to new richer languages for existing neighborhood semantics for modal logic. Our main results are relative completeness theorems for the resulting dynamic logic of evidence.
  43. Review: G. E. Hughes, M. J. Cresswell, A Companion to Modal Logic. [REVIEW]Johan Van Benthem - 1986 - Journal of Symbolic Logic 51 (3):824-826.
  44. A Simple Incomplete Extension of T Which is the Union of Two Complete Modal Logics with F.M.P.Roy A. Benton - 2002 - Journal of Philosophical Logic 31 (6):527-541.
    I present here a modal extension of T called KTLM which is, by several measures, the simplest modal extension of T yet presented. Its axiom uses only one sentence letter and has a modal depth of 2. Furthermore, KTLM can be realized as the logical union of two logics KM and KTL which each have the finite model property (f.m.p.), and so themselves are complete. Each of these two component logics has independent interest as well.
  45. The Philosophical Significance Modal Logic.Gustav Bergmann - 1960 - Mind 69 (276):466-485.
  46. Impossible Worlds.Francesco Berto - 2013 - Stanford Encyclopedia of Philosophy (2013).
    It is a venerable slogan due to David Hume, and inherited by the empiricist tradition, that the impossible cannot be believed, or even conceived. In Positivismus und Realismus, Moritz Schlick claimed that, while the merely practically impossible is still conceivable, the logically impossible, such as an explicit inconsistency, is simply unthinkable. -/- An opposite philosophical tradition, however, maintains that inconsistencies and logical impossibilities are thinkable, and sometimes believable, too. In the Science of Logic, Hegel already complained against “one of the (...)
  47. Non-Normal Worlds and Representation.Francesco Berto - 2012 - In Michal Peliš & Vít Punčochář (eds.), The Logica Yearbook. College Publications.
    World semantics for relevant logics include so-called non-normal or impossible worlds providing model-theoretic counterexamples to such irrelevant entailments as (A ∧ ¬A) → B, A → (B∨¬B), or A → (B → B). Some well-known views interpret non-normal worlds as information states. If so, they can plausibly model our ability of conceiving or representing logical impossibilities. The phenomenon is explored by combining a formal setting with philosophical discussion. I take Priest’s basic relevant logic N4 and extend it, on the syntactic (...)
  48. Glivenko Type Theorems for Intuitionistic Modal Logics.Guram Bezhanishvili - 2001 - Studia Logica 67 (1):89-109.
    In this article we deal with Glivenko type theorems for intuitionistic modal logics over Prior's MIPC. We examine the problems which appear in proving Glivenko type theorems when passing from the intuitionistic propositional logic Intto MIPC. As a result we obtain two different versions of Glivenko's theorem for logics over MIPC. Since MIPCcan be thought of as a one-variable fragment of the intuitionistic predicate logic Q-Int, one of the versions of Glivenko's theorem for logics over MIPCis closely related to that (...)
  49. Modal Logics of Metric Spaces.Guram Bezhanishvili, David Gabelaia & Joel Lucero-Bryan - 2015 - Review of Symbolic Logic 8 (1):178-191.
  50. Locales, Nuclei, and Dragalin Frames.Guram Bezhanishvili & Wesley H. Holliday - forthcoming - In Lev Beklemishev & Stéphane Demri (eds.), Advances in Modal Logic, Vol. 11. College Publications.
    It is a classic result in lattice theory that a poset is a complete lattice iff it can be realized as fixpoints of a closure operator on a powerset. Dragalin [9,10] observed that a poset is a locale (complete Heyting algebra) iff it can be realized as fixpoints of a nucleus on the locale of upsets of a poset. He also showed how to generate a nucleus on upsets by adding a structure of “paths” to a poset, forming what we (...)
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