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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 1959). 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. Juan C. Agudelo & Walter Carnielli (2011). Polynomial Ring Calculus for Modal Logics: A New Semantics and Proof Method for Modalities. 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 (...)
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  2. Patrick Allo (2013). Adaptive Logic as a Modal Logic. 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, (...)
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  3. Giambattista Amati & Fiora Pirri (1994). A Uniform Tableau Method for Intuitionistic Modal Logics I. Studia Logica 53 (1):29 - 60.
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  4. Martin Amerbauer (1996). Cut-Free Tableau Calculi for Some Propositional Normal Modal Logics. 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 (...)
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  5. Alan Ross Anderson (1955). Correction to a Paper on Modal Logic. Journal of Symbolic Logic 20 (2):150.
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  6. Alan Ross Anderson (1954). Improved Decision Procedures for Lewis's Calculus S4 and Von Wright's Calculus M. Journal of Symbolic Logic 19 (3):201-214.
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  7. G. Aldo Antonelli & Richmond H. Thomason (2002). Representability in Second-Order Propositional Poly-Modal Logic. 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.
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  8. Krzysztof R. Apt & Robert van Rooij (eds.) (2008). New Perspectives on Games and Interactions. 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 ...
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  9. Lennart Åqvist (1973). Modal Logic with Subjunctive Conditionals and Dispositional Predicates. Journal of Philosophical Logic 2 (1):1 - 76.
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  10. Lennart Aqvist (1970). Review: D. Makinson, On Some Completeness Theorems in Modal Logic. [REVIEW] Journal of Symbolic Logic 35 (1):135-136.
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  11. Lennart Åqvist (1964). Results Concerning Some Modal Systems That Contain S. Journal of Symbolic Logic 29 (2):79-87.
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  12. Sergei Artemov & Tudor Protopopescu (2013). Discovering Knowability: A Semantic Analysis. 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 (...)
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  13. Steve Awodey & Jess Hughes, Modal Operators and the Formal Dual of Birkhoff's Completeness Theorem.
    Steve Awodey and Jesse Hughes. Modal Operators and the Formal Dual of Birkhoff's Completeness Theorem.
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  14. Guillermo Badia (2016). Bi-Simulating in Bi-Intuitionistic Logic. 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 (...)
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  15. Ruth C. Barcan (1946). A Functional Calculus of First Order Based on Strict Implication. Journal of Symbolic Logic 11 (1):1-16.
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  16. Ruth C. Barcan (1946). The Deduction Theorem in a Functional Calculus of First Order Based on Strict Implication. Journal of Symbolic Logic 11 (4):115-118.
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  17. R. J. Baxter (1973). On Some Models of Modal Logics. Notre Dame Journal of Formal Logic 14 (1):121-122.
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  18. Charles A. Baylis (1939). 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] Journal of Symbolic Logic 4 (1):27.
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  19. Charles A. Baylis (1937). Lewis C. I.. Emch's Calculus and Strict Implication. Journal of Symbolic Logic 2 (1):46.
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  20. Charles A. Baylis & C. I. Lewis (1937). Emch's Calculus and Strict Implication. Journal of Symbolic Logic 2 (1):46.
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  21. Bernhard Beckert & Rajeev GorÉ (2001). Free-Variable Tableaux for Propositional Modal Logics. 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.
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  22. Kaja Bednarska & Andrzej Indrzejczak (2015). Hypersequent Calculi for S5: The Methods of Cut Elimination. Logic and Logical Philosophy 24 (3):277–311.
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  23. Ben Shalom Dorit (2003). One Connection Between Standard Invariance Conditions on Modal Formulas and Generalized Quantifiers. 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 (...)
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  24. Hanoch Ben-Yami (2014). The Quantified Argument Calculus. 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 (...)
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  25. Hanoch Ben-Yami (2014). Why Rigidity? 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 (...)
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  26. Ermanno Bencivenga & Peter W. Woodruff (1981). A New Modal Language with the Λ Operator. 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.
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  27. Roy A. Benton (2002). A Simple Incomplete Extension of T Which is the Union of Two Complete Modal Logics with F.M.P. 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.
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  28. Gustav Bergmann (1960). The Philosophical Significance Modal Logic. Mind 69 (276):466-485.
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  29. Francesco Berto (2013). Impossible Worlds. 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 (...)
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  30. Francesco Berto (2012). Non-Normal Worlds and Representation. 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 (...)
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  31. Guram Bezhanishvili (2001). Glivenko Type Theorems for Intuitionistic Modal Logics. 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 (...)
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  32. Guram Bezhanishvili & Wesley H. Holliday (forthcoming). Locales, Nuclei, and Dragalin Frames. 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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  33. G. M. Bierman & V. C. V. de Paiva (2000). On an Intuitionistic Modal Logic. Studia Logica 65 (3):383-416.
    In this paper we consider an intuitionistic variant of the modal logic S4 (which we call IS4). The novelty of this paper is that we place particular importance on the natural deduction formulation of IS4— our formulation has several important metatheoretic properties. In addition, we study models of IS4— not in the framework of Kirpke semantics, but in the more general framework of category theory. This allows not only a more abstract definition of a whole class of models but also (...)
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  34. Marta Bílková (2007). Uniform Interpolation and Propositional Quantifiers in Modal Logics. Studia Logica 85 (1):1-31.
    We investigate uniform interpolants in propositional modal logics from the proof-theoretical point of view. Our approach is adopted from Pitts’ proof of uniform interpolationin intuitionistic propositional logic [15]. The method is based on a simulation of certain quantifiers ranging over propositional variables and uses a terminating sequent calculus for which structural rules are admissible.
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  35. Robert Binkley (1968). The Surprise Examination in Modal Logic. Journal of Philosophy 65 (5):127-136.
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  36. Patrick Blackburn (2002). Modal Logic: Graph. Darst. Cambridge University Press.
    This modern, advanced textbook reviews modal logic, a field which caught the attention of computer scientists in the late 1970's.
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  37. Patrick Blackburn, Maarten de Rijke & Yde Venema (2002). Modal Logic. Cambridge University Press.
    This modern, advanced textbook reviews modal logic, a field which caught the attention of computer scientists in the late 1970's.
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  38. Patrick Blackburn & Maarten Marx (2002). Remarks on Gregory's “Actually” Operator. Journal of Philosophical Logic 31 (3):281-288.
    In this note we show that the classical modal technology of Sahlqvist formulas gives quick proofs of the completeness theorems in [8] (D. Gregory, Completeness and decidability results for some propositional modal logics containing "actually" operators, Journal of Philosophical Logic 30(1): 57-78, 2001) and vastly generalizes them. Moreover, as a corollary, interpolation theorems for the logics considered in [8] are obtained. We then compare Gregory's modal language enriched with an "actually" operator with the work of Arthur Prior now known under (...)
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  39. Patrick Blackburn, Johan van Benthem & Frank Wolter (eds.) (2007). Handbook of Modal Logic. Elsevier.
    The Handbook of Modal Logic contains 20 articles, which collectively introduce contemporary modal logic, survey current research, and indicate the way in which the field is developing. The articles survey the field from a wide variety of perspectives: the underling theory is explored in depth, modern computational approaches are treated, and six major applications areas of modal logic (in Mathematics, Computer Science, Artificial Intelligence, Linguistics, Game Theory, and Philosophy) are surveyed. The book contains both well-written expository articles, suitable for beginners (...)
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  40. Stephen Blamey & Lloyd Humberstone (1991). A Perspective on Modal Sequent Logic. Publications of the Research Institute for Mathematical Sciences 27 (5):763-782.
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  41. Robert Blanché (1957). Sur la Structuration du Tableau Des Connectifs Interpropositionnels Binaires. Journal of Symbolic Logic 22 (1):17-18.
  42. W. J. Blok (1979). An Axiomatization of the Modal Theory of the Veiled Recession Frame. Studia Logica 38 (1):37 - 47.
    The veiled recession frame has served several times in the literature to provide examples of modal logics failing to have certain desirable properties. Makinson [4] was the first to use it in his presentation of a modal logic without the finite model property. Thomason [5] constructed a (rather complicated) logic whose Kripke frames have an accessibility relation which is reflexive and transitive, but which is satisfied by the (non-transitive) veiled recession frame, and hence incomplete. In Van Benthem [2] the frame (...)
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  43. Susanne Bobzien (2014). Higher-Order Vagueness and Numbers of Distinct Modalities. Disputatio (39):131-137.
    This paper shows that the following common assumption is false: that in modal-logical representations of higher-order vagueness, for there to be borderline cases to borderline cases ad infinitum, the number of possible distinct modalities in a modal system must be infinite. (Open access journal).
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  44. Susanne Bobzien (1993). Chrysippus' Modal Logic and Its Relation to Philo and Diodorus. In K. Doering & Th Ebert (eds.), Dialektiker und Stoiker. Franz Steiner. pp. 63--84.
    ABSTRACT: The modal systems of the Stoic logician Chrysippus and the two Hellenistic logicians Philo and Diodorus Cronus have survived in a fragmentary state in several sources. From these it is clear that Chrysippus was acquainted with Philo’s and Diodorus’ modal notions, and also that he developed his own in contrast of Diodorus’ and in some way incorporated Philo’s. The goal of this paper is to reconstruct the three modal systems, including their modal definitions and modal theorems, and to make (...)
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  45. Susanne Bobzien (1986). Die stoische Modallogik (Stoic Modal Logic). Königshausen & Neumann.
    ABSTRACT: Part 1 discusses the Stoic notion of propositions (assertibles, axiomata): their definition; their truth-criteria; the relation between sentence and proposition; propositions that perish; propositions that change their truth-value; the temporal dependency of propositions; the temporal dependency of the Stoic notion of truth; pseudo-dates in propositions. Part 2 discusses Stoic modal logic: the Stoic definitions of their modal notions (possibility, impossibility, necessity, non-necessity); the logical relations between the modalities; modalities as properties of propositions; contingent propositions; the relation between the Stoic (...)
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  46. Giacomo Bonanno (2008). Belief Revision in a Temporal Framework. In Krzysztof Apt & Robert van Rooij (eds.), New Perspectives on Games and Interaction. Amsterdam University Press.
    The theory of belief revision deals with (rational) changes in beliefs in response to new information. In the literature a distinction has been drawn between belief revision and belief update (see [6]). The former deals with situations where the objective facts describing the world do not change (so that only the beliefs of the agent change over time), while the letter allows for situations where both the facts and the doxastic state of the agent change over time. We focus on (...)
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  47. Giacomo Bonanno (2002). Modal Logic and Game Theory: Two Alternative Approaches. Risk Decision and Policy 7:309-324.
    Two views of game theory are discussed: (1) game theory as a description of the behavior of rational individuals who recognize each other’s rationality and reasoning abilities, and (2) game theory as an internally consistent recommendation to individuals on how to act in interactive situations. It is shown that the same mathematical tool, namely modal logic, can be used to explicitly model both views.
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  48. George Boolos (1979). The Unprovability of Consistency: An Essay in Modal Logic. Cambridge University Press.
    The Unprovability of Consistency is concerned with connections between two branches of logic: proof theory and modal logic. Modal logic is the study of the principles that govern the concepts of necessity and possibility; proof theory is, in part, the study of those that govern provability and consistency. In this book, George Boolos looks at the principles of provability from the standpoint of modal logic. In doing so, he provides two perspectives on a debate in modal logic that has persisted (...)
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  49. George S. Boolos (1993). The Logic of Provability. Cambridge University Press.
    This book, written by one of the most distinguished of contemporary philosophers of mathematics, is a fully rewritten and updated successor to the author's earlier The Unprovability of Consistency. Its subject is the relation between provability and modal logic, a branch of logic invented by Aristotle but much disparaged by philosophers and virtually ignored by mathematicians. Here it receives its first scientific application since its invention. Modal logic is concerned with the notions of necessity and possibility. What George Boolos does (...)
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  50. George Boolos & Giovanni Sambin (1985). An Incomplete System of Modal Logic. Journal of Philosophical Logic 14 (4):351 - 358.
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