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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; Cocchiarella 2008; Fitting unknown; Hughes & Cresswell 1996 (highly recommended); Lemmon 1977
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  1. Giambattista Amati & Fiora Pirri (1994). A Uniform Tableau Method for Intuitionistic Modal Logics I. Studia Logica 53 (1):29 - 60.
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  2. 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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  3. Alan Ross Anderson (1955). Correction to a Paper on Modal Logic. Journal of Symbolic Logic 20 (2):150.
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  4. 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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  5. 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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  6. Lennart Åqvist (1973). Modal Logic with Subjunctive Conditionals and Dispositional Predicates. Journal of Philosophical Logic 2 (1):1 - 76.
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  7. Lennart Åqvist (1964). Results Concerning Some Modal Systems That Contain S. Journal of Symbolic Logic 29 (2):79-87.
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  8. 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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  9. 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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  10. R. J. Baxter (1973). On Some Models of Modal Logics. Notre Dame Journal of Formal Logic 14 (1):121-122.
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  11. 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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  12. Dorit Ben Shalom (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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  13. Hanoch Ben-Yami (forthcoming). The Quantified Argument Calculus. Review of Symbolic Logic:1-27.
    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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  14. 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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  15. 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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  16. Gustav Bergmann (1960). The Philosophical Significance Modal Logic. Mind 69 (276):466-485.
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  17. 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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  18. 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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  19. 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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  20. 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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  21. 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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  22. Robert Binkley (1968). The Surprise Examination in Modal Logic. Journal of Philosophy 65 (5):127-136.
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  23. 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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  24. 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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  25. 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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  26. Susanne Bobzien (1993). Chrysippus' Modal Logic and Its Relation to Philo and Diodorus. In K. Doering & Th Ebert (eds.), Dialektiker und Stoiker. Franz Steiner. 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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  27. 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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  28. 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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  29. 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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  30. Giacomo Bonanno (2000). Common Belief with the Logic of Individual Belief. Mathematical Logic Quarterly 46 (1):49-52.
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  31. George 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 (CUP, 1979). Modal logic is concerned with the notions of necessity and possibility. What George Boolos does is to show how the concepts, techniques and methods of modal logic shed brilliant light on the most important logical discovery of the twentieth century: the incompleteness theorems of Kurt Godel and the 'self referential' (...)
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  32. 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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  33. George Boolos & Giovanni Sambin (1985). An Incomplete System of Modal Logic. Journal of Philosophical Logic 14 (4):351 - 358.
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  34. Marco Borga (1983). On Some Proof Theoretical Properties of the Modal Logic GL. Studia Logica 42 (4):453 - 459.
    This paper deals with the system of modal logicGL, in particular with a formulation of it in terms of sequents. We prove some proof theoretical properties ofGL that allow to get the cut-elimination theorem according to Gentzen's procedure, that is, by double induction on grade and rank.
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  35. Milan Božić & Kosta Došen (1984). Models for Normal Intuitionistic Modal Logics. Studia Logica 43 (3):217 - 245.
    Kripke-style models with two accessibility relations, one intuitionistic and the other modal, are given for analogues of the modal systemK based on Heyting's prepositional logic. It is shown that these two relations can combine with each other in various ways. Soundness and completeness are proved for systems with only the necessity operator, or only the possibility operator, or both. Embeddings in modal systems with several modal operators, based on classical propositional logic, are also considered. This paper lays the ground for (...)
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  36. Torben Braüner (2002). Modal Logic, Truth, and the Master Modality. Journal of Philosophical Logic 31 (4):359-386.
    In the paper (Braüner, 2001) we gave a minimal condition for the existence of a homophonic theory of truth for a modal or tense logic. In the present paper we generalise this result to arbitrary modal logics and we also show that a modal logic permits the existence of a homophonic theory of truth if and only if it permits the definition of a socalled master modality. Moreover, we explore a connection between the master modality and hybrid logic: We show (...)
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  37. Manuel Bremer (2005). Book Reviews:Patrick Blackburn, Maarten de Rijke and Yde Venema, Modal Logic, Cambridge: Cambridge University Press, 2002, XXII + 554 Pp., US$53.00, ISBN 0-52152-714-7 (Paperback). [REVIEW] Minds and Machines 15 (1):126-129.
  38. David Bresolin, Joanna Golinska-Pilarek & Ewa Orlowska (2006). Relational Dual Tableaux for Interval Temporal Logics. Journal of Applied Non-Classical Logics 16 (3-4):251–277.
    Interval temporal logics provide both an insight into a nature of time and a framework for temporal reasoning in various areas of computer science. In this paper we present sound and complete relational proof systems in the style of dual tableaux for relational logics associated with modal logics of temporal intervals and we prove that the systems enable us to verify validity and entailment of these temporal logics. We show how to incorporate in the systems various relations between intervals and/or (...)
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  39. Baruch A. Brody (1972). De Re and de Dicto Interpretations of Modal Logic or a Return to an Aristotelean Essentialism. Philosophia 2 (1-2):117-136.
  40. Jan Broersen, Rosja Mastop, John-Jules Meyer & Paolo Turrini (2009). Determining the Environment: A Modal Logic for Closed Interaction. Synthese 169 (2):351 - 369.
    The aim of the work is to provide a language to reason about Closed Interactions, i.e. all those situations in which the outcomes of an interaction can be determined by the agents themselves and in which the environment cannot interfere with they are able to determine. We will see that two different interpretations can be given of this restriction, both stemming from Pauly Representation Theorem. We will identify such restrictions and axiomatize their logic. We will apply the formal tools to (...)
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  41. Eric M. Brown, Logic II: The Theory of Propositions.
    This is part two of a complete exposition of Logic, in which there is a radically new synthesis of Aristotelian-Scholastic Logic with modern Logic. Part II is the presentation of the theory of propositions. Simple, composite, atomic, compound, modal, and tensed propositions are all examined. Valid consequences and propositional logical identities are rigorously proven. Modal logic is rigorously defined and proven. This is the first work of Logic known to unite Aristotelian logic and modern logic using scholastic logic as the (...)
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  42. Carlos Caleiro, Luca Viganò & Marco Volpe (2013). On the Mosaic Method for Many-Dimensional Modal Logics: A Case Study Combining Tense and Modal Operators. [REVIEW] Logica Universalis 7 (1):33-69.
    We present an extension of the mosaic method aimed at capturing many-dimensional modal logics. As a proof-of-concept, we define the method for logics arising from the combination of linear tense operators with an “orthogonal” S5-like modality. We show that the existence of a model for a given set of formulas is equivalent to the existence of a suitable set of partial models, called mosaics, and apply the technique not only in obtaining a proof of decidability and a proof of completeness (...)
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  43. Michael J. Carroll (1978). An Axiomatization of S13. Philosophia 8 (2-3):381-382.
    Specifies an axiomatization of the system S13 of modal logic. Referenced in Cocchiarella & Freund "Modal Logic: an Introduction to its Syntax and Semantics", Oxford University Press, 2008.
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  44. Michael J. Carroll (1976). On Interpreting the S5 Propositional Calculus: An Essay in Philosophical Logic. Dissertation, University of Iowa
    Discusses alternative interpretations of the modal operators, for the modal propositional logic S5.
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  45. Alexander Chagrov (1997). Modal Logic. Oxford University Press.
    For a novice this book is a mathematically-oriented introduction to modal logic, the discipline within mathematical logic studying mathematical models of reasoning which involve various kinds of modal operators. It starts with very fundamental concepts and gradually proceeds to the front line of current research, introducing in full details the modern semantic and algebraic apparatus and covering practically all classical results in the field. It contains both numerous exercises and open problems, and presupposes only minimal knowledge in mathematics. A specialist (...)
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  46. David J. Chalmers & Brian Rabern (2014). Two-Dimensional Semantics and the Nesting Problem. Analysis 74 (2):210-224.
    Graeme Forbes (2011) raises some problems for two-dimensional semantic theories. The problems concern nested environments: linguistic environments where sentences are nested under both modal and epistemic operators. Closely related problems involving nested environments have been raised by Scott Soames (2005) and Josh Dever (2007). Soames goes so far as to say that nested environments pose the “chief technical problem” for strong two-dimensionalism. We call the problem of handling nested environments within two-dimensional semantics “the nesting problem”. We show that the two-dimensional (...)
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  47. Brian F. Chellas (1980). Another Proof for the Decidability of Four Modal Logics. Philosophia 9 (2):251-264.
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  48. Brian F. Chellas (1980). Modal Logic: An Introduction. Cambridge University Press.
    A textbook on modal logic, intended for readers already acquainted with the elements of formal logic, containing nearly 500 exercises. Brian F. Chellas provides a systematic introduction to the principal ideas and results in contemporary treatments of modality, including theorems on completeness and decidability. Illustrative chapters focus on deontic logic and conditionality. Modality is a rapidly expanding branch of logic, and familiarity with the subject is now regarded as a necessary part of every philosopher's technical equipment. Chellas here offers an (...)
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  49. Brian F. Chellas & Krister Segerberg (1996). Modal Logics in the Vicinity of S. Notre Dame Journal of Formal Logic 37 (1):1-24.
    We define prenormal modal logics and show that S1, S1, S0.9, and S0.9 are Lewis versions of certain prenormal logics, determination and decidability for which are immediate. At the end we characterize Cresswell logics and ponder C. I. Lewis's idea of strict implication in S1.
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  50. Brian F. Chellas & Krister Segerberg (1994). Modal Logics with the MacIntosh Rule. Journal of Philosophical Logic 23 (1):67 - 86.
    Having gained some idea of what MacIntosh logics there are, we conclude this paper with a remark about the totality of them. Let theterritory of a rule or condition be the class of all modal logics that have the rule or satisfy the condition. What is MacIntosh territory, the class of all normal logics with the MacIntosh rule, like? What is its structure?
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