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Summary Several kinds of semantics for modal logic have been proposed, the most popular of which is Kripke semantics. (This is sometimes called possible worlds semantics, although the formal semantics doesn’t require us to think of the entities in its domain as possible worlds.) In Kripke semantics, models comprise a domain of entities, variously called pointssituationsscenarios or possible worlds, with one or more accessible relations between them. The truth-value of a sentence at a possible world w may depend on the truth-value of other sentences at worlds accessible from w. In modal logics, for example, ‘necessarily, A’ is true at a worldw iff A is true at all worlds u accessible from w, and ‘possibly, A’ is true at w iff A is true at some world u accessible from w. This approach can be applied to other kinds of modalities, including ‘agent a knows that’ in modal epistemic logics. One can also give algebraic, topological and categorical semantics for modal logics, in place of Kripke semantics. These approaches have intrinsic mathematical interest, but have received less attention in the philosophical literature (perhaps because they do not provide an analysis of modal concepts in the way that possible worlds semantics does).
Key works Possible worlds semantics was first presented as a formal semantics for modal logic in Kripke 1959, 1963 and for intuitionistic logic in Kripke 1963Carnap 1947 was an important precursor to possible worlds semantics. Hintikka 1962, 1967 develops the possible worlds semantics and applies it to epistemic concepts. van Benthem 1983 is the classic investigation of the relationship between accessibility relations in the semantics and modal axioms. Lewis 1968 develops counterpart theory in the context of (first-order) possible worlds semantics. Lewis 1986 argues for an extreme realist philosophical interpretation of the possible worlds semantics.
Introductions Girle 2003 and Girle 2000 are introductory textbooks on possible worlds and modal logic. Priest 2001 is a general introduction to propositional modal, intuitionistic and relevant logics. Cresswell & Hughes 1996 is a classic textbook in modal logic. Sider 2010 includes a good presentation of quantified first-order logic.
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  1. Strategic Commitment and Release in Logics for Multi-Agent Systems.Thomas Ågotnes, Valentin Goranko & Wojciech Jamroga - manuscript
    In this paper we analyze how the semantics of the Alternating-time Temporal Logic ATL$^*$ deals with agents' commitments to strategies in the process of formula evaluation. In (\acro{atl}$^*$), one can express statements about the strategic ability of an agent (or a coalition of agents) to achieve a goal $\phi$ such as: ``agent $i$ can choose a strategy such that, if $i$ follows this strategy then, no matter what other agents do, $\phi$ will always be true''. However, strategies in \acro{atl} are (...)
  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. A Kripke Semantics for the Logic of Gelfand Quantales.Gerard Allwein & Wendy MacCaull - 2001 - Studia Logica 68 (2):173-228.
    Gelfand quantales are complete unital quantales with an involution, *, satisfying the property that for any element a, if a b a for all b, then a a* a = a. A Hilbert-style axiom system is given for a propositional logic, called Gelfand Logic, which is sound and complete with respect to Gelfand quantales. A Kripke semantics is presented for which the soundness and completeness of Gelfand logic is shown. The completeness theorem relies on a Stone style representation theorem for (...)
  4. Combination Semantics for Intensional Logics Part I Makings and Their Use in Making Combination Semantics Jerzy Perzanowski.Logique A. Analyse - 1999 - Logique Et Analyse 42:181.
  5. 'First-Order Modal Logic', to Appear in V. Hendricks & SA Pedersen, Eds.,'40 Years of Possible Worlds', Special Issue Of.H. Arlo-Costa - forthcoming - Studia Logica.
  6. Quantificational Logic and Empty Names.Andrew Bacon - 2013 - Philosophers' Imprint 13.
    The result of combining classical quantificational logic with modal logic proves necessitism – the claim that necessarily everything is necessarily identical to something. This problem is reflected in the purely quantificational theory by theorems such as ∃x t=x; it is a theorem, for example, that something is identical to Timothy Williamson. The standard way to avoid these consequences is to weaken the theory of quantification to a certain kind of free logic. However, it has often been noted that in order (...)
  7. Modal Logics for Parallelism, Orthogonality, and Affine Geometries.Philippe Balbiani & Valentin Goranko - 2002 - Journal of Applied Non-Classical Logics 12 (3-4):365-397.
    We introduce and study a variety of modal logics of parallelism, orthogonality, and affine geometries, for which we establish several completeness, decidability and complexity results and state a number of related open, and apparently difficult problems. We also demonstrate that lack of the finite model property of modal logics for sufficiently rich affine or projective geometries (incl. the real affine and projective planes) is a rather common phenomenon.
  8. Two Formal Systems for Situation Semantics.Barba Escriba Juan - 1991 - Notre Dame Journal of Formal Logic 33 (1):70-88.
  9. Can Counterfactuals Really Be About Possible Worlds?Stephen Barker - 2011 - Noûs 45 (3):557-576.
    The standard view about counterfactuals is that a counterfactual (A > C) is true if and only if the A-worlds most similar to the actual world @ are C-worlds. I argue that the worlds conception of counterfactuals is wrong. I assume that counterfactuals have non-trivial truth-values under physical determinism. I show that the possible-worlds approach cannot explain many embeddings of the form (P > (Q > R)), which intuitively are perfectly assertable, and which must be true if the contingent falsity (...)
  10. Branching Versus Divergent Possible Worlds.Jiri Benovsky - 2005 - Kriterion - Journal of Philosophy 19 (1):12-20.
    David Lewis' modal counterpart theory falls prey to the famous Saul Kripke's objection, and this is mostly due to his 'static' ontology (divergence) of possible worlds. This paper examines a genuinely realist but different, branching ontology of possible worlds and a new definition of the counterpart relation, which attempts to provide us with a better account of de re modality, and to meet satisfactorily Kripke's claim, while being also ontologically more 'parsimonious'.
  11. Possible Worlds Semantics: A Research Program That Cannot Fail?Johan Benthem - 1984 - Studia Logica 43 (4):379 - 393.
    Providing a possible worlds semantics for a logic involves choosing a class of possible worlds models, and setting up a truth definition connecting formulas of the logic with statements about these models. This scheme is so flexible that a danger arises: perhaps, any (reasonable) logic whatsoever can be modelled in this way. Thus, the enterprise would lose its essential tension. Fortunately, it may be shown that the so-called incompleteness-examples from modal logic resist possible worlds modelling, even in the above wider (...)
  12. Multimodal and Intuitionistic Logics in Simple Type Theory.Christoph Benzmueller & Lawrence Paulson - 2010 - Logic Journal of the IGPL 18 (6):881-892.
    We study straightforward embeddings of propositional normal multimodal logic and propositional intuitionistic logic in simple type theory. The correctness of these embeddings is easily shown. We give examples to demonstrate that these embeddings provide an effective framework for computational investigations of various non-classical logics. We report some experiments using the higher-order automated theorem prover LEO-II.
  13. Quantified Multimodal Logics in Simple Type Theory.Christoph Benzmüller & Lawrence C. Paulson - 2013 - Logica Universalis 7 (1):7-20.
    We present an embedding of quantified multimodal logics into simple type theory and prove its soundness and completeness. A correspondence between QKπ models for quantified multimodal logics and Henkin models is established and exploited. Our embedding supports the application of off-the-shelf higher-order theorem provers for reasoning within and about quantified multimodal logics. Moreover, it provides a starting point for further logic embeddings and their combinations in simple type theory.
  14. On Conceiving the Inconsistent.Francesco Berto - 2014 - Proceedings of the Aristotelian Society 114 (1pt1):103-121.
    I present an approach to our conceiving absolute impossibilities—things which obtain at no possible world—in terms of ceteris paribus intentional operators: variably restricted quantifiers on possible and impossible worlds based on world similarity. The explicit content of a representation plays a role similar in some respects to the one of a ceteris paribus conditional antecedent. I discuss how such operators invalidate logical closure for conceivability, and how similarity works when impossible worlds are around. Unlike what happens with ceteris paribus counterfactual (...)
  15. Believing in Semantics.John C. Bigelow - 1978 - Linguistics and Philosophy 2 (1):101--144.
    This paper concerns the semantics of belief-sentences. I pass over ontologically lavish theories which appeal to impossible worlds, or other points of reference which contain more than possible worlds. I then refute ontologically stingy, quotational theories. My own theory employs the techniques of possible worlds semantics to elaborate a Fregean analysis of belief-sentences. In a belief-sentence, the embedded clause does not have its usual reference, but refers rather to its own semantic structure. I show how this theory can accommodate quantification (...)
  16. Modal Logic As Dialogical Logic.Patrick Blackburn - 2001 - Synthese 127 (1):57-93.
    The title reflects my conviction that, viewed semantically, modal logic is fundamentally dialogical: this conviction is based on the key role played by the notion of bisimulation in modal model theory. But this dialogical conception of modal logic does not seem to apply to modal proof theory, which is notoriously messy. Nonetheless, by making use of ideas which trace back to Arthur Prior I will show how to lift the dialogical conception to modal proof theory. I argue that this shift (...)
  17. Modal Logic.Patrick Blackburn, Maarten de Rijke & Yde Venema - 2002 - Cambridge University Press.
    This modern, advanced textbook reviews modal logic, a field which caught the attention of computer scientists in the late 1970's.
  18. Modal Logic: A Semantic Perspective.Patrick Blackburn & Johan van Benthem - 1988 - Ethics 98:501-517.
    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 BASIC MODAL LOGIC . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.
  19. A Model-Theoretic Account of Columnar Higher-Order Vagueness.Susanne Bobzien - manuscript
  20. Quantificational Modal Logic with Sequential Kripke Semantics.Stefano Borgo - 2005 - Journal of Applied Non-Classical Logics 15 (2):137-188.
  21. Modelling Belief Revision Via Belief Bases Using Situation Semantics.Ayse Sena Bozdag - 2017 - Dissertation, Bogazici University
    The belief base approach to belief representation and belief dynamics is developed as an alternative to the belief set approaches, which are pioneered by the AGM model. The belief base approach models collections of information and expectations of an agent as possibly incomplete and possibly inconsistent foundations for her beliefs. Nevertheless, the beliefs of an agent are always consistent; this is ensured by a sophisticated inference relation. Belief changes take place on the information base instead of on the belief set, (...)
  22. Partial Worlds and Paradox.Elke Brendel - 1993 - Erkenntnis 39 (2):191 - 208.
    Since universal language systems are confronted with serious paradoxical consequences, a semantic approach is developed in whichpartial worlds form the ontological basis. This approach shares withsituation semantics the basic idea that statements always refer to certain partial worlds, and it agrees with the extensional and model-theoretic character ofpossible worlds semantics. Within the framework of the partial worlds conception a satisfactory solution to theLiar paradox can be formulated. In particular, one advantage of this approach over those theories that are based on (...)
  23. Again on Relativistic Semantics.Aldo Bressan - 1995 - Logic and Logical Philosophy 3:23-36.
    This paper has two parts: Part I is a continuation of the work [10] and as well as this it deals mainly with the logic of an auxiliary (semantical) theory, ST , in effect generally considered (by textbooks) within the semantics for a typical theory T of general relativity; and especially the modal features of this auxiliary theory are studied. Part II deals with the influence had by relativistic theories and especially by the new notion of space-time on pragmatic languages (...)
  24. Modalities in Linear Logic Weaker Than the Exponential “of Course”: Algebraic and Relational Semantics. [REVIEW]Anna Bucalo - 1994 - Journal of Logic, Language and Information 3 (3):211-232.
    We present a semantic study of a family of modal intuitionistic linear systems, providing various logics with both an algebraic semantics and a relational semantics, to obtain completeness results. We call modality a unary operator on formulas which satisfies only one rale (regularity), and we consider any subsetW of a list of axioms which defines the exponential of course of linear logic. We define an algebraic semantics by interpreting the modality as a unary operation on an IL-algebra. Then we introduce (...)
  25. Meaning and Necessity: A Study in Semantics and Modal Logic.Rudolf Carnap - 1947 - University of Chicago Press.
    This is identical with the first edition (see 21: 2716) except for the addition of a Supplement containing 5 previously published articles and the bringing of the bibliography (now 73 items) up to date. The 5 added articles present clarifications or modifications of views expressed in the first edition. (PsycINFO Database Record (c) 2009 APA, all rights reserved).
  26. On Interpreting the S5 Propositional Calculus: An Essay in Philosophical Logic.Michael J. Carroll - 1976 - Dissertation, University of Iowa
    Discusses alternative interpretations of the modal operators, for the modal propositional logic S5.
  27. A New Semantics for Positive Modal Logic.S. Celani & R. Jansana - 1997 - Notre Dame Journal of Formal Logic 38 (1):1-18.
    The paper provides a new semantics for positive modal logic using Kripke frames having a quasi ordering on the set of possible worlds and an accessibility relation connected to the quasi ordering by the conditions (1) that the composition of with is included in the composition of with and (2) the analogous for the inverse of and . This semantics has an advantage over the one used by Dunn in "Positive modal logic," Studia Logica (1995) and works fine for extensions (...)
  28. Epistemic Two-Dimensional Semantics.David J. Chalmers - 2004 - Philosophical Studies 118 (1-2):153-226.
  29. Modal Logic: An Introduction.Brian F. Chellas - 1980 - 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 (...)
  30. The Worlds of Possibility: Modal Realism and the Semantics of Modal Logic.Charles Chihara - 2001 - Philosophical Quarterly 51 (202):108-110.
  31. The Worlds of Possibility: Modal Realism and the Semantics of Modal Logic.Charles S. Chihara - 2001 - Mind 110 (439):736-740.
  32. The Worlds of Possibility: Modal Realism and the Semantics of Modal Logic.Charles S. Chihara - 1998 - Oxford University Press.
    A powerful challenge to some highly influential theories, this book offers a thorough critical exposition of modal realism, the philosophical doctrine that many possible worlds exist of which our own universe is just one. Chihara challenges this claim and offers a new argument for modality without worlds.
  33. Review of David Lewis, On the Plurality of Worlds. [REVIEW]Michael Clark - 1987 - Philosophical Books 28.
  34. Modal Logic: An Introduction to its Syntax and Semantics.Cocchiarella Nino & A. Freund Ma - 2008 - Oxford University Press.
    In this text, a variety of modal logics at the sentential, first-order, and second-order levels are developed with clarity, precision and philosophical insight.
  35. Elementary Canonical Formulae: A Survey on Syntactic, Algorithmic, and Modeltheoretic Aspects.W. Conradie, V. Goranko & D. Vakarelov - 2005 - In Renate Schmidt, Ian Pratt-Hartmann, Mark Reynolds & Heinrich Wansing (eds.), Advances in Modal Logic, Volume 5. Kings College London Publ.. pp. 17-51.
    In terms of validity in Kripke frames, a modal formula expresses a universal monadic second-order condition. Those modal formulae which are equivalent to first-order conditions are called elementary. Modal formulae which have a certain persistence property which implies their validity in all canonical frames of modal logics axiomatized with them, and therefore their completeness, are called canonical. This is a survey of a recent and ongoing study of the class of elementary and canonical modal formulae. We summarize main ideas and (...)
  36. Algorithmic Correspondence and Completeness in Modal Logic. IV. Semantic Extensions of SQEMA.Willem Conradie & Valentin Goranko - 2008 - Journal of Applied Non-Classical Logics 18 (2-3):175-211.
    In a previous work we introduced the algorithm \SQEMA\ for computing first-order equivalents and proving canonicity of modal formulae, and thus established a very general correspondence and canonical completeness result. \SQEMA\ is based on transformation rules, the most important of which employs a modal version of a result by Ackermann that enables elimination of an existentially quantified predicate variable in a formula, provided a certain negative polarity condition on that variable is satisfied. In this paper we develop several extensions of (...)
  37. Algorithmic Correspondence and Completeness in Modal Logic. V. Recursive Extensions of SQEMA.Willem Conradie, Valentin Goranko & Dimitar Vakarelov - 2010 - Journal of Applied Logic 8 (4):319-333.
    The previously introduced algorithm \sqema\ computes first-order frame equivalents for modal formulae and also proves their canonicity. Here we extend \sqema\ with an additional rule based on a recursive version of Ackermann's lemma, which enables the algorithm to compute local frame equivalents of modal formulae in the extension of first-order logic with monadic least fixed-points \mffo. This computation operates by transforming input formulae into locally frame equivalent ones in the pure fragment of the hybrid mu-calculus. In particular, we prove that (...)
  38. Meredith, Prior, and the History of Possible Worlds Semantics.B. Jack Copeland - 2006 - Synthese 150 (3):373-397.
    This paper charts some early history of the possible worlds semantics for modal logic, starting with the pioneering work of Prior and Meredith. The contributions of Geach, Hintikka, Kanger, Kripke, Montague, and Smiley are also discussed.
  39. The Genesis of Possible Worlds Semantics.B. Jack Copeland - 2002 - Journal of Philosophical Logic 31 (2):99-137.
    This article traces the development of possible worlds semantics through the work of: Wittgenstein, 1913-1921; Feys, 1924; McKinsey, 1945; Carnap, 1945-1947; McKinsey, Tarski and Jónsson, 1947-1952; von Wright, 1951; Becker, 1952; Prior, 1953-1954; Montague, 1955; Meredith and Prior, 1956; Geach, 1960; Smiley, 1955-1957; Kanger, 1957; Hintikka, 1957; Guillaume, 1958; Binkley, 1958; Bayart, 1958-1959; Drake, 1959-1961; Kripke, 1958-1965.
  40. Priorean Strict Implication, Q and Related Systems.Fabrice Correia - 2001 - Studia Logica 69 (3):411-427.
    We introduce a system PSI for a strict implication operator called Priorean strict implication. The semantics for PSI is based on partial Kripke models without accessibility relations. PSI is proved sound and complete with respect to that semantics, and Prior's system Q and related systems are shown to be fragments of PSI or of a mild extension of it.
  41. Adequacy Results for Some Priorean Modal Propositional Logics.Fabrice Correia - 1999 - Notre Dame Journal of Formal Logic 40 (2):236-249.
    Standard possible world semantics for propositional modal languages ignore truth-value gaps. However, simple considerations suggest that it should not be so. In Section 1, I identify what I take to be a correct truth-clause for necessity under the assumption that some possible worlds are incomplete (i.e., "at" which some propositions lack a truth-value). In Section 2, I build a world semantics, the semantics of TV-models, for standard modal propositional languages, which agrees with the truth-clause for necessity previously identified. Sections 3–5 (...)
  42. Physical theories and possible worlds.M. J. Cresswell - 1973 - Logique Et Analyse 16 (63):495.
    Formalized physical theories are not, as a rule, stated in intensional languages. Yet in talking about them we often treat them as if they were. We say for instance: 'Consider what would happen if instead of p's being true q were. In such a case r would be likely.' If we say this sort of thing, p, q and r appear to stand for the meanings of sentences of the theory, but meanings in some intensional sense. Now it is very (...)
  43. A New Introduction to Modal Logic.M. J. Cresswell & G. E. Hughes - 1996 - Routledge.
    This long-awaited book replaces Hughes and Cresswell's two classic studies of modal logic: _An Introduction to Modal Logic_ and _A Companion to Modal Logic_. _A New Introduction to Modal Logic_ is an entirely new work, completely re-written by the authors. They have incorporated all the new developments that have taken place since 1968 in both modal propositional logic and modal predicate logic, without sacrificing tha clarity of exposition and approachability that were essential features of their earlier works. The book takes (...)
  44. Worlds and Models in Bayart and Carnap.Max Cresswell - 2016 - Australasian Journal of Logic 13 (1).
    In the early days of the semantics for modal logic the `possible worlds' were thought of as models or interpretations. This was particularly so when the interpretation was of emph{logical} necessity or possibility, where this was understood in terms of validity. Arnould Bayart in 1958 may have been the first modal logician to argue explicitly against the identification of necessity and validity. This note contrasts his semantics with that provided by Rudolf Carnap in 1946, and examines Bayart's proof that if (...)
  45. From Modal Discourse to Possible Worlds.Maxwell J. Cresswell - 2006 - Studia Logica 82 (3):307-327.
    The possible-worlds semantics for modality says that a sentence is possibly true if it is true in some possible world. Given classical prepositional logic, one can easily prove that every consistent set of propositions can be embedded in a ‘maximal consistent set’, which in a sense represents a possible world. However the construction depends on the fact that standard modal logics are finitary, and it seems false that an infinite collection of sets of sentences each finite subset of which is (...)
  46. Embedded Counterfactuals and Possible Worlds Semantics.Charles B. Cross - 2016 - Philosophical Studies 173 (3):665-673.
    Stephen Barker argues that a possible worlds semantics for the counterfactual conditional of the sort proposed by Stalnaker and Lewis cannot accommodate certain examples in which determinism is true and a counterfactual Q > R is false, but where, for some P, the compound counterfactual P > (Q > R) is true. I argue that the completeness theorem for Lewis’s system VC of counterfactual logic shows that Stalnaker–Lewis semantics does accommodate Barker’s example, and I argue that its doing so should (...)
  47. From Worlds to Probabilities: A Probabilistic Semantics for Modal Logic.Charles B. Cross - 1993 - Journal of Philosophical Logic 22 (2):169 - 192.
    I give a probabilistic semantics for modal logic in which modal operators function as quantifiers over Popper functions in probabilistic model sets, thereby generalizing Kripke's semantics for modal logic.
  48. Studies in the Semantics of Modality.Charles Byron Cross - 1985 - Dissertation, University of Pittsburgh
    Possible worlds talk is, in my view, a metaphor, and what makes it a good metaphor is its capacity to be extended and elaborated in fruitful ways. The essays in this dissertation all concern ways of adding structure to the basic apparatus of possible worlds semantics--the Kripke frame--so as to make it bear more fruit. ;One way of adding structure is to think of possible worlds as histories. In "A Theory of Conditionals in the Context of Branching Time" Richmond Thomason (...)
  49. Literary Semantics and Possible Worlds = Literatursemantik Und Mögliche Welten.Károly Csúri - 1980 - Auctoritate Et Consilio Cathedrae Comparationis Litterarum Universarum Universitatis Szegediensis de Attila József Nominatae Edita.
  50. Neighbourhood Semantics and Generalized Kripke Models.Bernd Dahn - 1976 - Bulletin of the Section of Logic 5 (1):2-7.
    t is proved in [2] that generalized Kripke semantics has the same depth as the semantics of propositional matrices with exactly one designated el- ement and hence for modal logics has greater depth than the Boolean se- mantics . This result has been consider- ably extended by K. Bernhardt [1]. The aim of this note is to prove that generalized Kripke semantics for modal propositional logic with standard interpretation of classical propositional connectives has the same depth as the neighbourhood semantics.
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