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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. Hughes & Cresswell 1996 is a classic textbook in modal logic. Sider 2010 includes a good presentation of quantified first-order logic.
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  1. Gerard Allwein & Wendy MacCaull (2001). A Kripke Semantics for the Logic of Gelfand Quantales. 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 (...)
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  2. Andrew Bacon (2013). Quantificational Logic and Empty Names. Philosophers' Imprint 13 (24).
    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 $\exists xt = x$; 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 (...)
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  3. Stephen Barker (2011). Can Counterfactuals Really Be About Possible Worlds? 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 (...)
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  4. Jiri Benovsky (2005). Branching Versus Divergent Possible Worlds. Kriterion 19:12-20.
    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'.
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  5. Johan Benthem (1984). Possible Worlds Semantics: A Research Program That Cannot Fail? 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 (...)
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  6. John C. Bigelow (1978). Believing in Semantics. 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 (...)
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  7. Patrick Blackburn (2001). Modal Logic as Dialogical Logic. Synthese 127 (1-2):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 (notably the use of nominals, special proposition symbols which name worlds) I will show how to lift the dialogical (...)
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  8. 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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  9. Patrick Blackburn & Johan van Benthem (1988). Modal Logic: A Semantic Perspective. Ethics 98:501-517.
    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 BASIC MODAL LOGIC . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.
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  10. Susanne Bobzien, A Model-Theoretic Account of Columnar Higher-Order Vagueness.
    ABSTRACT: Paper currently being revised.
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  11. Elke Brendel (1993). Partial Worlds and Paradox. 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 (...)
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  12. 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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  13. Anna Bucalo (1994). Modalities in Linear Logic Weaker Than the Exponential “of Course”: Algebraic and Relational Semantics. [REVIEW] 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 (...)
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  14. 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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  15. S. Celani & R. Jansana (1997). A New Semantics for Positive Modal Logic. 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 (...)
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  16. David J. Chalmers (2004). Epistemic Two-Dimensional Semantics. Philosophical Studies 118 (1-2):153-226.
  17. 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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  18. Charles S. Chihara (1998). The Worlds of Possibility: Modal Realism and the Semantics of Modal Logic. 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.
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  19. Nino Cocchiarella & Ma A. Freund (2008). Modal Logic: An Introduction to its Syntax and Semantics. 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.
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  20. B. Jack Copeland (2006). Meredith, Prior, and the History of Possible Worlds Semantics. 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.
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  21. B. Jack Copeland (2002). The Genesis of Possible Worlds Semantics. 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.
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  22. Fabrice Correia (2001). Priorean Strict Implication, Q and Related Systems. 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.
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  23. Fabrice Correia (1999). Adequacy Results for Some Priorean Modal Propositional Logics. 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 (...)
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  24. Maxwell J. Cresswell (2006). From Modal Discourse to Possible Worlds. 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 (...)
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  25. Charles B. Cross (1993). From Worlds to Probabilities: A Probabilistic Semantics for Modal Logic. Journal of Philosophical Logic 22 (2):169 - 192.
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  26. Louis deRosset (2009). Possible Worlds II: Non-Reductive Theories of Possible Worlds. Philosophy Compass 4 (6):1009-1021.
    It is difficult to wander far in contemporary metaphysics without bumping into talk of possible worlds. And, reference to possible worlds is not confined to metaphysics. It can be found in contemporary epistemology and ethics, and has even made its way into linguistics and decision theory. What are those possible worlds, the entities to which theorists in these disciplines all appeal? Some have hoped that a theory of possible worlds can be used to reduce modality to non-modal terms. This paper (...)
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  27. Louis deRosset (2009). Possible Worlds I: Modal Realism. Philosophy Compass 4 (6):998-1008.
    It is difficult to wander far in contemporary metaphysics without bumping into talk of possible worlds. And reference to possible worlds is not confined to metaphysics. It can be found in contemporary epistemology and ethics, and has even made its way into linguistics and decision theory. What are those possible worlds, the entities to which theorists in these disciplines all appeal? This paper sets out and evaluates a leading contemporary theory of possible worlds, David Lewis's Modal Realism. I note two (...)
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  28. John Divers (2006). Possible-Worlds Semantics Without Possible Worlds: The Agnostic Approach. Mind 115 (458):187-226.
    If a possible-worlds semantic theory for modal logics is pure, then the assertion of the theory, taken at face-value, can bring no commitment to the existence of a plurality of possible worlds (genuine or ersatz). But if we consider an applied theory (an application of the pure theory) in which the elements of the models are required to be possible worlds, then assertion of such a theory, taken at face-value, does appear to bring commitment to the existence of a plurality (...)
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  29. John Divers (1995). Modal Fictionalism Cannot Deliver Possible Worlds Semantics. Analysis 55 (2):81--9.
  30. Kit Fine (2000). Semantics for the Logic of Essence. Journal of Philosophical Logic 29 (6):543-584.
    This paper provides a possible worlds semantics for the system of the author's previous paper 'The Logic of Essence'. The basic idea behind the semantics is that a statement should be taken to be true in virtue of the nature of certain objects just in case it is true in any possible world compatible with the nature of those objects. It is shown that a slight variant of the original system is sound and complete under the proposed semantics.
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  31. Graeme Forbes (1983). Physicalism, Instrumentalism and the Semantics of Modal Logic. Journal of Philosophical Logic 12 (3):271 - 298.
    The delicate point in the formalistic position is to explain how the non-intuitionistic classical mathematics is significant, after having initially agreed with the intuitionists that its theorems lack a real meaning in terms of which they are true (S. C. Kleene, 1952).
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  32. D. M. Gabbay & G. Malod (2002). Naming Worlds in Modal and Temporal Logic. Journal of Logic, Language and Information 11 (1):29-65.
    In this paper we suggest adding to predicate modal and temporal logic a locality predicate W which gives names to worlds (or time points). We also study an equal time predicate D(x, y)which states that two time points are at the same distance from the root. We provide the systems studied with complete axiomatizations and illustrate the expressive power gained for modal logic by simulating other logics. The completeness proofs rely on the fairly intuitive notion of a configuration in order (...)
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  33. Daniel Gallin (1975). Intensional and Higher-Order Modal Logic: With Applications to Montague Semantics. American Elsevier Pub. Co..
    CHAPTER 1. INTENSIONAL LOGIC §1. Natural Language and Intensional Logic When we speak of a theory of meaning for a natural language such as English, ...
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  34. Martin Gerson (1975). The Inadequacy of the Neighbourhood Semantics for Modal Logic. Journal of Symbolic Logic 40 (2):141-148.
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  35. Silvio Ghilardi (1991). Incompleteness Results in Kripke Semantics. Journal of Symbolic Logic 56 (2):517-538.
    By means of models in toposes of C-sets (where C is a small category), necessary conditions are found for the minimum quantified extension of a propositional (intermediate, modal) logic to be complete with respect to Kripke semantics; in particular, many well-known systems turn out to be incomplete.
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  36. Paul Gochet Et Eric Gillet (1999). Quantified Modal Logic, Dynamic Semantics and S 5. Dialectica 53 (3-4):243–251.
  37. Alessandro Giordani (2013). A New Semantics for Systems of Logic of Essence. Studia Logica (3):1-30.
    The purpose of the present paper is to provide a way of understanding systems of logic of essence by introducing a new semantic framework for them. Three central results are achieved: first, the now standard Fitting semantics for the propositional logic of evidence is adapted in order to provide a new, simplified semantics for the propositional logic of essence; secondly, we show how it is possible to construe the concept of necessary truth explicitly by using the concept of essential truth; (...)
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  38. L. F. Goble (1973). A Simplified Semantics for Modal Logic. Notre Dame Journal of Formal Logic 14 (2):151-174.
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  39. Benedikt Paul Göcke, Martin Pleitz & Hanno von Wulfen (2008). How to Kripke Brandom's Notion of Necessity. In Bernd Prien & David P. Schweikard (eds.), Robert Brandom. Analytic Pragmatist. ontos.
    In this paper we discuss Brandom's definition of necessity, which is part of the incompatibility sematnics he develops in his fifth John Locke Lecture. By comparing incompatibility semantics to standard Kripkean possible worlds semantics for modality, we motivate an alternative definition of necessity in Brandom's own terms. Our investigation of this alternative necessity will show that - contra to Brandom's own results - incompatibility semantics does not necessarily lead to the notion of necessity of the modal logic S5.
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  40. Joanna Golinska-Pilarek (2012). On Decidability of a Logic for Order of Magnitude Qualitative Reasoning with Bidirectional Negligibility. In Luis Farinas del Cerro, Andreas Herzig & Jerome Mengin (eds.), Logics in Artificial Intelligence. Springer. 255--266.
    Qualitative Reasoning (QR) is an area of research within Artificial Intelligence that automates reasoning and problem solving about the physical world. QR research aims to deal with representation and reasoning about continuous aspects of entities without the kind of precise quantitative information needed by conventional numerical analysis techniques. Order-of-magnitude Reasoning (OMR) is an approach in QR concerned with the analysis of physical systems in terms of relative magnitudes. In this paper we consider the logic OMR_N for order-of-magnitude reasoning with the (...)
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  41. Joanna Golinska-Pilarek & Emilio Munoz Velasco (2012). Reasoning with Qualitative Velocity: Towards a Hybrid Approach. In Emilio Corchado, Vaclav Snasel, Ajith Abraham, Michał Woźniak, Manuel Grana & Sung-Bae Cho (eds.), Hybrid Artificial Intelligent Systems. Springer. 635--646.
    Qualitative description of the movement of objects can be very important when there are large quantity of data or incomplete information, such as in positioning technologies and movement of robots. We present a first step in the combination of fuzzy qualitative reasoning and quantitative data obtained by human interaction and external devices as GPS, in order to update and correct the qualitative information. We consider a Propositional Dynamic Logic which deals with qualitative velocity and enables us to represent some reasoning (...)
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  42. Dominic Gregory (2006). Functionalism About Possible Worlds. Australasian Journal of Philosophy 84 (1):95 – 115.
    Various writers have proposed that the notion of a possible world is a functional concept, yet very little has been done to develop that proposal. This paper explores a particular functionalist account of possible worlds, according to which pluralities of possible worlds are the bases for structures which provide occupants for the roles which analyse our ordinary modal concepts. It argues that the resulting position meets some of the stringent constraints which philosophers have placed upon accounts of possible worlds, while (...)
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  43. Dominic Gregory (2001). The Worlds of Possibility: Modal Realism and the Semantics of Modal Logic. Charles S. Chihara. Mind 110 (439):736-740.
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  44. V. Halbach & P. Welch (2009). Necessities and Necessary Truths: A Prolegomenon to the Use of Modal Logic in the Analysis of Intensional Notions. Mind 118 (469):71-100.
    In philosophical logic necessity is usually conceived as a sentential operator rather than as a predicate. An intensional sentential operator does not allow one to express quantified statements such as 'There are necessary a posteriori propositions' or 'All laws of physics are necessary' in first-order logic in a straightforward way, while they are readily formalized if necessity is formalized by a predicate. Replacing the operator conception of necessity by the predicate conception, however, causes various problems and forces one to reject (...)
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  45. Volker Halbach, Hannes Leitgeb & Philip Welch (2003). Possible-Worlds Semantics for Modal Notions Conceived as Predicates. Journal of Philosophical Logic 32 (2):179-223.
    If □ is conceived as an operator, i.e., an expression that gives applied to a formula another formula, the expressive power of the language is severely restricted when compared to a language where □ is conceived as a predicate, i.e., an expression that yields a formula if it is applied to a term. This consideration favours the predicate approach. The predicate view, however, is threatened mainly by two problems: Some obvious predicate systems are inconsistent, and possible-worlds semantics for predicates of (...)
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  46. William H. Hanson & James Hawthorne (1985). Validity in Intensional Languages: A New Approach. Notre Dame Journal of Formal Logic 26 (1):9-35.
    Although the use of possible worlds in semantics has been very fruitful and is now widely accepted, there is a puzzle about the standard definition of validity in possible-worlds semantics that has received little notice and virtually no comment. A sentence of an intensional language is typically said to be valid just in case it is true at every world under every model on every model structure of the language. Each model structure contains a set of possible worlds, and models (...)
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  47. Sven Ove Hansson (2006). Ideal Worlds — Wishful Thinking in Deontic Logic. Studia Logica 82 (3):329 - 336.
    The ideal world semantics of standard deontic logic identifies our obligations with how we would act in an ideal world. However, to act as if one lived in an ideal world is bad moral advice, associated with wishful thinking rather than well-considered moral deliberation. Ideal world semantics gives rise to implausible logical principles, and the metaphysical arguments that have been put forward in its favour turn out to be based on a too limited view of truth-functional representation. It is argued (...)
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  48. Charles W. Harvey (1986). Husserl's Phenomenology and Possible Worlds Semantics: A Reexamination. [REVIEW] Husserl Studies 3 (3):191-207.
  49. Allen Hazen (1979). Counterpart-Theoretic Semantics for Modal Logic. Journal of Philosophy 76 (6):319-338.
  50. Meir Hemmo (1996). Possible Worlds in the Modal Interpretation. Philosophy of Science 63 (3):337.
    An outline for a modal interpretation in terms of possible worlds is presented. The so-called Schmidt histories are taken to correspond to the physically possible worlds. The decoherence function defined in the histories formulation of quantum theory is taken to prescribe a non-classical probability measure over the set of the possible worlds. This is shown to yield dynamics in the form of transition probabilities for occurrent events in each world. The role of the consistency condition is discussed.
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