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Higher Order Modal Logic

In Patrick Blackburn, Johan Van Benthem & Frank Wolter (eds.), Handbook of Modal Logic. Elsevier. pp. 621-653 (2006)

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  1. All Properties Are Divine or God Exists.Frode Bjørdal - 2018 - Logic and Logical Philosophy 3 (27):329-350.
    A metaphysical system engendered by a third order quantified modal logic S5 plus impredicative comprehension principles is used to isolate a third order predicate D, and by being able to impredicatively take a second order predicate G to hold of an individual just if the individual necessarily has all second order properties which are D we in Section 2 derive the thesis (40) that all properties are D or some individual is G. In Section 3 theorems 1 to 3 suggest (...)
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  • Sense and the Computation of Reference.Reinhard Muskens - 2004 - Linguistics and Philosophy 28 (4):473 - 504.
    The paper shows how ideas that explain the sense of an expression as a method or algorithm for finding its reference, preshadowed in Frege’s dictum that sense is the way in which a referent is given, can be formalized on the basis of the ideas in Thomason (1980). To this end, the function that sends propositions to truth values or sets of possible worlds in Thomason (1980) must be replaced by a relation and the meaning postulates governing the behaviour of (...)
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  • Deontic Modals and Hyperintensionality.Federico L. G. Faroldi - 2019 - Logic Journal of the IGPL 27 (4):387-410.
    In this paper I argue that deontic modals are hyperintensional, i.e. logically equivalent contents cannot be substituted in their scope. I give two arguments, one deductive and the other abductive. First, I show that the contrary thesis leads to falsity; second, I argue that a hyperintensional theory of deontic modals fares better than its rivals in terms of elegance, theoretical simplicity and explanatory power. I then propose a philosophical analysis of this thesis and outline some consequences. In Section 1 I (...)
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  • 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.
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  • Intensional Models for the Theory of Types.Reinhard Muskens - 2007 - Journal of Symbolic Logic 72 (1):98-118.
    In this paper we define intensional models for the classical theory of types, thus arriving at an intensional type logic ITL. Intensional models generalize Henkin's general models and have a natural definition. As a class they do not validate the axiom of Extensionality. We give a cut-free sequent calculus for type theory and show completeness of this calculus with respect to the class of intensional models via a model existence theorem. After this we turn our attention to applications. Firstly, it (...)
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  • Completeness in Hybrid Type Theory.Carlos Areces, Patrick Blackburn, Antonia Huertas & María Manzano - 2013 - Journal of Philosophical Logic (2-3):1-30.
    We show that basic hybridization (adding nominals and @ operators) makes it possible to give straightforward Henkin-style completeness proofs even when the modal logic being hybridized is higher-order. The key ideas are to add nominals as expressions of type t, and to extend to arbitrary types the way we interpret $@_i$ in propositional and first-order hybrid logic. This means: interpret $@_i\alpha _a$ , where $\alpha _a$ is an expression of any type $a$ , as an expression of type $a$ that (...)
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  • CIFOL: Case-Intensional First Order Logic: Toward a Theory of Sorts.Nuel Belnap & Thomas Müller - 2014 - Journal of Philosophical Logic 43 (2-3):393-437.
    This is part I of a two-part essay introducing case-intensional first order logic, an easy-to-use, uniform, powerful, and useful combination of first-order logic with modal logic resulting from philosophical and technical modifications of Bressan’s General interpreted modal calculus. CIFOL starts with a set of cases; each expression has an extension in each case and an intension, which is the function from the cases to the respective case-relative extensions. Predication is intensional; identity is extensional. Definite descriptions are context-independent terms, and lambda-predicates (...)
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  • Propositional Quantifiers in Labelled Natural Deduction for Normal Modal Logic.Matteo Pascucci - 2019 - Logic Journal of the IGPL 27 (6):865-894.
    This article concerns the treatment of propositional quantification in a framework of labelled natural deduction for modal logic developed by Basin, Matthews and Viganò. We provide a detailed analysis of a basic calculus that can be used for a proof-theoretic rendering of minimal normal multimodal systems with quantification over stable domains of propositions. Furthermore, we consider variations of the basic calculus obtained via relational theories and domain theories allowing for quantification over possibly unstable domains of propositions. The main result of (...)
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  • Axiomatization in the Meaning Sciences.Wesley H. Holliday & Thomas Icard - 2018 - In Derek Ball & Brian Rabern (eds.), The Science of Meaning: Essays on the Metatheory of Natural Language Semantics. Oxford, UK: Oxford University Press. pp. 73-97.
    While much of semantic theorizing is based on intuitions about logical phenomena associated with linguistic constructions—phenomena such as consistency and entailment—it is rare to see axiomatic treatments of linguistic fragments. Given a fragment interpreted in some class of formally specified models, it is often possible to ask for a characterization of the reasoning patterns validated by the class of models. Axiomatizations provide such a characterization, often in a perspicuous and efficient manner. In this paper, we highlight some of the benefits (...)
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  • Outlook-Based Semantics.Elizabeth Coppock - 2018 - Linguistics and Philosophy 41 (2):125-164.
    This paper presents and advocates an approach to the semantics of opinion statements, including matters of personal taste and moral claims. In this framework, ‘outlook-based semantics’, the circumstances of evaluation are not composed of a possible world and a judge ; rather, outlooks replace possible worlds in the role of circumstance of evaluation. Outlooks are refinements of worlds that settle not only matters of fact but also matters of opinion. Several virtues of the framework and advantages over existing implementations of (...)
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