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Notes on Logics of Metric Spaces

Studia Logica 85 (1):75-104 (2007)

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  1. The Modal Logic of Inequality.Maarten De Rijke - 1992 - Journal of Symbolic Logic 57 (2):566 - 584.
    We consider some modal languages with a modal operator D whose semantics is based on the relation of inequality. Basic logical properties such as definability, expressive power and completeness are studied. Also, some connections with a number of other recent proposals to extend the standard modal language are pointed at.
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  • Multi-Dimensional Modal Logic.Dimiter Vakarelov - 2000 - Bulletin of Symbolic Logic 6 (4):490-495.
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  • Modal Logic with Names.George Gargov & Valentin Goranko - 1993 - Journal of Philosophical Logic 22 (6):607 - 636.
    We investigate an enrichment of the propositional modal language L with a "universal" modality ■ having semantics x ⊧ ■φ iff ∀y(y ⊧ φ), and a countable set of "names" - a special kind of propositional variables ranging over singleton sets of worlds. The obtained language ℒ $_{c}$ proves to have a great expressive power. It is equivalent with respect to modal definability to another enrichment ℒ(⍯) of ℒ, where ⍯ is an additional modality with the semantics x ⊧ ⍯φ (...)
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  • Axiomatizations with Context Rules of Inference in Modal Logic.Valentin Goranko - 1998 - Studia Logica 61 (2):179-197.
    A certain type of inference rules in modal logics, generalizing Gabbay's Irreflexivity rule, is introduced and some general completeness results about modal logics axiomatized with such rules are proved.
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  • Hybrid Languages and Temporal Logic.P. Blackburn & M. Tzakova - 1999 - Logic Journal of the IGPL 7 (1):27-54.
    Hybridization is a method invented by Arthur Prior for extending the expressive power of modal languages. Although developed in interesting ways by Robert Bull, and by the Sofia school , the method remains little known. In our view this has deprived temporal logic of a valuable tool.The aim of the paper is to explain why hybridization is useful in temporal logic. We make two major points, the first technical, the second conceptual. First, we show that hybridization gives rise to well-behaved (...)
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  • The Modal Logic of Inequality.Maarten de Rijke - 1992 - Journal of Symbolic Logic 57 (2):566-584.
    We consider some modal languages with a modal operator $D$ whose semantics is based on the relation of inequality. Basic logical properties such as definability, expressive power and completeness are studied. Also, some connections with a number of other recent proposals to extend the standard modal language are pointed at.
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  • Derivation Rules as Anti-Axioms in Modal Logic.Yde Venema - 1993 - Journal of Symbolic Logic 58 (3):1003-1034.
    We discuss a `negative' way of defining frame classes in (multi)modal logic, and address the question of whether these classes can be axiomatized by derivation rules, the `non-ξ rules', styled after Gabbay's Irreflexivity Rule. The main result of this paper is a metatheorem on completeness, of the following kind: If Λ is a derivation system having a set of axioms that are special Sahlqvist formulas and Λ+ is the extension of Λ with a set of non-ξ rules, then Λ+ is (...)
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  • Interpolation for Extended Modal Languages.Balder ten Cate - 2005 - Journal of Symbolic Logic 70 (1):223-234.
    Several extensions of the basic modal language are characterized in terms of interpolation. Our main results are of the following form: Language ℒ' is the least expressive extension of ℒ with interpolation. For instance, let ℳ be the extension of the basic modal language with a difference operator [7]. First-order logic is the least expressive extension of ℳ with interpolation. These characterizations are subsequently used to derive new results about hybrid logic, relation algebra and the guarded fragment.
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  • Failure of Interpolation in Combined Modal Logics.Maarten Marx & Carlos Areces - 1998 - Notre Dame Journal of Formal Logic 39 (2):253-273.
    We investigate transfer of interpolation in such combinations of modal logic which lead to interaction of the modalities. Combining logics by taking products often blocks transfer of interpolation. The same holds for combinations by taking unions, a generalization of Humberstone's inaccessibility logic. Viewing first-order logic as a product of modal logics, we derive a strong counterexample for failure of interpolation in the finite variable fragments of first-order logic. We provide a simple condition stated only in terms of frames and bisimulations (...)
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  • Axiomatizing Distance Logics.Oliver Kutz, Holger Sturm, Nobu-Yuki Suzuki, Frank Wolter & Michael Zakharyaschev - 2002 - Journal of Applied Non-Classical Logics 12 (3-4):425-439.
    In [STU 00, KUT 03] we introduced a family of ‘modal' languages intended for talking about distances. These languages are interpreted in ‘distance spaces' which satisfy some of the standard axioms of metric spaces. Among other things, we singled out decidable logics of distance spaces and proved expressive completeness results relating classical and modal languages. The aim of this paper is to axiomatize the modal fragments of the semantically defined distance logics of [KUT 03] and give a new proof of (...)
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  • An Approach to Tense Logic.R. A. Bull - 1970 - Theoria 36 (3):282-300.
    The author's motivation for constructing the calculi of this paper\nis so that time and tense can be "discussed together in the same\nlanguage" (p. 282). Two types of enriched propositional caluli for\ntense logic are considered, both containing ordinary propositional\nvariables for which any proposition may be substituted. One type\nalso contains "clock-propositional" variables, a,b,c, etc., for\nwhich only clock-propositional variables may be substituted and that\ncorrespond to instants or moments in the semantics. The other type\nalso contains "history-propositional" variables, u,v,w, etc., for\nwhich only history-propositional variables may (...)
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  • An Ascending Chain of S4 Logics.Kit Fine - 1974 - Theoria 40 (2):110-116.
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  • Multi-Dimensional Modal Logic.Maarten Marx - 1997 - Kluwer Academic Publishers.
    Over the last twenty years, in all of these neighbouring fields, modal systems have been developed that we call multi-dimensional. (Our definition of multi ...
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  • A Logic for Metric and Topology.Frank Wolter & Michael Zakharyaschev - 2005 - Journal of Symbolic Logic 70 (3):795 - 828.
    We propose a logic for reasoning about metric spaces with the induced topologies. It combines the 'qualitative' interior and closure operators with 'quantitative' operators 'somewhere in the sphere of radius r.' including or excluding the boundary. We supply the logic with both the intended metric space semantics and a natural relational semantics, and show that the latter (i) provides finite partial representations of (in general) infinite metric models and (ii) reduces the standard '∈-definitions' of closure and interior to simple constraints (...)
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