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A Neighbourhood Frame for T with No Equivalent Relational Frame

Martin Gerson

Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 22 (1):29-34 (1976)

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Correspondence, Canonicity, and Model Theory for Monotonic Modal Logics.Kentarô Yamamoto - 2020 - Studia Logica 109 (2):397-421.details We investigate the role of coalgebraic predicate logic, a logic for neighborhood frames first proposed by Chang, in the study of monotonic modal logics. We prove analogues of the Goldblatt–Thomason theorem and Fine’s canonicity theorem for classes of monotonic neighborhood frames closed under elementary equivalence in coalgebraic predicate logic. The elementary equivalence here can be relativized to the classes of monotonic, quasi-filter, augmented quasi-filter, filter, or augmented filter neighborhood frames, respectively. The original, Kripke-semantic versions of the theorems follow as a (...) Download Export citation Bookmark
(1 other version)A normal logic that is complete for neighborhood frames but not for Kripke frames.Dov M. Gabbay - 1975 - Theoria 41 (3):148-153.details Download Export citation Bookmark 2 citations
Canonicity for intensional logics without iterative axioms.Timothy J. Surendonk - 1997 - Journal of Philosophical Logic 26 (4):391-409.details David Lewis proved in 1974 that all logics without iterative axioms are weakly complete. In this paper we extend Lewis's ideas and provide a proof that such logics are canonical and so strongly complete. This paper also discusses the differences between relational and neighborhood frame semantics and poses a number of open questions about the latter. Download Export citation Bookmark 5 citations
Mathematical modal logic: A view of its evolution.Robert Goldblatt - 2003 - Journal of Applied Logic 1 (5-6):309-392.details Download Export citation Bookmark 57 citations
The inadequacy of the neighbourhood semantics for modal logic.Martin Gerson - 1975 - Journal of Symbolic Logic 40 (2):141-148.details We present two finitely axiomatized modal propositional logics, one betweenTandS4 and the other an extension ofS4, which are incomplete with respect to the neighbourhood or Scott-Montague semantics.Throughout this paper we are referring to logics which contain all the classical connectives and only one modal connective □ (unary), no propositional constants, all classical tautologies, and which are closed under the rules of modus ponens (MP), substitution, and the rule RE (fromA↔Binfer αA↔ □B). Such logics are calledclassicalby Segerberg [6]. Classical logics which (...) Download Export citation Bookmark 19 citations
(1 other version)A normal logic that is complete for neighborhood frames but not for Kripke frames.Dov M. Gabbay - 1974 - Theoria 40 (3):148-153.details Download Export citation Bookmark 1 citation
Neighborhood Semantics for Modal Logic.Eric Pacuit - 2017 - Cham, Switzerland: Springer.details This book offers a state-of-the-art introduction to the basic techniques and results of neighborhood semantics for modal logic. In addition to presenting the relevant technical background, it highlights both the pitfalls and potential uses of neighborhood models – an interesting class of mathematical structures that were originally introduced to provide a semantics for weak systems of modal logic. In addition, the book discusses a broad range of topics, including standard modal logic results ; bisimulations for neighborhood models and other model-theoretic (...) Download Export citation Bookmark 37 citations

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