Abstract
In the modal literature various notions of “completeness” have been studied for normal modal logics. Four of these are defined here, viz. (plain) completeness, first-order completeness, canonicity and possession of the finite model property — and their connections are studied. Up to one important exception, all possible inclusion relations are either proved or disproved. Hopefully, this helps to establish some order in the jungle of concepts concerning modal logics. In the course of the exposition, the interesting properties of first-order definability and preservation under ultrafilter extensions are introduced and studied as well.
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References
J. F. A. K. van Benthem, Modal Correspondence Theory, dissertation, Department of Mathematics, University of Amsterdam, 1976.
—, Modal Logic as Second-Order Logic, report 77-04, 1977, Department of Mathematics, University of Amsterdam; expanded version to appear in Studia Logica Selected Topics Library 1980.
—, Canonical modal logics and ultrafilter extensions, The Journal of Symbolic Logic 44 (1979), pp. 1–8.
—, Two simple incomplete modal logics, Theoria 44 (1978), pp. 25–37.
C. C. Chang and H. J. Keisler, Model Theory, North-Holland, Amsterdam, 1973.
K. Fine, The logics containing S4.3, Zeitschrift für mathematische Logik und Grundlagen der Mathematik 17 (1971), pp. 371–376.
—, An incomplete logic containing S4, Theoria 40 (1974), pp. 23–29.
—, Some connections between elementary and modal logic, in: Proceedings of the Third Scandinavian Logic Symposium, Uppsala 1973, North-Holland, Amsterdam 1975, pp. 15–31.
R. I. Goldblatt, Metamathematics of modal logic, Reports on Mathematical Logic 6 (1976), pp. 41–77, and 7, pp. 21–52.
R. I. Goldblatt and S. K. Thomason, Axiomatic classes in propositional modal logic, in: Algebra and Logic, Springer Lecture Notes in Mathematics 450, Berlin (1974), pp. 163–173.
R. Harrop, On the existence of finite models and decision procedures for propositional calculi, Proceedings of the Cambridge Philosophical Society 54 (1958), pp. 1–13.
S.A. Kripke, A completensesz theorem in modal logic, The Journal of Symbolic Logic 24 (1959), pp. 1–14.
E. J. Lemmon and D. Scott, An Introduction to Modal Logic, Blackwell, Oxford 1966, 1977, (Edited by K. Segerberg.)
D. C. Makinson, A normal calculus between T and S4 without the finite model property, The Journal of Symbolic Logic 34 (1969), pp. 35–38.
M. Mortimer, Some results in modal model theory, The Journal of Symbolic Logic 39 (1974), pp. 496–508.
H. Sahlqvist, Completeness and correspondence in the First and Second Order semantics for modal logic, in: Proceedings of the Third Scandinavian Logic Symposium, Uppsala 1973, North-Holland, Amsterdam 1975, pp. 110–143.
K. Segerberg, An Essay in Classical Modal Logic, Department of Philosophy, University of Uppsala, 1971.
S. K. Thomason, An incompleteness theorem in modal logic, Theoria 40 (1974), pp. 30–34.
—, Reduction of second-order logic to modal logic, Zeitschrift für mathematische Logik und Grundlagen der Mathematik 21 (1975), pp. 107–114.
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van Benthem, J.F.A.K. Some kinds of modal completeness. Stud Logica 39, 125–141 (1980). https://doi.org/10.1007/BF00370316
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DOI: https://doi.org/10.1007/BF00370316