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QUANTIFIED MODAL LOGIC ON THE RATIONAL LINE

Published online by Cambridge University Press:  12 May 2014

PHILIP KREMER
PHILIP KREMER*
Affiliation:
Department of Philosophy, University of Toronto
*
*DEPARTMENT OF PHILOSOPHY, UNIVERSITY OF TORONTO SCARBOROUGH, 1265 MILITARY TRAIL, TORONTO, ON M1C 1A4, CANADA E-mail: kremer@utsc.utoronto.ca
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Abstract

In the topological semantics for propositional modal logic, S4 is known to be complete for the class of all topological spaces, for the rational line, for Cantor space, and for the real line. In the topological semantics for quantified modal logic, QS4 is known to be complete for the class of all topological spaces, and for the family of subspaces of the irrational line. The main result of the current paper is that QS4 is complete, indeed strongly complete, for the rational line.

Type
Research Article
Information
The Review of Symbolic Logic , Volume 7 , Issue 3 , September 2014 , pp. 439 - 454
Copyright
Copyright © Association for Symbolic Logic 2014 

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References

BIBLIOGRAPHY

Awodey, S., & Kishida, K. (2008). Topology and modality: The topological interpretation of first-order modal logic. Review of Symbolic Logic, 1, 146–166.CrossRefGoogle Scholar
Dragalin, A. G. (1988). Mathematical Intuitionism: Introduction to Proof Theory, Translations of Mathematical Monographs, Vol. 67, translated by Mendelson, E.. American Mathematical Society, Providence, RI. (Russian original, 1979)Google Scholar
Dugundji, J. (1966). Topology. Boston: Allyn and Bacon.Google Scholar
Engelking, R. (1989). General Topology. Berlin: Heldermann Verlag.Google Scholar
Gabbay, D. M., Shehtman, V. B., & Skvortsov, D. (2009). Quantification in nonclassical logic, Vol. 1. Amsterdam: Elsevier.Google Scholar
Goldblatt, R. (1980). Diodorean modality in Minkowski spacetime. Studia Logica, 39, 219–236.CrossRefGoogle Scholar
Hughes, G. E., & Cresswell, M. J. (1996). A New Introduction to Modal Logic. London: Routledge.Google Scholar
Kishida, K. (2006). Topological Semantics for First-Order Modal Logic. MSc Thesis, Carnegie-Mellon University.Google Scholar
Kremer, P. (2013). Strong completeness of S4 for any dense-in-itself metric space. Review of Symbolic Logic, 6, 545–570.Google Scholar
McKinsey, J. C. C. (1941). A solution of the decision problem for the Lewis systems S2 and S4, with an application to topology. The Journal of Symbolic Logic, 6, 117–134.Google Scholar
McKinsey, J. C. C., & Tarski, A. (1944). The algebra of topology. Annals of Mathematics, 45, 141–191.CrossRefGoogle Scholar
Rasiowa, H., & Sikorski, R. (1963). The Mathematics of Metamathematics, Państowowe Wydawnictwo Naukowe, Warsaw, 1963.Google Scholar
Steen, L., & Seebach, J. (1970). Counterexamples in Topology. Springer-Verlag, Holt, Rinehart and Winston.Google Scholar
van Benthem, J., Bezhanishvili, G., ten Cate, B., & Sarenac, D. (2006). Multimodal logics of products of topologies. Studia Logica, 84, 369–392.Google Scholar