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STRONG COMPLETENESS OF S4 FOR ANY DENSE-IN-ITSELF METRIC SPACE

Published online by Cambridge University Press:  20 June 2013

PHILIP KREMER
PHILIP KREMER*
Affiliation:
Department of Philosophy, University of Toronto
*
*DEPARTMENT OF PHILOSOPHY, UNIVERSITY OF TORONTO, 170 ST. GEORGE STREET, TORONTO ON, CANADA, M5R 2M8 E-mail: kremer@utsc.utoronto.ca
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Abstract

In the topological semantics for modal logic, S4 is well-known to be complete for the rational line, for the real line, and for Cantor space: these are special cases of S4’s completeness for any dense-in-itself metric space. The construction used to prove completeness can be slightly amended to show that S4 is not only complete, but also strongly complete, for the rational line. But no similarly easy amendment is available for the real line or for Cantor space and the question of strong completeness for these spaces has remained open, together with the more general question of strong completeness for any dense-in-itself metric space. In this paper, we prove that S4 is strongly complete for any dense-in-itself metric space.

Type
Research Article
Information
The Review of Symbolic Logic , Volume 6 , Issue 3 , September 2013 , pp. 545 - 570
Copyright
Copyright © Association for Symbolic Logic 2013 

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