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QUANTIFIED INTUITIONISTIC LOGIC OVER METRIZABLE SPACES

Published online by Cambridge University Press:  08 April 2019

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

In the topological semantics, quantified intuitionistic logic, QH, is known to be strongly complete not only for the class of all topological spaces but also for some particular topological spaces — for example, for the irrational line, ${\Bbb P}$, and for the rational line, ${\Bbb Q}$, in each case with a constant countable domain for the quantifiers. Each of ${\Bbb P}$ and ${\Bbb Q}$ is a separable zero-dimensional dense-in-itself metrizable space. The main result of the current article generalizes these known results: QH is strongly complete for any zero-dimensional dense-in-itself metrizable space with a constant domain of cardinality ≤ the space’s weight; consequently, QH is strongly complete for any separable zero-dimensional dense-in-itself metrizable space with a constant countable domain. We also prove a result that follows from earlier work of Moerdijk: if we allow varying domains for the quantifiers, then QH is strongly complete for any dense-in-itself metrizable space with countable domains.

Keywords

03B20quantified intuitionistic logictopological semanticscompleteness
Type
Research Article
Information
The Review of Symbolic Logic , Volume 12 , Issue 3 , September 2019 , pp. 405 - 425
Copyright
Copyright © Association for Symbolic Logic 2019 

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