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REALIZABILITY SEMANTICS FOR QUANTIFIED MODAL LOGIC: GENERALIZING FLAGG’S 1985 CONSTRUCTION

Published online by Cambridge University Press:  04 April 2016

BENJAMIN G. RIN  and
SEAN WALSH
BENJAMIN G. RIN*
Affiliation:
Institute for Logic, Language and Computation at the University of Amsterdam
SEAN WALSH*
Affiliation:
University of California, Irvine
*
*FACULTEIT DER NATUURWETENSCHAPPEN, WISKUNDE EN INFORMATICA INSTITUTE FOR LOGIC, LANGUAGE, AND COMPUTATION UNIVERSITEIT VAN AMSTERDAM P.O. BOX 94242 1090 GE AMSTERDAM E-mail: benjamin.rin@gmail.com
†DEPARTMENT OF LOGIC AND PHILOSOPHY OF SCIENCE 5100 SOCIAL SCIENCE PLAZA UNIVERSITY OF CALIFORNIA, IRVINE IRVINE, CA 92697-5100, U.S.A. E-mail: swalsh108@gmail.com or walsh108@uci.edu
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Abstract

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A semantics for quantified modal logic is presented that is based on Kleene’s notion of realizability. This semantics generalizes Flagg’s 1985 construction of a model of a modal version of Church’s Thesis and first-order arithmetic. While the bulk of the paper is devoted to developing the details of the semantics, to illustrate the scope of this approach, we show that the construction produces (i) a model of a modal version of Church’s Thesis and a variant of a modal set theory due to Goodman and Scedrov, (ii) a model of a modal version of Troelstra’s generalized continuity principle together with a fragment of second-order arithmetic, and (iii) a model based on Scott’s graph model (for the untyped lambda calculus) which witnesses the failure of the stability of nonidentity.

Type
Research Article
Information
The Review of Symbolic Logic , Volume 9 , Issue 4 , December 2016 , pp. 752 - 809
Copyright
Copyright © Association for Symbolic Logic 2016 

References

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