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STABLE MODAL LOGICS

Published online by Cambridge University Press:  27 September 2018

GURAM BEZHANISHVILI ,
NICK BEZHANISHVILI  and
JULIA ILIN
GURAM BEZHANISHVILI*
Affiliation:
Department of Mathematical Sciences, New Mexico State University
NICK BEZHANISHVILI*
Affiliation:
Institude for Logic, Language and Computation, University of Amsterdam
JULIA ILIN*
Affiliation:
Institude for Logic, Language and Computation, University of Amsterdam
*
*DEPARTMENT OF MATHEMATICAL SCIENCES NEW MEXICO STATE UNIVERSITY LAS CRUCES, NM 88003, USA E-mail: guram@nmsu.edu
†INSTITUTE FOR LOGIC, LANGUAGE AND COMPUTATION UNIVERSITY OF AMSTERDAM P.O. BOX 94242, 1090 GE AMSTERDAM, THE NETHERLANDS E-mail: N.Bezhanishvili@uva.nl
‡INSTITUTE FOR LOGIC, LANGUAGE AND COMPUTATION UNIVERSITY OF AMSTERDAM P.O. BOX 94242, 1090 GE AMSTERDAM, THE NETHERLANDS E-mail: ilin.juli@gmail.com
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Abstract

Stable logics are modal logics characterized by a class of frames closed under relation preserving images. These logics admit all filtrations. Since many basic modal systems such as K4 and S4 are not stable, we introduce the more general concept of an M-stable logic, where M is an arbitrary normal modal logic that admits some filtration. Of course, M can be chosen to be K4 or S4. We give several characterizations of M-stable logics. We prove that there are continuum many S4-stable logics and continuum many K4-stable logics between K4 and S4. We axiomatize K4-stable and S4-stable logics by means of stable formulas and discuss the connection between S4-stable logics and stable superintuitionistic logics. We conclude the article with many examples (and nonexamples) of stable, K4-stable, and S4-stable logics and provide their axiomatization in terms of stable rules and formulas.

Keywords

03B4503B55modal logicmodal consequence relationcanonical formulacanonical rulefinite model propertyaxiomatizationintuitionistic logic
Type
Research Article
Information
The Review of Symbolic Logic , Volume 11 , Issue 3 , September 2018 , pp. 436 - 469
Copyright
Copyright © Association for Symbolic Logic 2018 

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References

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