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INSTANTIAL NEIGHBOURHOOD LOGIC

Published online by Cambridge University Press:  19 December 2016

JOHAN VAN BENTHEM ,
NICK BEZHANISHVILI ,
SEBASTIAN ENQVIST  and
JUNHUA YU
JOHAN VAN BENTHEM*
Affiliation:
Institute for Logic, Language and Computation, University of Amsterdam; Department of Philosophy, Stanford University; and Changjiang Scholar Program, Tsinghua University
NICK BEZHANISHVILI*
Affiliation:
Institute for Logic, Language and Computation, University of Amsterdam
SEBASTIAN ENQVIST*
Affiliation:
Institute for Logic, Language and Computation, University of Amsterdam
JUNHUA YU*
Affiliation:
Department of Philosophy, Tsinghua University
*
*INSTITUTE FOR LOGIC, LANGUAGE AND COMPUTATION UNIVERSITY OF AMSTERDAM, P.O. BOX 94242 1090 GE AMSTERDAM, THE NETHERLANDS DEPARTMENT OF PHILOSOPHY STANFORD UNIVERSITY, CA USA and CHANGJIANG SCHOLAR PROGRAM TSINGHUA UNIVERSITY BEIJING 100084 CHINA E-mail: [email protected]
†INSTITUTE FOR LOGIC, LANGUAGE AND COMPUTATION UNIVERSITY OF AMSTERDAM, P.O. BOX 94242 1090 GE AMSTERDAM, THE NETHERLANDS E-mail: [email protected]
‡INSTITUTE FOR LOGIC, LANGUAGE AND COMPUTATION UNIVERSITY OF AMSTERDAM, P.O. BOX 94242 1090 GE AMSTERDAM, THE NETHERLANDS E-mail: [email protected]
§DEPARTMENT OF PHILOSOPHY TSINGHUA UNIVERSITY BEIJING 100084 CHINA E-mail: [email protected]
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Abstract

This paper explores a new language of neighbourhood structures where existential information can be given about what kind of worlds occur in a neighbourhood of a current world. The resulting system of ‘instantial neighbourhood logic’ INL has a nontrivial mix of features from relational semantics and from neighbourhood semantics. We explore some basic model-theoretic behavior, including a matching notion of bisimulation, and give a complete axiom system for which we prove completeness by a new normal form technique. In addition, we relate INL to other modal logics by means of translations, and determine its precise SAT complexity. Finally, we discuss proof-theoretic fine-structure of INL in terms of semantic tableaux and some expressive fine-structure in terms of fragments, while discussing concrete illustrations of the instantial neighborhood language in topological spaces, in games with powers for players construed in a new way, as well as in dynamic logics of acquiring or deleting evidence. We conclude with some coalgebraic perspectives on what is achieved in this paper. Many of these final themes suggest follow-up work of independent interest.

Keywords

03B45modal logicneigbourhood semanticsbisimulationsaxiomatizationcompletenesstranslationstopological semanticsgame logiccoalgebraic logic
Type
Research Article
Information
The Review of Symbolic Logic , Volume 10 , Issue 1 , March 2017 , pp. 116 - 144
Copyright
Copyright © Association for Symbolic Logic 2016 

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