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CONDITIONAL BELIEFS: FROM NEIGHBOURHOOD SEMANTICS TO SEQUENT CALCULUS

Published online by Cambridge University Press:  28 June 2018

MARIANNA GIRLANDO ,
SARA NEGRI ,
NICOLA OLIVETTI  and
VINCENT RISCH
MARIANNA GIRLANDO*
Affiliation:
Aix-Marseille University; Department of Philosophy, University of Helsinki
SARA NEGRI*
Affiliation:
Department of Philosophy, University of Helsinki
NICOLA OLIVETTI*
Affiliation:
Aix-Marseille University
VINCENT RISCH*
Affiliation:
Aix-Marseille University
*
*AIX-MARSEILLE UNIVERSITY UNIVERSITÉ DE TOULON CNRS, LIS MARSEILLE, FRANCE and DEPARTMENT OF PHILOSOPHY UNIVERSITY OF HELSINKI HELSINKI, FINLAND E-mail: marianna.girlando@univ-amu.fr
†DEPARTMENT OF PHILOSOPHY UNIVERSITY OF HELSINKI HELSINKI, FINLAND E-mail: sara.negri@helsinki.fi
‡AIX-MARSEILLE UNIVERSITY UNIVERSITÉ DE TOULON CNRS, LIS MARSEILLE, FRANCE E-mail: nicola.olivetti@univ-amu.fr
§AIX-MARSEILLE UNIVERSITY UNIVERSITÉ DE TOULON CNRS, LIS MARSEILLE, FRANCE E-mail: vincent.risch@univ-amu.fr
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Abstract

The logic of Conditional Beliefs (CDL) has been introduced by Board, Baltag, and Smets to reason about knowledge and revisable beliefs in a multi-agent setting. In this article both the semantics and the proof theory for this logic are studied. First, a natural semantics for CDL is defined in terms of neighbourhood models, a multi-agent generalisation of Lewis’ spheres models, and it is shown that the axiomatization of CDL is sound and complete with respect to this semantics. Second, it is shown that the neighbourhood semantics is equivalent to the original one defined in terms of plausibility models, by means of a direct correspondence between the two types of models. On the basis of neighbourhood semantics, a labelled sequent calculus for CDL is obtained. The calculus has strong proof-theoretic properties, in particular admissibility of contraction and cut, and it provides a decision procedure for the logic. Furthermore, its semantic completeness is used to obtain a constructive proof of the finite model property of the logic. Finally, it is shown that other doxastic operators can be easily captured within neighbourhood semantics. This fact provides further evidence of the naturalness of neighbourhood semantics for the analysis of epistemic/doxastic notions.

Keywords

03B4203F0554A99conditional doxastic logicepistemic logicneighbourhood semanticsplausibility semanticsAlexandroff correspondencelabelled sequent calculusstructural proof theorycompletenessdecidabilitycanonical models
Type
Research Article
Information
The Review of Symbolic Logic , Volume 11 , Issue 4 , December 2018 , pp. 736 - 779
Copyright
Copyright © Association for Symbolic Logic 2018 

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