Topology and duality in modal logic
type: Item
instance of: scholarly article
title: Topology and duality in modal logic
zbMATH Open document ID: 0643.03014
DOI: 10.1016/0168-0072(88)90021-8
author: Giovanni Sambin
author: Virginia Vaccaro
published in: Annals of Pure and Applied Logic
publication date: 1988-01-01T00:00:00Z
review text: There are two major approaches to the semantics of propositional modal logics: the algebraic, as exemplified by modal algebras (i.e., Boolean algebras with an extra unary operation); and the relational, as exemplified by Kripke frames (i.e., sets with a binary operation). Each approach gives rise to an associated first order language, and much research in this area has centered around completeness theorems roughly of the form: sentence \(\sigma\) is a theorem of a set \(\Gamma\) of sentences relative to such-and-such a modal logic if and only if \(\sigma\) is a consequence of \(\Gamma\) is every modal algebra (or frame) that satisfies such-and-such (perhaps first order) conditions. Of course there are many variations on this theme, and it is natural to assume any general theory that connects the two semantical approaches mentioned above has potential for shedding significant light on completeness phenomena, as well as being of intrinsic interest. The paper under review not only gives a quite readable and leisurely introduction to the important issues of the subject, but delineates a new connection between the algebraic and relational approaches, said connection taking the form of an adjoint pair of functors between certain categories. (This adjunction extends known duality theorems that arise from an analysis of the classical duality of M. H. Stone. This is where topology enters the picture.) The paper is divided into three chapters. Chapter I introduces the reader to modal semantics, discusses the ideas upon which duality theory is based, and points out the relevance of these ideas to completeness theorems. Chapter II develops the technical machinery leading to the adjoint functor theorem alluded to above. Finally, Chapter III gives applications of the theory developed in Chapter II; one such is a simple description of the structure of modal classes of ``generalized'' Kripke frames.
Mathematics Subject Classification ID: 03B45
Mathematics Subject Classification ID: 18A40
zbMATH DE Number: 4047687
zbMATH Keywords: general frames
zbMATH Keywords: valuations
zbMATH Keywords: duality
zbMATH Keywords: adjoint functors
zbMATH Keywords: semantics of propositional modal logics
zbMATH Keywords: modal algebras
zbMATH Keywords: adjoint pair of functors
zbMATH Keywords: completeness theorems
zbMATH Keywords: Kripke frames
reviewed by: Paul Bankston
MaRDI profile type: Publication
full work available at URL: 0168-0072(88)90021-8
OpenAlex ID: W2071467583
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description: scientific article; zbMATH DE number 4047687