Multi-dimensional modal logic
type: Item
instance of: scholarly article
title: Multi-dimensional modal logic
zbMATH Open document ID: 0942.03029
author: Maarten Marx
author: Yde Venema
published in: Applied Logic Series
publication date: 1998-02-25T00:00:00Z
review text: This is an advanced textbook on extending modal logic to higher dimensions. It originates from the dissertations of the two authors at the University of Amsterdam, one of the capitals of modal logic in the world; in fact, one can feel its inspiring influence (including relations with other centres) throughout the book. It is advantageous to the reader to have gained experience of (modal) logic to a certain extent already before reading the book. Although all the required material is listed, its presentation is rather concise with only few concessions to the needs of beginners. Moreover, not only the fundamental principles of modal logic are presupposed, but also more advanced topics like Sahlqvist theory. Finally, due to the origin of many systems considered in the book, an algebraic point of view of modal logic is always present; so the reader will surely make utilizable his knowledge of basic algebra. The book is divided into six chapters followed by two appendices. The appendices contain the prerequisites mentioned above and should be read just after the introductory Chapter 1. They not only provide one with the designations chosen by the authors, but also contain an interestingly arranged and worth reading extract from the essentials of modal logic, by the way. Chapter 2 deals with two-dimensional modal logic. First, several modalities are introduced and interpreted as relations over squares \(U\times U\), where \(U\) is some base set; among these are, e.g.: decomposing the square into lines which are in parallel with an axis, reversing, and composition. Afterwards, the two-dimensional modal logic of squares is examined with respect to two modal language types of ascending expressiveness. Corresponding completeness proofs are given, introducing the `step-by-step method' crucial for realizing maximal consistent sets of formulas. This method is refined appropriately and used again and again in the book. The modal logic of composition is treated in a separate secti…
Mathematics Subject Classification ID: 03B45
Mathematics Subject Classification ID: 03-02
zbMATH DE Number: 1120788
zbMATH Keywords: multi-dimensional modal logic
zbMATH Keywords: expressiveness
zbMATH Keywords: axiomatizability of classes of multi-dimensional frames
zbMATH Keywords: mosaic method
zbMATH Keywords: step-by-step method
zbMATH Keywords: arrow logic
zbMATH Keywords: modal logic of intervals
zbMATH Keywords: modal logics of relations
zbMATH Keywords: multi-dimensional semantics
reviewed by: Bernhard Heinemann
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1120788