Multi-dimensional modal logic
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 section, and a certain two-dimensional temporal logic as well. Besides the keyword topics also the question of decidability comes into play for the first of these systems. Each of the chapters 3-6 continues (at least) one section of Chapter 2. The subject of Chapter 3 is arrow logic, a `modal logic of arrows'. The latter represent in particular various kinds of `dynamics'. Arrows are taken as original objects rather than pairs in the `plane' \(U\times U\); nevertheless, there are close connections with the modal logic of composition. In Chapter 4 a slight modification of a modal logic of intervals due to \textit{J. Y. Halpern} and \textit{Y. Shoham} [Proc.\ Logic in Computer Science, Washington, IEEE Computer Society Press, 279-292 (1986)] is treated in the framework of two-dimensional modal logic. Modal logics of dimension \(n\geq 2\) are studied in Chapter 5. This chapter is entitled `Modal Logics of Relations' since a modalization of first-order predicate calculus over a relational type is provided (among other things). It turns out that conditions differ considerably in dimensions \(n>2\) from those in dimension \(n=2\), as it is often the case in mathematics. In the final chapter the authors show how every modal language can be given a multi-dimensional semantics. The issues of the authors' investigations first and foremost concern those topics mentioned as keywords above: expressiveness of the languages and (non-)axiomatizability of the corresponding multi-modal logics considered in the course of the book. In addition, decidability and the Craig interpolation property are (dis)proved in most cases. Several tables help the reader to orientate himself. Step-by-step constructions constitute an important method in completeness proofs, as has already been indicated; another one, called `mosaic method', supports these procedures to a great extent. Besides, negative definability and unorthodox axiomatics as well as weakening the semantics have been applied in order to obtain such results. The book contains convincing motivations, is mainly well-structured, and clearly and elegantly written; however, some more details would have been appreciated here and there. In short, it is highly recommended to people from a background of modal logic. Meanwhile one can already discover the effects of the book in the literature, for instance on determining the complexity of multi-dimensional modal logics.
axiomatizability of classes of multi-dimensional frames
scientific article; zbMATH DE number 1120788