Springer, 2007, XXII, 1058 Pages, ISBN 978-1-4020-5586-7.

F.4.1 Mathematical logic - first-order logic, modal logic, temporal logic, spatial logic

I.2.4 Knowledge representation formalisms and methods - modal logic, temporal logic, spatial logic

G.2.3 Discrete mathematics applications - geometry

A.2 Reference

Just as the study of time brings in properties of structures, such as the real line, the infinite binary tree and other more esoteric worlds, the study of space brings in aspects like topology, geometry and more abstract structures, like toposes. In the first chapter, the editors address the issue of introducing what spatial logic is, and make the analogy to its more famous counterpart, temporal logic. Six of the next seven articles in this volume elucidate this aspect. Particularly, I would like to mention two: "Modal logics of space" by van Benthem (one of the editors) and Bezhanishvili, and "Topology and epistemic logic" by Parikh, Moss and Steinsvold.

The subject matter of both chapters is virtually the same: defining modal logics which deal with space and topology. Van Benthem and Bezhanishvili take the modal correspondence approach (as in van Benthem's well-known chapter in [1]). They begin with what various modal logic axioms correspond to when given different kinds of spatial and topological semantics and move on to connections with other logics: epistemic, first-order, linear, arrow and hybrid. Parikh et al start from the topological end: they fix the mathematical structures and ask what modal, epistemic and doxastic logics can say about them. For example, can completeness be proven? I found reading these two chapters and watching the interplay between them very rewarding. Put together they serve as a superb introduction to this area. The combination of a good mathematical introduction and a research-level survey is the hallmark of most of this book.

Pratt-Hartmann's chapter on mereotopology, Bennett and Düntsch's chapter on a relation algebraic treatment, and Balbiani, Goranko, Kellerman and Vakarelov's chapter on axiomatizing geometry are in the more conventional first-order setting. Vickers's piece on locales and toposes deals with higher-order structures. The remaining chapter in this part, by Renz and Nebel, is a wonderful introduction to spatial reasoning algorithms by first introducing algorithms for temporal reasoning, and then generalizing them.

At this stage, after taking a rather detailed look at spatial logic from various viewpoints, the book turns to look at its counterpart again. When put together, what could spatial and temporal logic be like? There are three chapters on this aspect. The first is by Kontchakov, Kurucz, Wolter and Zakharyaschev, and the other is by Kremer and Mints---two variations on the same theme are offered. The first chapter is encyclopedic, with a focus on decidability aspects of combination logics. This is a natural concern, since the two dimensions of space and time allow easy coding of undecidable problems. The second article fixes one particular logic, an extension of Pratt's propositional dynamic logic for topological semantics, and explores how far completeness, the finite model property and decidability can be carried through. The chemistry between the two articles is not as sparkling as the articles mentioned earlier, but it is only fair to say that this area of research is far less developed.

The third chapter, a tour de force, is by Andréka, Madarász and Németi, from the Hungarian school. It's nothing less than an attempt to prove Einstein's relativity theory of space-time (both special and general) from axioms, entirely within logic. I confess I understood very little by the time I got to black holes, wormholes and timewarps, but this article promises to be richly rewarding on a second reading. It merits a seminar course by itself, for interested graduate students.

I enjoyed reading Smyth and Webster's article on discrete models. It is a refreshing look at how discrete mathematics can approach the area of spatial logic. The article by Geerts and Kuijpers talks about the connection to databases, especially the GIS kind. I would have liked to learn a little more about real algebraic geometry from the algorithmic side. The article by Bloch, Heijmans and Ronse, "Mathematical morphology," is an introduction to the mathematics of space from an image-processing perspective. Varzi rounds off the book with a philosophical perspective on the part-whole relation.

The editors of this handbook deserve to be congratulated for commissioning and bringing together top-notch contributions on a wide variety of concerns in philosophy, logic and computing science under the rubric of spatial logic. The idea of grouping together several articles on a subfield has worked very well. The index seems adequate, and I can't think of any aspect of the subject which has been left out. Apart from being a reference, this handbook also serves to cover the entire field of spatial logics.

[1] Gabbay, D.; Guenthner, F.
*Handbook of philosophical logic.* D. Reidel, Boston, MA, 1984.