Modern modal logic originated as a branch of philosophical logic in which the concepts of necessity and possibility were investigated by means of a pair of dual operators ( and [diamond]) that are added to a propositional or first-order language. The field owes much of its flavor and success to the introduction in the 1950s of the "possible-worlds" semantics in which the modal operators are interpreted via some "accessibility relation" connecting possible worlds. In subsequent years, modal logic has received attention as an attractive approach towards formalizing such diverse notions as time, knowledge, or action. Nowadays, modal logics are applied in various disciplines, ranging from economics to linguistics and computer science. Consequently, there is by now a large variety of modal languages, with an even greater wealth of interpretations. For instance, many applications require a poly-modal framework consisting of a language with a family of modal operators and a semantics in which the corresponding accessibility relations are connected somehow.
But despite this enormous diversity in appearance and applications, there is still a huge trunk to the tree of modal logic. In particular, the mathematical degree of sophistication that the field has reached over the years allows a nice general and abstract perspective on modal logic. Until recently, this perspective had not been put forward in an advanced textbook or monograph, and it is this gap that Chagrov and Zakharyaschev aim to fill with their book. Now it is true, as the authors mention in their introduction, that "modal logic is too extensive a field to be covered comprehensively only by one book," and they have chosen to restrict themselves entirely to the mathematical side of modal logic. That is, the book contains no discussion of the philosophical background or relevance of modal logic; in fact, it does not discuss applied modal logic at all.
Its aim is to abstract from particular systems of modal logic, instead providing a systematic treatment of important methods for determining whether a given logic has certain desirable properties such as Kripke completeness or decidability. The word logic is to be taken in the technical sense here, meaning a set of formulas that is closed under certain inference rules such as Modus Ponens, Necessitation, and Substitution. This is not to say that a syntactic or axiomatic viewpoint prevails: most of the logics appearing in the book are specified semantically as the collection of formulas that are valid in a certain class of (algebraic or relational) structures, many of the properties of modal logics that the authors discuss are semantically oriented, and most of the proofs in the book use semantic methods.
But even within these constraints, the authors have made a number of considerable restrictions concerning the volume's content. For instance, the reader will not find any material on modal predicate logic or on the proof theory of modal logic. This is quite understandable, since these are quite specific topics requiring monographs on their own. But some of the other choices I found quite debatable, such as the authors' exclusive focus on logics in uni-modal languages. There are many interesting and important problems concerning the interaction of distinct modal operators and none of these have found their way into this book. Another questionable (near) omission concerns the notion of a bisimulation between two models; this notion, which is of fundamental importance in modal logic, is only mentioned in one of the exercises. And finally, while the authors provide a fairly systematic discussion of modal logic from an algebraic perspective, the equally useful relation with first-order logic and its model theory has received a rather stepmotherly treatment.
Before I give a more detailed overview of the book's contents, it should be stressed that its title does not reflect its content properly, since intuitionistic logic and its so-called superintuitionistic extensions play just as leading a role as modal logics. This is quite natural from the authors' technical perspective--if it were not well known already that modal and superintuitionistic logics have many characteristics in common, this book would quite convincingly display it. But for the reader who is primarily interested in one of these two fields, a clearer separation and signposting of the two tracks would have been desirable.
Contents. The book consists of eighteen chapters divided over five parts. In the first part the authors introduce classical, intuitionistic, and modal propositional logic, and some of the most important properties pertaining to these logics, such as decidability, independent axiomatizability, and interpolation. They primarily take a semantic perspective on logics, concentrating on models, frames, and operations on these structures. But there is also a syntactic approach based on semantic tableaux and Hilbert-style calculi, and the chapters contain completeness results for some well-known logics. In the last chapter of part 1 the authors formulate their program, which is to develop a general theory providing tools for solving problems not for each modal or superintuitionistic logic individually, but for big classes of these at once.
Part 2 starts with discussing some of these tools, namely the method of canonical models for proving Kripke completeness, and the method of filtration for showing that a logic is finitely approximable (which is the authors' terminology for "having the finite model property"). The second chapter of this part provides a number of negative results, displaying modal logics lacking some desirable properties such as finite approximability, Kripke completeness, or canonicity.
In the third part of the book, the authors lay the semantic foundations for investigating modal and superintuitionistic logics. They introduce adequate semantics for modal logics in the form of modal algebras and of general frames, and likewise for superintuitionistic logics. The connections, both between the algebraic and generalized Kripke semantics, and between the modal and intuitionistic semantics, are discussed in detail. The last chapter of part 3 develops a powerful technique (for transitive frames only) of a somewhat different flavor; roughly, the gist of this method is that the geometry of a general frame refuting a given modal formula can be described quite elegantly in terms of the frame's relation to a finite set of finite Kripke frames.
Parts 4 and 5 then put this machinery to work. Part 4 contains six chapters, subsequently providing a detailed discussion of the following properties: Kripke completeness, finite approximability, tabularity, Post completeness, interpolation, and some disjunction properties. Finally, the last part of the book focuses on algorithmic aspects of modal logics, discussing not only the decidability and complexity of logics themselves but also the question whether it is decidable whether an effectively presented logic has a certain property.
Review. In the preface the authors express their hope that they have found "a reasonable compromise between a textbook and a monograph." I think that they have succeeded quite well: the book is certainly well-structured and any student or researcher mastering its content will be well equipped to tackle technical problems in modal logic. Each chapter has an ample supply of good exercises and finishes with adequate and interesting historical notes. The exposition of the material is clear, although a caveat is in order: this work is certainly not suitable for mathematical novices--although the book does not presuppose any background knowledge, one certainly needs mathematical sophistication and experience to follow the proofs.
Working their way through the book, many readers will encounter asecond, and less pardonable, hurdle: the poverty of the book's navigation system. While the authors use a wealth of terminology and notation, they provide no such thing as a list of symbols, and only a rudimentary index. This severely reduces the value of Modal Logic as a work of reference, and discourages any reading of it other than from cover to cover. Fortunately, this fault could easily be remedied in a second edition.
There are some other shortcomings as well: for instance, the authors' decision to confine themselves to modal languages with one diamond only makes some of their examples rather involved. And finally, the authors do not waste many words on motivation, at times leaving the impression that issues are interesting because they have been investigated instead of the other way around.
Nevertheless, despite my qualms concerning the book's content and its shortcomings in presentation, I think this is a great work and I am very glad to have it. A modern textbook on the mathematics of modal logic was long due, and this work fills the gap perfectly. I expect that it will become one of the standard references in the field of modal logic.
University of Amsterdam
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|Publication:||The Philosophical Review|
|Article Type:||Book Review|
|Date:||Apr 1, 2000|
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