Review: Alan Bundy, The Computer Modelling of Mathematical Reasoning

phisms, and still others by spaces preserving some sort of structure definable in the language of Boolean algebra. Following a short overview that highlights some of the underlying themes, the book is divided into two parts. Part I presents an extensional system that treats the categories mentioned above and handles constructions such as passives, reflexives, and relative clauses. One of the new aspects of this work is that semantic subcategories can be described in Boolean terms. For example, the extensional adjectives (e.g. tall, female) are interpreted by restricting functions (f(x) g x) on the NP algebra. This is because such adjectives validate entailment patterns exemplified by Every tall thief is a thief. Non-restricting adjectives such as alleged fail to validate such patterns. A mathematically interesting feature of the extensional system is that the development requires a new theorem of Boolean algebra. Let A and B be complete atomic Boolean algebras, and let A* be the powerset algebra of A. For every function / from the principal ultrafilters of A into B, there is a unique complete homomorphism g from A* to B extending /'. This result is used many times in defining and studying semantic spaces. The uses of this theorem constitute an original application of mathematical tools to linguistics. This book contains several other applications not discussed here. Part II of the book extends the system to consider intensional phenomena such as sentential complements (via bar categories! and de re -de dictu ambiguity. The extension of the semantics uses possible worlds. It leads to some technical problems that, as the authors point out, are not fully resolved. Judgments concerning intensional entailments are often tentative, and this causes the reader to question the value of a model-theoretic approach to intensional semantics. In balance, I think it is fair to say that the most fruitful application of the Boolean approach has been in the extensional system. It might be useful to list briefly some of the ways in which BSNL differs from Montague grammar and its extensions. As mentioned above, these include the introduction of ideas from Boolean algebra and the ontological shift from entities to properties. Another significant difference is the lack of an intermediary level of intensional logic. That is, the entire fragment is interpreted directly without recourse to a translation into a separate language (as in PTQ, for example). This is an advantage, making the system easier to understand and work with. Although not stressed in the book, there are some problem areas arising from this work that might be of interest to logicians. The study of quantifiers in natural language by Keenan and Stavi (and related work of Barwise and Cooper) is certainly one of them. Another possible project might be to investigate the meaning-preserving properties of non-classical connectives—all of the sentential connectives are interpreted classically in BSNL or to see whether a meaningful interpretation of non-classical logic is possible in the framework of Boolean semantics. A final problem area concerns the semantic spaces. Even for very small sets of basic entities or properties, the sizes of some of the interpretation spaces (e.g. of PP) are very large. Indeed, they are superexponential in the size of the set of basic entities. They are smaller than the spaces of Montague grammar, but one should look for further restrictions with an eye toward increasing the explanatory power of a semantic system and making it computationally cogent. This book would be of value to anyone with an interest in these areas. It will also be useful to someone interested in the interaction of syntax and semantics because Keenan and Faltz use semantics as a guide for syntax. This book will be accessible to a mathematically capable reader with an interest in semantics. The ideal reader would be familiar both with elementary Boolean algebra and with higher-type functions as they occur in Montague grammar. A background in semantics is not presumed. The authors discuss alternative and competing analyses of some phenomena, but these sections can be omitted without losing the thread of the book. Since the authors are principally concerned with the presentation of a fragment, they do not address methodological issues. It would have been useful to have done so, and to have more explicit about the advantages of the Boolean approach and about semantic problems that (in their judgment) will require different methods. LAWRENCE S. MOSS