Mathematical Methods in Linguistics

1. All other texts on the mathematics of language are now obsolete. Therefore, instead of going on about what a wonderful job Partee, ter Meulen, and Wall (henceforth, PMW) have done in some ways (breadth of coverage, much better presentation of formal semantics than is usual in books on mathematics of language, etc.), I will leave the lily ungilded, and focus on some points where the book under review could be made far better than it actually is. 2. Perhaps my main complaint concerns the treatment of the connections between the mathematical methods and the linguistics. This whole question is dealt with rather unevenly, and this is reflected in the very structure of the book. The major topics covered, corresponding to the book's division into parts (which are then subdivided into chapters) are set theory, logic and formal systems, algebra, "English as a formal language" (this is the heading under which compositionality, lambda-abstraction, generalized quantifiers, and intensionality are discussed), and finally formal language and automata theory. Now, the "English as a formal language" part deals with a Montague-style treatment of this language, but it does not go into contemporary syntactic analyses of English, not even ones that are mathematically precise and firmly grounded in formal language theory. Having praised the book for its detailed discussion of the uses of formal semantics in linguistics, I must damn its cavalier treatment of the uses of formal syntax. Thus, there is no mention anywhere in it of generalized phrase structure grammar or X-bar syntax or almost anything else of relevance to modern syntactic theory. Likewise, although the section on set theory deals at some length with nondenumerable sets, there is no mention of the argument of Langendoen and Postal (1984) that NLs are not denumerable. Since this is perhaps the one place in the literature where set theory and linguistics meet, one does not have to be a fan of Langendoen and Postal to see that this topic should be broached. 3. Certain important theoretical topics, usually ones at the interface of mathematics and linguistics, are presented sketchily and even misleadingly; for example, the compositionality of formal semantics, the generative power of transformational grammar, the nonregularity and noncontext freeness of NLs, and (more generally) the question of what kinds of objects one can prove things about. Let us begin with the principle of compositionality (i.e., that "the meaning of a complex expression is a function of the meanings of its parts and of the syntactic rules by which they are combined"). PMW claim that "construed broadly and vaguely