I came to this book as a sociolinguist with an interest in semantics and some undergraduate background in mathematics. I was drawn in first by the title – the mathematical nature of formal semantics has always intrigued me, as have the parallels between structural linguistics more generally and mathematical approaches to formalizing abstract relationships. The second draw for me was the promo blurb on the back cover: ‘Mathematical Structures in Language introduces a number of mathematical concepts that are of interest to the working linguist. The areas covered include basic set theory and logic, formal languages and automata, trees, partial orders, lattices, Boolean structure, generalized quantifier theory, and linguistic invariants’ that is ‘[i]deal for advanced undergraduate and graduate students of linguistics’. I anticipated a book that would introduce these concepts in an accessible way, aimed at linguists who are intelligent but not necessarily up to speed with the mechanics and argumentation of mathematics. Unfortunately, that is not exactly what this book does.

It does cover the subject areas listed on the back, and in considerable detail, but the style and structure of the book as a whole are overwhelmingly those of a mathematics book that addresses language topics, rather than a linguistics book that addresses mathematical topics. The first two chapters do aim to introduce technical aspects of mathematical concepts (including argumentation, proofs, and the basics of formal logic) but these seem pitched much more as a refresher than as an introduction. The exercises throughout the book in principle could compensate for the lack of detailed explanation of core concepts, and indeed this approach is very evocative of most of the mathematics books I have encountered in my life. However, with no model answers provided (neither in the book nor, as far as I can tell, anywhere online), even keen and motivated linguists who lack sufficient training in mathematics will struggle to use this book for self-directed learning. Becoming adept at producing formal (or even informal) proofs in mathematics is a process that requires feedback, however minimal; with no opportunity to have mistakes pointed out and to develop argumentation skills, the naïve reader is likely to find a lot of this book hard going.

This is not to say that the content of the book itself is fundamentally flawed. The roles of formal logic and mathematical reasoning in structural linguistics – particularly in semantics (a primary concern in this volume) but also in syntax, morphology, and phonology – are important dimensions of formal analysis, and ones that are easy to overlook. In fact, Keenan & Moss cover this territory well: they apply a mathematical framework to the analysis of small-scale phenomena from reduced (i.e. hugely simplified and small) sets of language data, not to show that that they have solved the problem of modeling natural language, but instead to show that it is, in principle, possible to model them productively. It is in this sense that the text is introductory, and in this respect, it does what it sets out to do: namely, to demonstrate the value of their approach, and to suggest ways that this approach can enrich linguistic analysis more broadly.

The preface sets out the book’s two main purposes: (i) ‘to present mathematical background to help the student of linguistics formulate generalizations concerning the structure of natural languages’ (ix), which makes up the meat of the first five chapters; and (ii) to ‘study our formal models to derive further generalizations that were unsuspected, perhaps even unformulable, in the absence of a way to say them’ (ix), which is covered throughout the remaining eight chapters.

The book opens with an introductory discussion in Chapter 1 about why thinking about language in a structurally systematic way mirrors how mathematics thinks about relations. Here we are introduced to some of the linguistic concepts that will recur throughout the remainder of the book (e.g. iteration and recursion, relative clauses, DPs), and some of the mathematical concepts as well (e.g. functions, Boolean compounding). Chapter 2 introduces set theory and the mechanisms of mathematical proofs; and this chapter is inevitably where the lack of mathematical support for non-mathematicians begins to be felt. It includes some remarks about setting up proofs (Section 2.12) and provides a handful of sample proofs with limited explanation which nevertheless anticipate a fairly high degree of mathematical literacy on the part of readers. Chapter 3 begins to bring mathematical reasoning to bear on the description of sound systems, using phonology as an illustrative example of the utility of set theory when it comes to describing language phenomena – in this case, phonological features and their distributions. Chapter 4 moves from phonology to syntax, and explores the structural properties of trees and hierarchical (ordered) relations of constituents within trees. The mathematical concept of isomorphism is introduced here, as a relation between sets that preserves structures, which is relied upon in subsequent discussions. Chapter 5 extends this relational approach to syntax by defining some beginning rules for a grammar for (reduced) English, which is abstractly thought of as a set of lexical items and a set of rules for deriving expressions out of those lexical items. The initial focus is on declarative sentences: a handful of types of lexical items are introduced (although not always clearly defined in linguistic terms until later) as well as two rules: merge and coord, defined as ‘partial functions from $n$ -tuples of possible expressions to possible expressions’ (135). This framing of familiar syntactic processes (merging and coordinating) in the language of mathematical formalism serves to illustrate the inter-translatability of linguistics and mathematics.

Although there is no structural break after Chapter 5, Chapter 6 feels conceptually like the start of a second section of the book. This is where the focus seems to shift from showing that mathematics can be applied to linguistics to exploring what that shift can do, analytically. The principles of Context-Free Grammars (CFGs) (or ‘phrase structure rules’ as I learned them – e.g. Gazdar et al. Reference Gazdar, Klein, Pullum and Sag1985, Pollard & Sag Reference Pollard and Sag1994) are explored here, as are agreement systems and reduplication – which add another level of complexity to the grammatical categories, as well as introducing associated subcategory and agreement rules (161). Chapter 7 shifts from CFGs to regular languages and their associated ‘simple grammars’, which Keenan & Moss argue are more useful for some types of linguistic analyses (183). Their introduction of finite-state automata is nicely intuitive, although the following discussion of simple grammars relies quite heavily on the reader having come fully to grips with set theory as covered in previous chapters. Keenan & Moss do not claim that natural languages are regular languages, or that simple grammars (or indeed CFGs) can fully represent natural language, but rather that there are aspects of natural language that can be productively analyzed using these tools.

Starting in Chapter 8, there is a close engagement with the field of semantics, which is perhaps in some ways the most intuitive target for the mathematical representation of linguistics, given the prevalence of $\unicode[STIX]{x1D706}$ -calculus in generative semantics. These chapters focus to a large extent (although not exclusively) on DPs, and the various requirements and restrictions that they place on any mathematical formalism of natural language grammars. This approach generally works well, as it allows the authors to problematize earlier generalizations and add complexity to the (simplistic) models that they develop. Chapter 8 considers compositionality, truth conditions, and presupposition; it presents four properties of Sentential Logic (i.e. propositional logic) and applies these to a ‘small fragment of English’ (239) as an illustration of how semantic interpretation might work. Chapter 9 examines Boolean connectives and negation (either/or, neither/nor, both/and) in more detail, and introduces lattices as a way to demonstrate the complexity of equivalences between pairs of sentences, such that He wrote a novel or a play and He wrote a novel or he wrote a play can be shown to be equivalent, but Some student both laughed and cried and Some student laughed and some student cried cannot. Chapter 10 moves from Sentential Logic to First-Order Logic (FOL), and tackles issues of scope as they relate to quantifiers, negation, and the interpretation of objects (e.g. how wide or narrow is the reading of ‘every teacher’ in sentences such as Some student praised every teacher). Some of the general linguistic properties of FOL are discussed, and the syntax and semantics of the $\unicode[STIX]{x1D706}$ -operator (as a function creator, rather than as a pre-defined function in itself) are brought into the discussion of scope interpretation, within a framework of function application and function composition. Chapter 11 explores negative polarity items and monotonicity (upward- and downward-entailment), while Chapter 12 takes a closer look at quantifiers more broadly. The generalizations from these two chapters are extended to cover more than DPs, but as noted above, the cyclical focus on DPs works well to tie these disparate strands of mathematical and semantic reasoning together.

Chapter 13 concludes the book with an acknowledgment that the (reduced) grammars discussed up to this point are clearly not adequate to model natural languages, but that the overall aim of the book (and the mathematics program more generally) has been to develop a grammar ‘with sufficient generality to support general, ideally universal claims that adequate grammars of all natural languages meet’ (391). To this end, they introduce and extend Keenan & Stabler’s (Reference Keenan and Stabler2003) notion of structural invariants – linguistic objects ‘that cannot be changed without changing the structure of expressions’ (398) – as a way to generalize certain types of relations (e.g. anaphora) across typologically different languages. They end with some empirical linguistic lines of inquiry that could likely benefit from their mathematical approach, and share some observations on linguistic invariants that invite closer inspection. The book itself ends a bit abruptly, but it leaves the reader in a position to think further on some of the implications and possibilities of mathematical formulations in the general field of linguistics.

Although the content is linguistically interesting, it is couched in linguistically-inaccessible language and mathematical formalism, making it a decidedly non-introductory text for the interested linguist. The linguistic concepts are introduced at a more foundational level than the mathematical ones, so – promotional jacket material aside – it raises a question of who the book is actually intended for. Linguists without a background in mathematics will be likely to struggle with the mechanics of the approach and the style of the writing; and it is not clear that there are any new purely mathematical insights, so it is equally uncertain whether mathematicians will be particularly drawn in by the material. On balance, the intended audience is more likely to be computer science students with an interest in computational linguistics and/or the syntax of programming languages – and indeed, Keenan & Moss’s discussion of FOL signals this natural alignment: ‘FOL is well studied and has many appealing linguistic properties, to the point where it is not foolish to think of mathematical logic as a mode of linguistic analysis – but the languages studied are mathematical ones …and increasingly programming languages in computer science’ (304). Although they limit this observation to the limited case of FOL, the fact is that this book as a whole seems pitched for a specialist audience.

This is not precisely a failing of the book per se, but it certainly suggests that the jacket material should be revised to more accurately represent the material to potential readers. It was an interesting but technically challenging book to read, and I confess that I gave up trying to do the exercises halfway through Chapter 3 (although I probably would have stuck it out for at least another few chapters, with access to an answer key). As a text for linguists, this book has conceptual potential, but the execution makes it less than ideal for anyone not already comfortable and confident in the mathematics milieu.