Journal of Logic, Language and Information8: 259–261, 1999. 259

Book Review

Basic Model Theory, Kees Doets, Studies in Logic, Language, and Information, Stanford, CA: CSLIPublications, 1996. Price: £32.50 (hardback), £11.95 (paperback), viii + 130 pp., ISBN 1-57586-049-X (hardback), 1-57586-048-1 (paperback).

Model theory is a beautiful subject, and a fundamental one, but few would describe it as straightfor-ward. Understanding it involves mastering a wide variety of model building techniques, while at thesame time acquiring some sort of unifying perspective; it is not easy to do this, and the longstandinglack of any intermediate level text compounds the difficulties. Kees Doets’s new book is importantbecause it fills this gap, and does so in a way that will be accessible to an interdisciplinary audience.

Complete beginners to model theory are already well served. For example, (Enderton, 1972), aclassic introduction to mathematical logic, contains a good discussion of the basic results of modeltheory (the completeness, compactness and Löwenheim–Skolem theorems) and there is even a beau-tiful little book for beginners (Bridge, 1977), solely devoted to the model theory of first-order logic.This little book succeeds admirably in the tasks it sets itself. It lucidly explains the role structures playin mathematical thinking, draws the distinction between structures and descriptions, and offers proofsof completeness, compactness and Löwenheim–Skolem that are hard to beat for clarity. However itdoes not go much further than this (elementary extensions, for example, are introduced only in thelast few pages) and will not really equip students to cope with the kinds of model theoretic argumentsthey are likely to encounter in (say) theoretical computer science.

So – until the appearance of Doets’s book – the student who wanted more has been forced togo straight to advanced texts. Some of these are excellent. For example, there is the classic (Changand Keisler, 1990), now in its third edition. Carefully written, detailed, and divided into an easierand a more advanced track, this was for many yearsthesource for model theoretic enlightenment.Nonetheless, unless your goal is to become a serious model theorist, you will probably find that youuse it simply as a reference for various model construction techniques; parts of the book demand asubstantial background in set theory, and if you are primarily interested in using model theory as atool, many of its concerns will seem irrelevant. Then there is the newer (Hodges, 1993). Like Changand Keisler, this is encyclopedic in scope – but it is more engagingly written, more up-to-date, morevaried in its selection of topics. It beautifully conveys the panorama of modern model theory, andprobably appeals to a wider audience than does Chang and Keisler; nonetheless, there is a vast gapbetween this book and, say, the Enderton and Bridge texts. In short, given enough mathematicalmaturity and a decent background in set theory (or the determination to pick up the prerequisites),it is certainly possible to learn a lot of model theory from the texts currently available. But if yourinterest in model theory is more applied (perhaps you are a computer scientist interested in databases,a linguist interested in descriptive complexity, or a modal logician interested in the expressive powerof extended modal languages), the task facing you is daunting: you somehow have to jump into thisadvanced literature, extract a coherent picture of what is relevant, and assemble a usable toolkit.

Kees Doets’s new book goes a long way towards dealing with these problems. For a start, it isa genuinelyintermediatelevel book: it is not a beginner’s book in disguise (for example, it coversomitting types and recursively saturated models). But neither is it an advanced book pretending to be

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user friendly; if you have had a standard introduction to mathematical logic, youcanread this book,and if you do, you will learn a lot. Moreover, it has a number of pedagogically attractive features.

First of all, Doets gets straight down to the business of doing model theory. After a 9-page intro-ductory chapter (which introduces first-order languages and gives the Tarski satisfaction definition),Chapter 2 immediately launches into a discussion of the fundamental relations between models:isomorphism, elementary extension, and elementary embeddability. A number of key examples aregiven, and by p. 19 the reader has seen proofs of the Downward Löwenheim–Skolem theorem andthe Elementary Chain Lemma. Crucially, Chapter 2 doesnot discuss deduction and the completenesstheorem. Pedagogically, this is an excellent decision. It subtly reinforces the message that modeltheory is aboutstructures(a message that can be frustratingly hard to convey to students, who oftenequate “logic” with “proof theory”), and enables easy material that is not usually covered in a firstcourse in logic (such as the Elementary Chain Lemma) to be swiftly reached. Incidentally, whenDoets does prove the completeness theorem (which he does in Appendix A after commenting thatthe topic “does not belong to model theory proper”), he does so in style. Instead of following traditionand proving the completeness of an axiomatic system, he proves the completeness of a Prawitz-stylenatural deduction system. Again this is sensible – after all, if you are going to take time out to look atdeduction, you may as well do it elegantly – and I suspect that many readers familiar with Henkin-style completeness proofs for axiomatic systems will benefit from seeing Henkin machinery appliedto natural deduction.

Second, much of the book explores model theory via Ehrenfeucht–Fraïssé games; in fact Chap-ter 3 (pp. 21–49) is solely devoted to this topic. It is hard to think of a better way to lead thestudent deeper into the model theoretic waters. For a start, Ehrenfeucht–Fraïssé games are simpleand intuitive. Moreover, they are important: they are a key technique in finite model theory, and areincreasingly widely used in many branches of applied logic. The chapter introduces the basic idea,explains its logical content, gives some simple applications, and then goes on to discuss monadicsecond order logic and infinitary logic. The emphasis on Ehrenfeucht–Fraïssé is one of the mostattractive features of the book; many readers will find it a useful source of background information.

Third, in Chapter 4 (pp. 51–92), the last, and longest chapter in the book, Doets offers a compactand accessible introduction to most of the basic model construction techniques you are likely toencounter: compactness, diagrams, ultraproducts, omitting types, saturation and recursive saturation.I like the way Doets delays introducing these techniques until after Ehrenfeucht–Fraïssé games havebeen covered; the student who has worked through the game theoretic material will be in a far betterposition to appreciate these new tools, and less likely to miss the wood for the trees. Doets alsogives an interesting selection of applications, including Vaught’s theorem, the Robinson consistencytheorem, Craig interpolation, Beth definability, Lindström’s Theorem, and a proof that fixed pointlogic satisfies the downward Löwenheim–Skolem Theorem.

Finally, the book manages to maneuver its way around a real stumbling block: understandingmodel theory requires some knowledge of set theory, and unfortunately, the required set theoreticpreliminaries are rarely covered in computer science departments (indeed, many mathematics depart-ments do not cover them either). However, Doets manages to make the book relatively self-containedin this respect. For a start, there is a short and refreshingly down-to-earth appendix on set theory,covering the set theoretic axioms, orderings, ordinals, cardinals, the axiom of choice, inductive def-initions, Ramsey’s theorem, and games. Further, Doets’s examples subtly convey the “geometrical”ideas underlying simple order-types. This is particularly so in the chapter on Ehrenfeucht–Fraïssé;because the examples discussed here often involve relatively simple structures such as linear order-ings and trees, many important set theoretic intuitions should simply “rub-off” even on relativelyinexperienced readers. Moreover, later in the book Doets gently introduces the idea of models ofset theory containing non-standard natural numbers, and links them to the concepts of recursive-saturation and resplendency. The book will not replace an introduction to set theory, but it willsensitize the reader to the role set theory plays in logic. It is a natural companion to Roitman’s

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model theoretically oriented introduction to set theory (Roitman, 1990).

All in all, a useful book. It succeeds in making a wide range of model theoretic ideas accessible atan intermediate level. Moreover, despite having to simultaneously convey a certain amount of settheoretic background, it maintains a level of clarity which will make it usable by a wide audience:mathematical linguists, computer scientists, applied logicians, and formally minded philosophersshould all find this book approachable. Moreover, it is an excellent springboard for studying modeltheory further.

One thing left me rather unhappy: I thought that the ratio of exercises to text was sometimes toohigh. Of course, this is a matter of taste – many will swear that the best way of learning a technicalsubject is simply to work out as much as possible for yourself. There is much to be said for this;you certainly cannot learn model theory (or indeed any other branch of mathematics) without doingproofs – but I think that leaving too much to the student is a mistake. For a start, it requires an awfullot of time. Moreover it is possible for a student to studiously work a lot of exercises and still missimportant insights and generalizations: sometimes a well chosen word, a metaphor, or a picture, canbring about a deeper understanding.

By and large the book gives excellent guidance (on several occasions I had anOf course! Whydidn’t I see that before?reaction, a sure sign of good instruction). Nonetheless, on occasions the textstruck me as too cramped, too semaphoric, with too much interesting work being left to the reader.For example, I thought the material on pp. 40–49, the closing pages on Ehrenfeucht–Fraïssé gameswas too dense; I think many students will bounce off it, and that is a shame.

But this is a relatively minor blemish. The fact remains, that using only 92 pages of text and2 appendices, Doets has done something important: he has explained a lot of useful model theory interms an interdisciplinary audience can understand. This is a considerable achievement, and the bookdeserves to be widely bought and read. Indeed, I hope that at some stage Doets will consider bringingout a second edition; strengthening the presentation of the weaker parts, perhaps by integrating moreexercises into the text, or by giving answers to some of the harder exercises, could convert an alreadygood book into something even more pedagogically effective.

References

Bridge, J., 1977,Beginning Model Theory: The Completeness Theorem and Some Consequences,Oxford Logic Guides, Vol. 1, Oxford: Oxford University Press.

Chang, C. and Keisler, H., 1990,Model Theory, 3rd edition, Amsterdam: North Holland.Enderton, H., 1972,A Mathematical Introduction to Logic, New York: Academic Press.Hodges, W., 1993,Model Theory, Encyclopedia of Mathematics and Its Applications, Vol. 42,

Cambridge: Cambridge University Press.Roitman, J., 1990,Introduction to Modern Set Theory, New York: John Wiley and Sons.

Patrick BlackburnComputerlinguistikUniversity of SaarlandD-66041 SaarbruckenGermanyE-mail: patrick@coli.uni-sb.de