You are here

Mathematical Logic

Ian Chiswell and Wilfrid Hodges
Oxford University Press
Publication Date: 
Number of Pages: 
Oxford Texts in Logic 3
[Reviewed by
Michael Berg
, on

I wonder if this terrific book will become popular at American colleges and universities.

This is a valid question, I think, because the approach currently prevalent in our undergraduate mathematics departments is geared toward bringing an ever larger set of majors (or minors) to the study of upper division material: “theoretical mathematics,” in the unfortunate parlance of so many frightened students who typically go on to claim that if it were not for that unfortunate aspect of the program, all would be well with their GPAs. In other words, we have opted for a populist pedagogical approach, causing a stratification of our undergraduate mathematics programs and producing a host of bizarre departmental realities, all of recent vintage.

We have to face the heavy pedagogical problem, for example, of courses in real analysis or group theory, populated both by a handful of future colleagues, i.e. theorem provers, and, at the same time, by a complementary set of youngsters whose relationship to proving theorems is largely adversarial. Thus, in recent decades many of our mathematics departments have been transformed from ateliers where apprentices learn the mathematical art to training grounds for mathematical social workers, and proper nurturing of future mathematicians must be relegated to a few advanced courses and seminars.

Concomitantly, due to this effective dilution of the pool of gifts distributed among today’s mathematics majors, transition courses have proliferated in undergraduate mathematics teaching, all in the cause of bringing greater numbers of lower division students to upper division mathematics as painlessly as possible. The upshot is that we often find ourselves watering our upper division courses down to the point where we postpone to senior seminars what we ourselves learned in first-semester junior level courses three decades ago (in my own case).

Given, then, that our undergraduate mathematics departments are afflicted by this lamentable state of affairs, is there a place for a textbook in mathematical logic that aims to take a student “from truth tables to the Completeness Theorem,” as Ian Chiswell and Wilfrid Hodges claim in their Preface to Mathematical Logic? The book presents a course aimed at third year undergraduate majors in the United Kingdom, entailing “a standard syllabus in propositional and predicate logic,” pointing to a treatment both of material every mathematics major should know and much more eclectic fare (though I believe that every cultured mathematician ought to know about Gödel’s theorems). So there is overlap with what is generally taught this side of the pond in the aforementioned transitional courses, but this lasts for only a chapter (the first, on “natural deduction”) and a half (the second, on “propositional logic”): in very short order Chiswell and Hodges go on to develop topics belonging properly to the subject of mathematical logic. Therefore the book is emphatically not suited to our “introduction to proofs” courses, raising the question: what then is it suited for, here in the Colonies?

An answer to this question must take into account the fact that most, if not all, American university programs in mathematical logic present the subject as firmly attached, if not ancillary, to set theory, which is to say that the usual sequence proceeds from, say, Zermelo-Fraenkel to model theory to Gödel and then to special topics (or something close to this). From this perspective the principal asset of Chiswell and Hodges’ book is also its greatest liability: set theory as such gets only a few pages of coverage (around page 200), and pretty tangentially at that. Therefore, the book can fit into our standard curriculum only after set theory has been given a good deal of separate à priori coverage, replete with the needed logical buttressing, and this suggests that, in truth, a more advanced text than the book under review should be used.

In my own experience at UCLA in the 1970’s we started with a quarter’s development of set theory à la Zermelo and Fraenkel, followed by two quarters on logic, the sources being the prototype of Herbert Enderton’s fine introductory text, A Mathematical Introduction to Logic, A. H. Schoenfeld’s Mathematical Logic, and then J. Donald Monk’s Model Theory. Chiswell and Hodges pitch their treatment at a measurably lower level than that of these books, leading me to answer the question of their book’s utility in the regular American set theory and logic sequence in the negative.

However, that still leaves irregular courses, so to speak, and in this context Mathematical Logic stands out as a gem! For a senior seminar or a reading course in logic (but not set theory) the book is irresistible. Mature students are called for, given that Chiswell and Hodges state, explicitly, that their four aims are (1) that “the mathematics should be clean, direct, and correct,” (2) that “students should be given something relevant they can do with it, … at least a calculation,” (3) that “links should be made to other areas,” and (4) that “the needs of students and teachers who prefer a formal treatment, as well as those who prefer an intuitive one,” will be taken into account. Coming in at 250 pages, Mathematical Logic is compact even if it is not inappropriately fast paced. The authors present the often somewhat alien notions characteristic to the subject attended by marvelous asides and examples from familiar mathematics, considerably softening the blow for the novice, and the idiosyncratic machinery of logic proper is developed carefully and thoroughly, supplemented by examples and exercises of the right sort and number: it is clear that Chiswell and Hodges are fine teachers of this (still) rather exotic and austere subject.

The book is oriented toward model theory, as it should be, using Hintikka’s construction “because it is more hands-on.” The treatment of propositional calculus is terrific, being heavily influenced by Frege, as is illustrated by the authors’ use of parsing trees, albeit in a variant form vis à vis Frege’s original notation. Quantifier free logic is discussed at length, as is the centrally important subject of first order sentential calculus, and the book closes with a discussion of some of the dramatic lynchpins from 20th century mathematical logic, including completeness and compactness theorems (for which the authors prefer the language of relations) and a fine “postlude” on Gödel, Matiyasevich, Church, and then some. Indeed, the four postlude pages constitute a beautiful culmination of what has come before, given that in such a compact space Chiswell and Hodges present, in order, a proof of Gödel’s diagonalization theorem, the resolution of Hilbert’s tenth problem by Matiyasevich, and proofs of Church’s undecidability theorem for first order logic, and proofs of the famous original undecidability and incompleteness theorems of Gödel — and each of these proofs is only a paragraph in length: a marvel of exposition!

Finally, qua style, Mathematical Logic is crisply written and is a pleasure to read, and it is even adorned by a large number of pictures of the major players in the game. I am slated to teach a tutorial in the near future to a philosophy graduate student with strong mathematical leanings who in fact hails from Oxford: Chiswell and Hodges’ book is at the very top of the reading list.

Michael Berg is Professor of Mathematics at Loyola Marymount University in California.

1. Prelude
2. Informal natural deduction
3. Propositional logic
4. First interlude: Wason's Selection Task
5. Quantifier-free logic
6. Second interlude: The Linda Problem
7. First-order logic
8. Postlude
A. The natural deduction rules
B. Denotational semantics
C. Solutions to some exercises