In the last few days, I’ve got two newly published introductory logic books, both relatively short and aimed at similar audiences.
One is Mathematical Logic by Ian Chiswell and Wilfrid Hodges (OUP). This is notionally targetted at third year maths undergraduates — which these days, in most UK universities, sadly isn’t saying very much. It would also e.g. be a terrific book to put in the hands of philosophy students who have done a first logic course using trees, and who now need to know about natural deduction, understand the formal semantics of quantificational logic, and get as far as the completeness and the LS theorems. As Chiswell and Hodges go along, they also say something about diophantine sets, and mention Matiyasevich’s Theorem, which enables them to get out an incompleteness theorem for almost no extra work.
The other book is The Mathematics of Logic by Richard Kaye (CUP) which is aimed perhaps at somewhat more sophisticated students with a wider mathematical background, but it is very good at signalling what are big ideas and what are boring technicalities. It starts off with a few chapters, e.g. on König’s Lemma, showing how the sort of ideas that will later turn up in e.g. completeness proofs are mathematically interesting in their own right. Incidentally, Kaye uses, as his way of laying out formal proofs, a Fitch-type system — which I think is the right choice if you really do want to stick as closely as possible to the ‘natural deductions’ of the mathematician in the street, though I’m not sure I’d have chosen quite his rules. And the “bonus” in Kaye’s book is not an incompleteness theorem but a chapter on non-standard analysis.
The two books (pretty unsurpringly given the authors) seem at least on a rapid glance through to be splendid! Anyone teaching logic will want to “borrow” ideas from both, and any good student at the right level ought to read both.
A comment on our times. Neither book, I imagine, could be entered for RAE purposes [for non-UK readers, the Research Assessment Exercise by which UK departments are ranked, and which determines the level of government funding that the university gets to support that department], since neither book would count as “research”. Yet the future of logic as a subject depends much more on having lively and accessible books such as these enthusing the next generation of students than it does on the publication of another research article or two that gets read by nine people …