For occasional readers, here are links to some of the perhaps more interesting 2023 posts on logic books here, which you might have missed!

I posted a series of comments on each of two substantial and wide ranging books on mathematical logic. First, Joseph Mileti’s *Modern Mathematical Logic* (CUP 2023, 502 pp.) is announced as aimed at advanced undergraduates or beginning graduates. Despite the title, the coverage is rather old-school and the approach thoroughly conventional. Mileti starts with basic first-order logic (though there’s no real proof theory). Then there’s a little model theory, entry-level axiomatic set theory, some computability theory, and the book ends with a treatment of incompleteness. But there are, of course, some terrific texts on the separate topics here, and I’m left quite unconvinced that there is any particular virtue in having the whole menu served up between one set of covers. And, though there are some nice sections, I can’t especially recommend Mileti’s presentations of FOL, or of elementary set theory, etc., as compared with some familiar standalone books. For a little more, I’ve wrapped up my various blog posts into single page here.

Jeremy Avigad’s *Mathematical Logic and Computation* (CUP 2023, 513 pp.) is a much more interesting book. In part because — despite Avigad’s intentions and despite the many virtues of the book — this isn’t really a book for beginners. The first seven chapters, some 190 pages, form a book within the book, on core FOL topics but with an unusually and distinctively proof-theoretic flavour. This is very well worth reading, especially if you already know enough (though the exposition is often *very* brisk, and the amount of motivational chat is variable and sometimes minimal). Then the book moves on to formal arithmetic and computational topics. So, for example, Chapter 9 is the most detailed and accessibly helpful treatment of Primitive Recursive Arithmetic that I know. On the other hand, Ch. 11 on computability is a fast-track introduction to the basics of the theory of partial recursive functions together with a look at Turing machines, and gets to Rice’s theorem in just ten pages, which tells you how *very* fast things go. I found myself repeatedly remarking on the differences in level/speed (sometimes quite radical) between different chapters, and quite often between sections within a chapter. Does this book in fact have a number of different archaeological layers, with different parts having their ultimate origins in handouts for differently paced, different level courses? I wonder! But if you are prepared for a pretty uneven ride, there is a great deal of highly interesting material here: you’ll just need to be primed to a suitable level (different for different episodes) to really appreciate it. Here’s a page putting together my blog posts on Avigad.

I was (to my surprise) disappointed by Greg Restall and Shawn Standefer’s *Logical Methods *(MIT, 2023, 270 pp.) The book’s Preface starts “Welcome to *Logical Methods*, an introduction to logic for philosophy students …”. And the text does indeed seem to start right from scratch. But Restall’s web-page for the book says “The text was developed through years of teaching intermediate (second-year) logic at the University of Melbourne.” While their Amazon blurb says “suitable for undergraduate courses and above.” Which suggests a rather unstable focus. The treatment of propositional logic is heavily skewed towards proof-theoretic methods. There’s one example of a truth-table; but we actually get a full-on, ten-page, proof of normalizability for intuitionistic propositional logic (starting as early as p. 53 in the book). This is in fact very accessibly done. But I honestly can’t imagine too many thinking that *this* is where they want their beginning philosophy students to be concentrating, so early in their logical encounters! After the chapters on PL, we get a tranche of modal, done before students see a quantifier. Again I can’t imagine too many agreeing that this is the order in which they want their students to meet topics, and the treatment is pretty uneven too. I said a bit more about *Logical Methods* in these blog posts.

I was late to getting round to reading the papers in the collection *Categories for the Working Philosopher* edited by Elaine Landry (originally published by OUP in 2017). It is the usual sort of mixed bag, with little sign that the editor had tried to impose a reasonably consistent level of accessibility and philosophical relevance, and some pieces seem quite out of place. There are eighteen papers, of which I was glad to have looked at perhaps half a dozen at most. I confess I started pretty sceptical about claims about the wider significance of category theory (once we go beyond the world of pure mathematics/logic — and perhaps functional programming): and on the evidence of this book, I remain as sceptical. Here, anyway, are my five blog posts on the collection.

I did enjoy the latest logical addition to the Cambridge Elements series — Penelope Maddy and Jouko Väänänen have written a very interesting contribution on *Philosophical Uses of Categoricity Arguments*. From their Introduction: “Mathematicians and philosophers have appealed to categoricity arguments in a surprisingly varied range of contexts. One familiar example calls on second-order categoricity in an attempt to show that the Continuum Hypothesis, despite its formal independence, has a determinate truth value, but this doesn’t exhaust the uses of categoricity even in set theory, not to mention its appearance in various roles in discussions of arithmetic. Here, we compare and contrast a sampling of these deployments to get a sense of when these arguments tend to succeed and when they tend to fail. Our story begins with two historical landmarks, Dedekind and Zermelo, on arithmetic and set theory, respectively, and ends with leading contemporary writers, Charles Parsons and the coauthors Tim Button and Sean Walsh, again on arithmetic and set theory, respectively. In between, we pause over the well-known contribution of Georg Kreisel. In each case, we ask: What does the author set out to accomplish, philosophically? What do they actually do (or what can be done), mathematically? And does what’s done (or can be done) accomplish what they set out to do? We find this focus on context illuminating: these authors have qualitatively different philosophical goals, and what works for one might not work for another.” Well worth looking at. If you want a spoiler, a report of Maddy and Väänänen’s score card for their various authors, see this short blog post.

I recently looked at Justin Khoo’s *The Meaning of If* published last year by OUP. Anyone writing an intro logic book (and I still hanker after a third edition of *IFL*) wants — or ought to want — to have something sensible to say about the relation of ‘if’ and ‘⊃’, though some do duck the task. So I’m always interested to see what people are writing these days on the topic of conditionals. But I can’t say I got much out of this. One of the phenomena here is that, however ‘if’s work in the wider world, in mathematics regimenting them by a connective ‘⊃’ governed by the usual rules (acceptable to classical logic and constructive logic alike) seems to work a treat, at least once we distinguish plain ‘if’s from the ‘imply’s we regiment using turnstiles. But there isn’t a word about this in Khoo’s book (you look in vain for anything about mathematics, or indeed about “conditional proof”, or “supposition”, and so on). So whatever the virtues of this book — which I confess didn’t impress me — it will probably be of no real interest to logicians.

Finally, I’ll quickly mention again another book which I did little more than mention in an earlier blog post. The editors Maria Hämeen-Anttila and Jan von Plato write in their short Preface

“If there is one “must” to be cleared in the enormous mass of the *Kurt Gödel Papers *kept at the Firestone Library of Princeton University, it is the series of four notebooks titled *Resultate Grundlagen*. Gödel wrote these 368 pages between 1940 and 1942, except for the ﬁrst 33 and last 12 pages. There is a continuous page numbering and the same goes for the theorems. It has been a great fortune for us to meet the task of transcribing, translating, and editing these notebooks.” So here we have the result, published at a quite extortionate price by Springer, as *Kurt Gödel, Results on Foundations*. I didn’t get much out of it myself. But the editors announce that Akihiro Kanamori has a forthcoming essay on *The remarkable set theory in Gödel’s 1940–42 Resultate Grundlagen*, “an essay that explains how Gödel had arrived at numerous results independently discovered by others later, sometimes much later, in an anticipation of the development of set theory from 1942 on, the year Gödel left formal work in logic and foundations”. So maybe I’ll be able to more usefully revisit Gödel’s notebooks with Kanamori as guide in due course.

Rowsety MoidRe

Results on Foundations, the “mention in an earlier blog post” link does not work for ordinary mortals, for it is an ‘edit’ link. Instead: Kurt Gödel: Results on Foundations. [Thanks, corrected!]And something I mentioned there re

That appears to be what’s presented in this talk:

Akihiro Kanamori – The Set Theory in Gödel’s Resultate Grundlagen (Gödel Conference)

Logic DudeWould you ever consider writing a Proof Theory textbook, given that there’s so little satisfactory material?

Peter SmithI wouldn’t go as far as saying there is “so little satisfactory material” — but perhaps you have to pick your way through a wandering path through various books. If I had world enough and time, an intro to proof theory could be a fun project. But that’s a

verybig “if”!