Logic

Beaney’s Tractatus translation

I’m not sure what prompted me to send off for a copy of the new translation of the Tractatus by Michael Beaney (it is, though, a very inexpensive paperback from Oxford World Classics, for which OUP are to be thanked).

A quick description. The initial apparatus is almost a hundred pages. There is a sixty page Introduction, very much aimed at the new student reader. Then an eighteen page Note on the Text which goes into probably unnecessary detail. There is a daunting Bibliography and a short Wittgenstein Chronology. Then comes the Translation — but no German text on facing pages on the rather feeble grounds that the text is readily available online. (Given the choice, I’d have thought that many readers might have preferred the read-once Note on the Text to be mostly an online supplement, and the might-want-to-consult-often German original to be included.) There’s an Appendix giving the top-level numbered propositions again. And then twenty pages of Explanatory Notes, which often  comment on the original German, making the absence of the original again an oddity. And finally there is a two-part Glossary, first German-English and then English-German.

How useful is the Introduction? Much more importantly, how good is the Translation?

On the first, I’d say the Introduction is not particularly good or clear. Perhaps it tries to do too much in too short a space. The student new to the Tractatus would do much better to read the (albeit rather longer but wonderfully clear) chapters in Antony Kenny’s still remarkable 1973 Wittgenstein or perhaps the main chapters of Roger White’s still short 2006 Wittgenstein’s Tractatus Logico-Philosophicus. 

As for the Translation, I’m in no position to really judge. But, almost at random, here’s the Pears/McGuinness rendering of 4.026:

The meanings of simple signs (words) must be explained to us if we are to understand them. With propositions, however, we make ourselves understood.

Beaney has

The meanings of simple signs (words) must be explained to us for us to understand them. With propositions, however, we communicate.

The repeated “us” in the first sentence is unnecessarily ugly. And Beaney adds a note on the second sentence “lit. we make ourselves understood, connecting with the use of ‘verstehen’ (‘understand’) in the previous sentence” which makes the departure from Pears/McGuinness seem a bit puzzling. Is Beaney’s version an improvement?

Again, here’s the Pears/McGuinness translation of 4.041

This mathematical multiplicity, of course, cannot itself be the subject of depiction. One cannot get away from it when depicting.

And Beaney:

This mathematical multiplicity, of course, cannot in it turn be depicted. One cannot get outside it in depiction.

Is the second sentence even English?

And so it goes. I’m not immediately bowled over. However, it is evident that a great deal of thought and widely-sought advice has gone into shaping the translation, and many of the notes on translation look well-judged. Beaney’s final explanatory note is a rather engaging two-page essay on why he has, for example, rendered the famed last sentence of the Tractatus as “Of what one cannot speak, about that one must be silent.”

As to content of the Tractatus itself, about that I must indeed be silent!

The Study Guide, corrected reprint

There is now a corrected update of the Beginning Mathematical Logic Study Guide. The list of known typos for the 2022 printing was getting embarrassingly long; so I’ve taken the opportunity to correct these plus the one thinko noted on the corrections page. Otherwise little has changed, apart from some minor rephrasing here and there. There’s certainly no need to rush to order a new copy if you already have one!

The corrected PDF is now available for download. The print-on-demand version should update to match in due course (depending on whether your local Amazon has any small stocks of the current printing to clear). So if you were thinking of buying yourself a copy as a New Year’s treat, you might want to hold off for a week or two.

I plan to intermittently work on a revised second edition over the coming year, improving some of the topic overviews, and — this will take some time and effort! — revisiting some of the sets of recommendations. Still, in reading through parts of the Guide while preparing the corrected reprint, I mostly still thought reasonably well of it in its current state; and so I hope this very minor update will serve for the moment.

Big Red Logic Books: 2024 plans

If you are new here, then here is the default page about the Big Red Logic Books

As I’ve noted before, self-publishing seemed exactly appropriate for the Big Red Logic Books. They are aimed at students, so why not make them available as widely as can be? — free to download as PDFs, for those happy to work from their screens, and at minimal-cost as print-on-demand paperbacks for the significant number who prefer to work from a physical copy. I posted reports of how things went in 2021 and 2022, half-hoping to encourage a few others to adopt the same sort of publishing model (though of course recognizing that those in early or mid career need the status points that come from conventional book publication). And I offered to give advice on the nuts and bolts of self-publishing to anyone interested. But response came there none. So I won’t bother to give a detailed report for sales and downloads in 2023. Rather, here are just a few headlines, and some thoughts about what comes next. Taking the books in the order of first publication on Logic Matters:

An Introduction to Gödel’s Theorem (2020: corrected reprint of CUP 2nd edition of 2013). Sales and downloads in 2023 slightly down on 2022 — but still almost 600 paperbacks sold in the year. I’m inclined to leave well alone, as many readers like the book as it is! (No, I’m not making a fortune! — the paperback prices are set so that total royalties are now zero for some books and pennies for others, together approximately covering the cost of keeping Logic Matters online.)

An Introduction to Formal Logic (2020: corrected reprint of CUP 2nd edition). Sales up over 20% at over 1500, downloads up over 55% compared with the previous year. Perhaps two or three more lecturers are using it as a course text. The absolute figures aren’t great, but then there are so many other intros to logic to choose from. There’s part of me that would like to one day write a third edition, or rather write a somewhat different Another Introduction … But whatever happens, I’ll leave this version available and in print, as it would be so annoying for those who have adopted the text if I dropped it!

Gödel Without (Too Many) Tears (2021, and then a second edition in late 2022). I thought that this much shorter book would for many be much preferred to IGT. However, after initially high sales for GWT, there now seems to be a steady pattern of the bigger book having 50% more sales and downloads. Unexpected, but I’m happy for IGT to be doing so well.

Beginning Mathematical Logic (2022) This descendant of the Teach Yourself Logic Study Guide is by far the most downloaded of the books. But it also sold well over 600 copies in paperback in 2023, to my genuine surprise. A considerable success then — but I suppose it is a text without obvious competitors.

Category Theory I (2023) New in August, and monthly sales and downloads already comparable to those of IGT. Again a cheering surprise since I have no standing on this topic, and it is only half a book — where, you might ask, is a finished second part?

So that’s the state of play at the turn of the year. What comes next? Obviously I need to finish the promised Category Theory II. But in fact I’ve changed my mind about what should go in Part I and what in Part II, pulling some chapters on functors into Part I, and moving the elementary discussion of toposes into Part II. The new edition of Category Theory I is on my desk as I write this, waiting to be proof-read. And I hope Part II will be print-ready by the end of February, though I’ll continue posting drafts as I go along.

I then want to return to BML, which needs an end-to-end rewrite (perhaps particularly on first-order logic where I want to rethink my recommendations). But that is going to take some time — a new edition of Beginning Mathematical Logic in 2025, Deo volente? But in the meantime, I ought quickly to do a revised reprint at least to correct a lot of known typos, and to add a page about some books published since early 2022.

That should all keep the grey cells ticking over. Watch this space …

Some logic book notes, 2023

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.

Maddy and Väänänen on categoricity arguments

There’s a new short book in the Cambridge Elements series — Penelope Maddy and Jouko Väänänen have written a very interesting contribution on Philosophical Uses of Categoricity Arguments. Here’s 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.

Their scorecard? “Dedekind has successfully achieved his goal” (p. 6), and “In the end … Zermelo accomplished more than he set out to do -– and ultimately more than he could have realized at the time – so this application of categoricity arguments must be counted as a resounding success” (p. 15). As for Kreisel, properly read “determinateness of CH wasn’t his target in the first place. At his actual goal – elucidating the independence phenomenon – he succeeds” (p. 21). Next, “In the end, there seems room for doubt that our shared concept [of number], Parsons’s own Hilbertian intuition of the endless sequence of strokes, is as clear and determinate as we think it is. And if there is this room for doubt, formal categoricity theorems don’t seem to be the kind of thing that might conceivably help. Given these open questions, both mathematical and philosophical, Parsons’s appeal to categoricity arguments to establish “the uniqueness of the natural numbers” can’t yet be judged a success.” (p. 38, after a particularly useful discussion.) Finally, “We conclude that Button and Walsh have not succeeded in establishing that internalist … concerns over the status of CH are “difficult to sustain” (p. 49).

Along the way, we get pointers to some significant first-order results due to Väänänen, and the book concludes

Perhaps unsurprisingly, we think the first-order theorems do make an important philosophical point: an outcome that was thought to require secondorder resources – namely, categoricity theorems – can actually be achieved by suitable first-order means. … This is a useful discovery, which supports our general moral: a bit of mathematics that fails at one task might succeed (and even be aimed) at another.

I hope that’s enough to pique your interest in what does seem to be one of the best so far of the logic/philosophy of mathematics Elements; I enjoyed a quick first reading — it is only 50 small pages — and will want to return to think more carefully about some of the interpretations and arguments.

(A minor but welcome point: unlike some earlier Elements, this looks to have been properly LaTeXed so the symbols aren’t garbled.)

This little book should be readily available if your library has a suitable Cambridge Core subscription. And until the end of today the CUP version is freely available for download here. But there is also (as pointed out in a comment below) a version which looks to be more or less identical on the arXiv here.

What to read?

What recent, new, and forthcoming logic/phil. maths  books have caught your attention?

This first little book, forthcoming in January from CUP in the very mixed quality Cambridge Elements series, looks promising. Maddy, at least, reliably writes interestingly and well (hopefully, she keeps her co-author from getting too mired in technicalities). And the topic is a great one. “This Element addresses the viability of categoricity arguments in philosophy by focusing with some care on the specific conclusions that a sampling of prominent figures have attempted to draw … It begins with Dedekind, Zermelo, and Kreisel, casting doubt on received readings of the latter two and highlighting the success of all three in achieving what are argued to be their actual goals. These earlier uses of categoricity arguments are then compared and contrasted with more recent work of Parsons and the co-authors Button and Walsh. …  the Element concludes that categoricity arguments have been more effective in historical cases that reflect philosophically on internal mathematical matters than in recent questions of pre-theoretic metaphysics.” So this is certainly on my list.

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. 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 be of no real interest to logicians.

Erik Stei’s Logical Pluralism and Logical Consequence was published early this year (at a disgraceful price) by CUP. From the blurb: “The logical pluralist challenges the philosophical orthodoxy that an argument is either deductively valid or invalid by claiming that there is more than one way for an argument to be valid. In this book, Erik Stei defends logical monism, provides a detailed analysis of different possible formulations of logical pluralism, and offers an original account of the plurality of correct logics that incorporates the benefits of both pluralist and monist approaches to logical consequence.” OK, that looks as if it should be just up my street, as the topic is basic and important and I’m all for calming down debates by trying to draw out what each side has got right. My first impressions, though, on reading early pages have so far not been that encouraging. But I’ll certainly try again, and let you know.

On my desk right now, though, is Introducing String Diagrams: The Art of Category Theory by Ralf Hinze and Dan Marsden, recently published by CUP. This is a comp. sci. book in origin, and it is taking me a while to get the measure of it. But the book comes much praised, so I shall press on in the hope of pennies starting to drop with satisfying clunks …

Meanwhile, I’ve revised a couple chapters of my own entry-level (and hyper-conservative?) Category Theory II. Both these chapters, one on categories of categories and one on functor categories, have been much revised and in places simplified, so I hope work much better. There remain two groups of chapters to revise, one group on the Yoneda lemma and related stuff, one group on adjunctions. Fun topics. I remain quite undecided, though, about how things will go after this initial round of revisions of old material.

Kurt Gödel: Results on Foundations

If you have access to a library which subscribes to Springer Link, you should be able to download an e-copy of this very recent addition to the growing list of editions of Gödel’s various notebooks. (If you don’t have good library access, then tough — Springer are price-gouging at £111.50 for the PDF, and more for the print-on-demand version.)

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.

And later, in their introductory essay

Resultate Grundlagen [RG] is a collection of results Gödel considered finished. … Close to two thirds of RG deal with set theory … Next to set theory, RG contains results on arithmetic and recursive functions. Type theory is one clearly separate topic, and so is what Gödel called “positive logic.” The latter relates to intuitionism which was one of Gödel’s permanent interests from the early 1930s on. This interest is clearly seen in [RG] with about one part in four devoted to intuitionistic logic and its interpretation.

So that tells us two things. First, about the topics of the RG notebooks themselves. And second, inadvertently, that the language of this edition is sometimes only an approximation to good English. Evidently, Springer’s contribution to the publishing of this book didn’t run to a native-speaker copy-editor. This matters, I think, for two reasons. First, readers for whom English is not their first language will stumble. Second, the editors have (oddly to my mind) not given their transcription of Gödel’s obsolete German shorthand in a parallel text (surely an achievement worth preserving for future researchers): so occasionally the reader might wonder whether seemingly odd or stuttering phrasing is in the original or is a result of rendering into clumsy English. In fact the editors write

RG is a polished shorthand text when compared with such sources of preliminary work as [other notebooks]. There are next to no cancellations, but there are additions that often result in awkward sentence structures. The question is to what extent such passages should get improved in translation.

Given this sort of issue, why indeed not pre-empt a reader’s questions with a parallel text, as in the canonical edition of the Works?

On the key set-theoretic content, the editors write

After the transcription and translation work was done, we were lucky to find in Akihiro Kanamori a reader without comparison of Gödel’s results on foundations. … Aki took up the task and presented us with a splendid 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.

Which is good to know; but since Kanamori’s essay isn’t included in the book as an introduction (and isn’t yet available elsewhere), the rest of us will have to wait a little for a knowledgeable guide to Gödel’s achievement in RG. All that said, it remains astonishing to find how productive Gödel was in those years when he was publishing so little. Fascinating but frustrating to dip into.

Mileti, Modern Mathematical Logic, Chs 7–10

To continue. Chapter 7 of Mileti’s MML is titled “Model theory”. Of the five sections, the first three can’t be recommended. In particular, §7.2 makes such heavy weather of that fun topic, nonstandard models of arithmetic and analysis. There are so many alternative treatments which will be more accessible and give a more intuitive sense of what’s going on. By contrast, I thought §7.4 on quantifier elimination did a better-than-often job at explaining the key ideas and working through examples. §7.5 on algebraically closed fields worked pretty well too.

And now we get two chapters on set theory, together amounting to almost a hundred pages. There’s a major oddity. The phrase “cumulative hierarchy” is never mentioned: nor is there any talk of sets being found at levels indexed by the ordinals. The usual V-shaped diagram of the universe with ordinals running up the spine is nowhere to be seen. I do find this very strange — and not very ‘modern’ either! There are minor oddities too. For example, the usual way of showing that the Cartesian product of A and B (defined as the set of Kuratowski pairs \langle a, b\rangle) is a set according to the ZFC axioms is to use Separation to carve it out of the set \mathcal{P}(\mathcal{P}(A \cup B)) in the obvious way. Mileti instead uses an unobvious construction using Replacement. Why? A reader might well come away from the discussion with the impression that Replacement is required to get Cartesian products and hence all the constructions of relations and functions which depend on that. (I rather suspect that Mileti isn’t much interested in ‘modern’ finer-tuned discussions of what depends on what, such as the question of  which set-theoretic claims really do depend on something as strong as replacement.)

So: Chapter 8, without explicitly mentioning the cumulative hierarchy (let alone the possibility of potentially more natural axiomatisations in terms of levels) gives us ZFC, and the usual sort of story about how to develop arithmetic and analysis in set theory. The mentioned oddities apart it is generally OK: but the recommendations for entry-level set theory in the Beginning Mathematical Logic Guide do the job better and in a friendlier way. However I should mention that, at the end of the chapter, §8.7 on models, sets and classes, does do the job of explaining the role of class talk rather nicely.

Chapter 9 is on ordinals, cardinals, and the axiom of choice; and I thought this chapter worked comparatively well. (Perhaps the perceived unevenness is all in my mind! And I know from my own efforts in writing long-ish books that maintaining a consistent level of approachability, of proportions of helpful less formal chat around the more formal stuff and so on, is difficult. I can only report how I am finding the book — and, as they say, your mileage may vary.)

Finally in this group, Chapter 10 is much shorter, just two sections on “Set-theoretic methods in model theory”. The first, just four pages, is on sizes of models; and then the second is an opaque and to my mind misjudged ten pages on ultraproducts.

And so it goes: as with the earlier chapters, a mixed bag.

Mileti, Modern Mathematical Logic, Chs 1–3

Towards the end of last year, within a week or two of publishing Jeremy Avigad’s Mathematical Logic and Computation (a bumpy ride, but very well worth having), CUP also released another textbook, Joseph Mileti’s Modern Mathematical Logic. I’d earlier seen a substantial set of notes that Mileti had posted online, and (to be frank) wasn’t over-impressed; so I haven’t been rushing to read this. But I thought I would now take a look at the book version, with a view to seeing whether there are any chapters which I’d want to mention or even recommend in the next iteration of the Beginning Mathematical Logic Study Guide.

Level and coverage? MML is announced as aimed at advanced undergraduates or beginning graduates (by US standards, anyway), though the book is distinctly less ambitious than Avigad’s. Mileti says he assumes familiarity with some basic abstract algebra; however, this seems perhaps more needed to best appreciate some illustrative examples rather than as necessary background for grasping core content. The coverage is broadly conventional, starting with basic first-order logic (though with the opposite emphasis to Avigad: there’s no real proof theory). Then there’s a little model theory, entry-level axiomatic set theory, some computability theory, and a treatment of incompleteness. At this point, then, at least just glancing at the table of contents and diving into the first chapters, I’m not at all sure quite what makes this a book on especially modern mathematical logic in either topics or general approach.

I rather liked the tone of the short Introduction; and going through the next couple of chapters, there is friendly signposting and some nice turns of phrase. But …

But Chapter 2, the first substantial chapter, is thirty pages on ‘Induction and Recursion’. We get a pretty dense treatment of what Mileti calls “generating systems”, three different ways of defining the set of generated whatnots, proofs that these definitions come to the same, then a criterion for free generating systems, a proof we can do recursive definitions over the free systems, and so on. This is all done in what strikes me as a rather heavy-handed way which could be pretty off-putting as a way of starting out. Many students, I would have thought, will just feel they have been made to labour unnecessarily hard at this point for small returns. And when the very general apparatus is applied e.g. in the next chapter to prove, e.g., unique parsing results, I don’t think that what we get is more illuminating than a more local argument. (I suppose my pedagogic inclination in such cases is to motivate a general proof idea by proving an interesting local case first and then, at an appropriate point later, saying “Hey, we can generalize …”.) I note, by the way, that by the end of §2.2 the reader is already supposed to know about countable sets and accept without demur that a countable union of countable sets is countable.

Chapter 3, the next fifty pages, is on propositional logic. A minor complaint is that the arrow connective is initially introduce in the preface as meaning “implies” (oh dear), and then we get not a word of discussion of the truth-functional treatment of the connective unless my attention flickered. But my main beef here is on the chosen formal proof system. This is advertised as natural deduction, but it is a sequent system, where on the left of sequents we get sequences rather than sets (why?). And although the rules are set out in a way that would naturally invite tree-shaped proofs, they are actually applied to produce linear proofs (why?). Moreover, the chosen rule-set is not happily motivated. We have disjunctive syllogism rather than a proper vE rule; double negation elimination is called ¬E; removing double negation elimination doesn’t give intuitionistic logic. OK Mileti  isn’t going to be interested in proof theory; but he should at least have chosen a modern(!) proof system with proof-theoretic virtues!

As for completeness, we get the sort of proof that (a) involves building up a maximal consistent set starting from some given wffs by going along looking at every possible wff in turn to see if it can next be chucked into our growing collection while maintaining consistency, rather than the sort of proof that (b) chucks in simpler truth-makers only as needed, Hintikka style. We are not told what might make the Henkin strategy better than the more economical Hintikka one.

To finish on a positive note, perhaps the best/most interesting thing in this chapter is the final section (and the accompanying exercises) on compactness for propositional logic, which gives a nice range of applications.

To be continued

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