This and that

Logical Methods — on propositional logic

I have now had a chance to read the first part of Greg Restall and Shawn Sandefer’s Logical Methods, some 113 pages on propositional logic.

I enjoyed this well enough but I am, to be frank, a bit puzzled about the intended readership. 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. And indeed, a significant amount of the material here, as we’ll see in a moment, is at what strikes me as a decidedly non-introductory level.

Certainly, things that can (and often should!) give pause to a philosophy student encountering formal logic for the first time are often skated over at speed. For example, when we do propositional logic, just what is the relation between the formal systems and our everyday inferences using the ordinary-language connectives? So, exactly what are these dratted “p”s and “q”s doing? On p.8 we are told that “declarative sentences express propositions”, and that we are going to be looking at propositional languages “where there are declarative sentences”. But then are also immediately told that our formal language is just designed “to express the forms of propositions combined with [the connectives]” (my emphasis). So do the “p”s and “q”s get interpretations as expressing propositions or not?

On p. 9 we are baldly told that “disjunctions will always be inclusive in this text” without a moment’s discussion of how things might or might not stand in ordinary language. And later, the much more vexed question of how the logician’s conditional might be related to the ordinary language conditional is relegated to a “challenge question” on p. 32. I wonder: if we don’t say rather more about the ordinary-language logical apparatus, how do we rack up a persuasive score sheet of the costs and benefits of various alternative formal choices? (Teachers using this book with real beginners might well be adding quite a bit of appropriate classroom chat on such matters as they go along — but I’m thinking here of a student reader taking the book “neat”.)

Again, the beginning reader is given just one worked example of a truth-table test for validity in action. And nothing is said e.g. about standard heuristics to speed things up (as in “you don’t need to work further on a line where the conclusion is true because that can’t give us a counterexample”) Yes, yes, of course truth-table testing complicated examples is as boring as heck. But surely(?) we do want our beginning students to be just a bit more au fait with how things can work out in practice.

So already, I’m not sure how well this is going to work with real beginners. But there are more serious worries. Restall and Sandefer advertise their book as presenting “proof construction on equal footing with model building” — but in fact that briskness over truth-tables is just one sign that their presentation is really skewed to emphasize proof-theoretic ideas. And so, long before we ever hear about the classical truth-functional interpretation of the connectives, we are tangling with why we might want detour-free proofs in a Gentzen-style natural deduction system. (By the way, much as though I like the elegance of Gentzen trees, I’m yet to be really persuaded that they trump Fitch-style proofs for introducing ND to students.)

And now, not only is the — I agree! — reasonably intuitive idea of a detour-free proof canvassed, 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). 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!

Now, I don’t want to carp, so let’s now recalibrate our expectations, and think of this as in fact a second-level text with some brisk reminders of the more elementary stuff. Then, on positive side, it can be said that the normalization proof and other parts of the discussion of Gentzen style ND are very accessibly done. So I can e.g. well foresee the relevant sections getting into the next edition of the Study Guide as warmly recommended reading on entry-level proof theory. But yes, for me at least, that is where this material really belongs, a step or two up from a first introductory text for philosophers. Call me old-fashioned!


I note that the text was typeset by the authors (and some of their aesthetic choices are a bit wonky!). But that does raise a question. I do wonder why, in 2023, since they have a nice PDF to hand, they have gone done the route of conventional publication when they could have got the book into so many more students’ hands by going down the free-PDF-plus-cheapo-print-on-demand route? Just saying.

Jonathan Raban, 1942–2023

I have hugely admired Jonathan Raban ever since I first read Old Glory forty years ago, and have praised his books here before. For example, I wrote this a few years ago (editing a little):


Now, how did that happen? Wanting to re-read Jonathan Raban’s Coasting, his evocation of a journey of discovery sailing round Britain, first published over thirty years ago, I just found a couple of days ago that our copy has disappeared from the shelves. A mystery:  I wonder what happened to it. So it is very good to discover that the book has recently been republished, along with four other books by Raban, by Eland, in rather handsomely produced paperbacks. I remember the book as being extraordinarily well written. The blurb on our new copy tells me that  Raban here “moves seamlessly between awkward memories of childhood as the son of a vicar, a vivid chronicle of the shape-shifting sea and incisive descriptions of the people and communities he encounters. As he faces his terror of racing water, eddies, offshore sandbars and ferries on a collision course, so he navigates the complex and turbulent waters of his own middle age. Coasting is a fearless attempt to discover the meaning of belonging and of his English homeland.” Which indeed is how I recall the book. I look forward to it!

I’ve been put in mind to read Coasting again because I have recently been re-reading with great  enjoyment two of Raban’s other books that are on our shelves (as it happens, another two of the five that have been republished by Eland). First there was For Love & Money, which is subtitled “Writing, Reading, Travelling: 1969–87”, and which reprints some early reviews and occasional pieces. The writing is consistently humane and insightful, but more than that, it is just so beautifully readable (the number of times I thought, “I wish I could write even half as well”). And then there was Raban’s early masterpiece, Old Glory from 1981, notionally recounting his voyage down the Mississippi in a small boat.

I say “notionally” as this complex work is lightly disguised as a straight travel book, a literal recounting of a journey taken. But the one-time English literature lecturer warns us clearly enough. One of the epigraphs of the book is from T. S. Eliot (writing of the  Mississippi), starting “I do not know much about gods; but I think that the river/Is a strong brown god …” The other epigraph is from Jean François Millet: “One man may paint a picture from a careful drawing made on the spot, and another may paint the same scene from memory, from a brief but strong impression; and the last may succeed better in giving the character, the physiognomy of the place, though all the details may be inexact.”

So we are set up for this to be a mythic tale (“the river/Is a strong brown god”)  and for the “Jonathan Raban” who features as the narrator and his adventures to be a very loose rendition of the author and his own journey (“all the details may be inexact”). And a mythic tale is indeed what we get, an ordeal by water, with auguries and signs, battles fought, even a princess won (but also lost, for this is a flawed epic, and the journey ends in emptiness as the river becomes the sea). But woven together with this are encounters with old American myths of frontiers and journeys. And, presciently — so striking, reading again now, nearly forty years later —  Raban notes the “deep, unsatisfied capacity for hero-worship” that makes many Americans long for a “strong” leader, a saviour. This is a many layered book, artful in the artlessness of its transparent prose. Wonderful.


And yes, I did indeed hugely enjoy re-reading Coasting. But Raban’s best book is surely Passage to Juneau (and this is saying something, for he is one of the very finest writers of non-fiction prose in English of recent decades). This is his 1999 book, notionally about sailing his 35ft ketch up the Inside Passage, from his home in Seattle to Juneau in Alaska. But like all great travel books, it is about so very much more — the voyage of Captain Vancouver that he is retracing, about the original Indian inhabitants and their relationship to the sea, about the idea of the sublime, about the death of his father, about his feelings for his young daughter, about other escapees he encounters at the edge of the world. And as ever “journeys hardly ever disclose their true meaning until after — and sometimes years after — they are over”. A masterpiece. Read it.


Jonathan Raban died a few days ago, aged 80. A stroke some years ago kept him confined to a wheelchair, an ironic fate, as the Guardian obituarist notes, for a writer who saw his journeys as “a means of escape, freedom and solitude, I could be happy … in a way I couldn’t be at home”. But he kept writing. And as another Guardian tribute tells us, “Raban’s memoir Father and Son is due to be published this autumn. … He is survived by his daughter, who said that on his last day he saw a bald eagle swooping and playing in the wind from his hospital window, before it flew off over Puget Sound, off Washington state.”

Self-publishing and the Big Red Logic Books

One way of increasing the chance of your books actually being read is to make them freely downloadable in some format, while offering inexpensive print-on-demand paperback versions for those who want them. Or at least, that’s a publication model which has worked rather well for me in the last couple of years. Here’s a short report of how things went during 2022, and then just a few general reflections which might (or might not) encourage one or two others to adopt the same model!


As I always say, the absolute download stats are very difficult to interpret, because if you open a PDF in your browser on different days, I assume that this counts as a new download — and I can’t begin to guess the typical number of downloads per individual reader (how many students download-and-save, how many keep revisiting the download page? who knows?). But here is the headline news:

PDF downloadsPaperback sales
Intro Formal Logic112211112
Intro Gödel’s Theorems7432627
Gödel Without Tears4394677
Beginning Mathematical Logic25863493

No doubt, the relative download figures, comparing books and comparing months, are more significant: and these have remained very stable over the year, with about a 10% increase over the previous year.

As for paperback sales of the first three books, these too remain very steady month-by-month, and the figures are very acceptable. So we have proof-of-concept: even if a text is made freely available, enough people prefer to work from a printed text to make it well worthwhile setting up an inexpensively priced paperback. (In addition there’s also a hardback of IFL which sold 150 copies over the year, and a hardback of the first edition of GWT sold 40 copies up to end of October, before being replaced by a new hardback edition.)

The BML Study Guide was newly paperbacked at the beginning of the year, not with any real expectation of significant sales given the rather particular nature of the book. Surprisingly, it is well on course to sell over 500 copies by its first anniversary.

Obviously, an author wants their books to conquer the world — why isn’t just everyone using IFL? —  but actually, I’m pretty content with these statistics.


To repeat what I said when giving an end-of-year report at the beginning of last January, I don’t know what general morals can be drawn from my experiences with these four books. Every book is what it is and not another book, and every author’s situation is what it is.

But providing an open-access PDF plus a very inexpensive but reasonably well produced paperback is obviously a fairly ideal publication model for getting stuff out there. I’d be delighted, and — much more importantly — potential readers will be delighted, if rather more people followed the model.

Yes, to produce a book this way, you need to be able to replicate in-house some of the services provided e.g. by a university press. But volunteer readers — friends, colleagues and students — giving comments and helping you to spot typos will (if there is a reasonable handful of them) probably do at least as good a job as paid publisher’s readers, in my experience. Writers of logic-related books, at any rate, should be familiar enough with LaTeX to be able to do a decent typographical job (various presses make their LaTeX templates freely available — you can start from one of those if you don’t feel like wrangling with the memoir class to design a book from scratch). Setting up Amazon print-on-demand is a doddle. You’ll need somehow to do your own publicity. But none of these should be beyond the wit of most of us!

The major downside of do-it-yourself publishing, of course, is that you don’t get the very significant reputational brownie points that accrue from publication by a good university press. And we can’t get away from it: job-prospects and promotions can turn on such things. So they will matter a great deal in early or mid career.

But for those who are well established and nearer the end of their careers, or for the idle retired among us … well, you might well pause to wonder a moment about the point of publishing a monograph with OUP or CUP (say) for £80, when you could spread the word to very many more readers by self-publishing. It seems even more pointless to publish a student-orientated book of one kind or another at an unaffordable price. So I can only warmly encourage you to explore the do-it-yourself route. (I’m always happy to respond to e-mailed queries about the process.)


Finally, I can somewhat shamefacedly add a last row to the table above, about work in (stuttering) progress towards an announced but as yet far from finished paperback:

PDF downloadsPaperback sales
Beginning Category Theory7482N/A

This download figure is embarrassing because, as I’ve said before, I know full well these notes are in a really rackety state. But I can’t bring myself to abandon them. So my logical New Year’s resolution is to spend the first six weeks of the year getting at least Part I of these notes (about what happens inside categories) into a much better shape. I just need to really settle at last to the task and not allow myself so many distractions. Promises, promises. Watch this space.

Super-infinite

Not, then, a maths post on the infinite, but rather a brief but very warm recommendation for Katherine Rundell’s biography of John Donne.

Having no young children around, we have a good rule which helps to keep Christmas relaxed — no exchanges of presents (I can really recommend that too!). But we might wander down to the bookshops a day or two before, and get ourselves a couple of new books we might particularly enjoy reading over the holidays. I went to Heffer’s planning to buy Irene Vallejo’s much praised Papyrus. However, browsing around, I found myself gripped by the first few pages of Super-Infinite: The Transformations of John Donne. So that’s what we bought. And I have now read it with great enjoyment over the last few days. I can certainly see why the book won the Baillie Gifford Prize for Non-Fiction.

Her pieces in the London Review of Books in recent years show that Katherine Rundell can write wonderful prose, of great zest and imaginative power. And she has been a prize-winning author of books for children. She brings her immense gifts for story-telling and her stylistic panache to her academic work as an English scholar, here recounting Donne’s quite extraordinary life and so illuminating his poetry.

Rundell’s title Super-infinite comes from one of Donne’s own new formations, and the paradox is impatient with the limitations of language and what it can be used to describe. The book stages an often thrilling meeting between Donne and Rundell, and its basis is this supple, flexible wordplay in which sentences jump between registers and tones, between this world and the next, between everything and something more. On a portrait of the young Donne, soulful and seductive: ‘He wore a hat big enough to sail a cat in, a big lace collar, an exquisite moustache.’ On his despairs and griefs, and the deaths of his family members and too many of his young children: ‘He was a man who walked so often in darkness that it became for him a daily commute.’ On the violence of his style: ‘He wanted to wear his wit like a knife in his shoe.’ Phrases such as these do the work which literary critics are sometimes afraid to do. They attest to love, and perform that love, and in so doing Rundell returns us to a Donne who is new yet old, nicely paradoxical, his own man and everyone’s.

I can’t put it better than that, from from Daniel Swift’s review in the Spectator. I couldn’t put Super-Infinite down and was even left wanting more.

So now, having devoured the book so quickly, I suppose I will need to wander now down to Heffer’s again, to pick up a copy of Papyrus.

A Christmas card

Another year, another Christmas card. Like last year, this is a picture to be found down the road at the Fitzwilliam Museum, a Nativity long attributed to Bastiano Mainardi, a member of Domenico Ghirlandaio’s workshop and his brother-in-law. But latterly, the Fitz has claimed to see in it the hand of the master himself. Though I do wonder!


Ghirlandaio has been in my mind recently, in part prompted by reading Paola Tinagli’s Women in Renaissance Art (which, among much else, has interesting things to say about the iconography of the great frescos in Santa Maria Novella). And I was prompted to read that book by the references that Maggie O’Farrell gives at the end of her wonderful novel The Marriage Portrait.

The portrait is of the young Lucrezia de’ Medici. Here she is:

Lucrezia is taking her seat at the long dining table, which is polished to a watery gleam and spread with dishes, inverted cups, a woven circlet of fir. Her husband is sitting down, not in his customary place at the opposite end but next to her, close enough that she could rest her head on his shoulder, should she wish; he is unfolding his napkin and straightening a knife and moving the candle towards them both when it comes to her with a peculiar clarity, as if some coloured glass has been put in front of her eyes, or perhaps removed from them, that he intends to kill her. She is sixteen years old, not quite a year into her marriage.

How can you not read on? By some way, my favourite novel of this last year. So that’s my warm recommendation of a book to treat yourself to this Christmastide!


Distracting fictions of one kind or another have been necessary more than ever, no? As Eliot says, “Humankind cannot bear very much reality.” To mention just one reality befalling the world, who would have predicted, twelve months ago, that we would be three hundred days into a simmering land war in Europe? Grim times.

But just for a few days let’s try not to dwell too much on that! May you and yours indeed have a very happy Christmas, and I earnestly hope your New Year is a peaceful one.

Why is it so difficult to introduce type theory to the rest of us?

One episode in the Beginning Mathematical Logic Study Guide which I must radically revise in the next edition is the final section on type theories, which was tacked on almost as an afterthought. But it will, indeed, take quite a bit of work to organize a better overview of what is a messy area, and devotees of varieties of type theory are not always the clearest of advocates to help us along.

Egbert Rijke has an Introduction to Homotopy Type Theory coming out soon with CUP. This is a textbook aimed at quite a wide readership; “it introduces the reader to Martin-Löf’s dependent type theory, to the central concepts of univalent mathematics, and shows the reader how to do mathematics from a univalent point of view. … The book is entirely self-contained, and in particular no prior familiarity with type theory or homotopy theory is assumed.” Which sounds promising. And a late pre-print can now be downloaded: here’s the link. This may very well be very useful to some readers of this blog, depending on your background. And all credit to Rijke for making his text freely available on the arXiv.

The first part of the book is, as announced, on Martin-Löf’s version of type theory. I’ve dived into this 98 page introduction hopefully. But I can’t, to be honest, say that I have wildly enjoyed the experience — Rijke makes it no easier than others for a conservatively-minded logician to happily find their way in. He acknowledges “Type theory [or at least, type theory of this stripe] can be confusing for people who are new to the subject,” but this means that many of us could do with rather more explanatory chat than we get here. For a small but not insignificant example, right at the outset we are told without further ado that “The expression 𝑎 : 𝐴 is … not considered to be a proposition, i.e., something which one can assert about an arbitrary element and an arbitrary type, but it is considered to be a judgment, i.e., an assessment that is part of the construction of the element 𝑎 : 𝐴.” Is the distinction between a proposition and a judgement transparently clear to you? No. Me neither. (Amusingly, I asked ChatGPT to give a simple explanation of the difference between propositions and judgements in Martin-Löf, and it was a lot clearer. On its first attempt. Though it lost the plot a bit when the question was re-asked.)

Well, meaning is use and all that, and eventually the mists clear somewhat as the notions of proposition and judgement get used later in Rijke. More generally, if you have in fact already encountered a bit of type theory, his explanations will probably serve well for revision and consolidation. But we still await (OK, I still await) a really introductory text on dependent-type-theory-for-old-fashioned-logicians.

Off to Boston Spa

A small package of hardback copies of Gödel Without (Too Many) Tears has just arrived. One copy I’ll give to the Moore Library here (the university’s main library for the mathematical sciences); another will go to my long-ago college (once a Trinity mathmo, always a Trinity mathmo). A third will be sent off to Boston Spa, because you are legally required to deliver a copy of any book you publish to the British Library.

I suppose it quite likely that this third copy will disappear into the “Additional Storage Building” there, soon never to be seen again. In his strangely rambling but sporadically gripping Bibliophobia, Brian Cummings sets the scene:

From high in the roof, the book robot swings down in an arc in a vertical plane, 10 m or more in a single movement, between stacks ranking 20 m high. It pauses, chooses a stack, then hovers, humming all the time, hunting for the book that it is programmed to find. The scale of the building is difficult for a human to take in. The void is 24 m high by 24 m wide by 64 m long. In any case, the room is not designed for humans. Oxygen levels are kept at 14–15 per cent, which is similar to trying to breathe at the top of a Himalayan mountain. Visitors watch from a special cage, advised to leave after fifteen minutes for their own safety. This library is not a human environment, for it is designed for habitation by books and robots alone.

This is a library for the twenty-first century. … Since space at the main site [of the British Library] at St Pancras in London is at a premium, it is proposed to house books 262 km away and transport them between the Reading Rooms in both locations. Eventually, it is planned that seven million items will be held at the Additional Storage Building in Boston Spa. All of this makes logical sense …

The robotic crane adds a volume to a pile that it is assorting in a plastic bucket. In time, it delivers this to an airlock at the end of the room. There are 140,000 bar-coded containers. It is only at this moment that human intervention takes over, as staff retrieve items from the airlock to send to Reading Rooms, a maximum of 48 hours from ordering to arrival at a London desk. Yet if a human librarian wanted to enter the vault to retrieve a book, using either gigantic ladders or high-wire trapeze artists, it would not be feasible. The books are no longer on fixed shelves ear-marked for their location. Only the crane knows where the books are. It never puts a book back as it found it, absorbing the used item back into the system in the order in which it comes, then remembering where it was. The books are engaged in an eternal game of musical buckets, finding new neighbours, and slotting in accordingly. Only humans need shelf marks any longer: the shelves have gone. The retrieval system is fully automated …

Logical, but the stuff of Borgesian nightmare, squirrelling away everything in a limitless library, including the unread and the unreadable, published in their tens of thousands a year. Libraries, you feel (well, I feel), should be places of more homely comfort — whether in a private room or two with friendly familiars on the walls, or in a congenial public space for browsing along the shelves and reading and working with others similarly occupied.

I’m sure no one would notice if GWT2 never arrived at Boston Spa. But duty calls.

Meeting Leibniz on the white road.

The non-fiction, non-logic, book which I have most enjoyed this year is the one I have just finished, Edmund de Waal’s remarkable The White Road (from 2105). I was enticed by the jacket description — the author “sets out on a quest – a journey that begins in the dusty city of Jingdezhen in China and travels on to Venice, Versailles, Dublin, Dresden, the Appalachian Mountains of South Carolina and the hills of Cornwall to tell the history of porcelain. Along the way, he meets the witnesses to its creation; those who were inspired, made rich or heartsick by it, and the many whose livelihoods, minds and bodies were broken by this obsession. It spans a thousand years and reaches into some of the most tragic moments of recent times.” It is indeed a quite remarkable read, intriguing and then — when it touches on some of the human disasters of the last century — distressing and moving. As you would expect from The Hare with Amber Eyes or Letter to Camondo, it is quite wonderfully well written — de Waal’s often short, beautifully crafted, paragraphs as carefully arranged as those white porcelain pots of his in their vitrines. Which makes the later pages all the more telling.


I hadn’t expected to bump into Leibniz on the road that takes us to seventeenth century Chinese porcelain and the attempts of the Jesuit missionaries to discern its secrets. But then I didn’t know that one of the few books Leibniz published in his lifetime was the Novissima Sinica — the very latest news from China! Unlike a Descartes who thinks that what is most fundamental is common to us all and is available through rational reflection, Leibniz is committed to the idea that the growth of knowledge will come from bringing together different perspectives, different ways of thinking, about our shared universe. So he has an extensive correspondence with the Jesuit mission, and (to quote de Waal) writes that

all this activity around China … is ‘un commerce de lumière’; enlightenment stretching both ways. This is a tremendous idea and a beautiful image, one of an equality of concerns, a correspondence of civilisations, of light.

As Leibniz writes to his friend Sophie, the wife of the elector of Hanover:

I will thus have a sign placed at my door with these words: bureau of address for China, because everyone knows that one only has to address me in order to learn some news. And if you wish to know about the great philosopher Confucius … or the drink of immortality which is the philosopher’s stone of that country, or some things which are a little more certain, you only have to order it.

He is one of the gatekeepers. If you want to know about Chinese mathematics, the I Ching as a coding of chance events, Chinese characters and their relationship to hieroglyphs, you go to him. Leibniz has been to visit Father Francesco Grimaldi in Rome, just back from the emperor Kangxi’s court, and written up copious notes on fireworks, glass and metal. I realise to my surprise that my hero, the father of rationalism, is anxious to keep ahead in this new, congested field of China Studies.

So Leibniz knows about the problem of porcelain. Later, he corresponds with the mathematician Ehrenfried Walter von Tschirnhaus, who has a sideline experimenting on how to create porcelain here in Europe, and almost gets there. And the extraordinary story continues, that was all new to me, as I followed The White Road. 

Mastodon, maybe: Twitter, not so much

Twitter could be pretty informative and entertaining, and I’d occasionally tweet posts myself. But the Elon Musk takeover seems to be Very Bad News on a number of fronts. So, like many, I’m backing off for the time being, though I’ve not actually deleted my account yet — the Musk era might or might not last. Hence I’ve removed the Twitter feed from the footer that shows on Logic Matters pages (when using a computer or tablet browser).

Instead, I’ll post occasional tweet-sized music-related snippets in the same place — about new CDs, old CDs I’ve been re-listening to, YouTube video links, etc.

Again like many, I’ve joined a Mastodon server, and there’s now a link in the footer to @PeterSmith@mathstodon.xyz. I’m seeing more mathsy stuff and fewer cats and otters. Still exploring who to follow in the wider fediverse. But so far, I’m enjoying the atmosphere and the occasional distractions.

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