Peter Smith's Blog, page 50

Kripke on diagonalization

Having thought a bit more about Kripke’s short note on diagonalization, linked in the last post, it seems to me that the situation is this, in rough headline terms.

How do we get from a Diagonalization Lemma to the incompleteness theorem? The usual route takes two steps

(1) The Lemma tells us that for the right kind of theory T, there is a fixed point G in T for the negation of T‘s provability predicate.

(2) We then invoke Theorem X:  if G is a fixed point for the negation of the provability predicate Prov for T, then (i) if T is consistent, it can’t prove G, and (ii) if T is omega-consistent, it can’t prove not-G.

The usual proof for the Diagonalization Lemma invoked in (1) is, as Kripke says, (not hard but) a little bit indirect and tricksy. So Kripke offers us a variant Lemma which has the form: for the right kind of the theory T, there is a fixed point in TK for any T-predicate where TK is T augmented with lots of constants and axioms involving them. The axioms are chosen to make the variant Lemma trivial. But now the application of Theorem X becomes more delicate. We get a fixed point for the negation of TK‘s provability predicate and apply Theorem X to get an incompleteness in TK. And we then have to bring that back to T by massaging away the constants. Not difficult, of course, but equally not very ‘direct’.

So you either go old-school, prove the original Diagonalization Lemma for T in its tricksy way, and directly apply Theorem X. Or you go for Kripke’s variant which more directly uses wffs which are ‘about’ themselves, but have to indirectly use Theorem X, going via TK, to get incompletness for the theory T we start off from. You pays your money and you takes your choice.

For a worked out version of these headline remarks see the last section of the revised draft Diagonalization Lemma chapter for Gödel Without Tears. Have I got this right?

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New Kripke note on Gödel

Here’s a short new paper by Kripke on proving the first incompleteness theorem.

First impression is that this gives an interesting little twist rather than significant novelty: but fun all the same!

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The Pavel Haas Quartet, concert at Brno



These troubled times call for more music. So here are the Pavel Haas Quartet at the Janáček festival in Brno a few days ago. Immensely enjoyable. They play Martinu’s  7th Quartet (starting at 2.15); Janáček’s ‘Kreutzer Sonata’ Quartet (at 27.30); and Dvořák’s String Quintet No. 3 (at 53.30). These are characteristically fine performances, and well filmed too. The PHQ were asked by John Gilhooly of Wigmore Hall to perform a cycle of the Martinu quartets, and are still listed as performing the first two of the planned concerts there later this month; but I doubt that they will now happen, given the situation with the pandemic both here and in Czechoslovakia. So it is very good to have the chance of hearing them play one of those quartets here.

Followers of the PHQ’s fortunes will know that Jiří Kabát left abruptly at the beginning of the year, and that their violist since has again (temporarily?) been their founder member Pavel Nikl who so sadly had to leave the quartet for family reasons a few years ago. I think the additional player for the Quintet is the violist of the Zemlinsky Quartet.

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Logic: A Study Guide

The Teach Yourself Logic study guide has, as I said a couple of posts ago, grown over the years in a really rather haphazard and disorganized way. Looking at it again, more carefully,  the guide really need to be rewriten from the ground up. And, to add to the guide’s usefulness, it would be very good to begin each chapter/major section with a short essay (up to half a dozen pages, say) giving some orientation, briefly surveying the relevant area of logic.

So all that is what I plan to do. And it should be fun to put it together. However, it will be really quite time-consuming, writing the essays and revisiting the large literature to re-assess my various current recommendations. So I intend to work on TYL’s planned descendant — Logic: A Study Guide — in intermittent stages over the coming months, posting the new chapters for comments as I go along.

I’ve made a start. And now will be a very good time to make suggestions for improvement for the early chapters on the more elementary material (i.e. the core math logic topics covered in what are now Chapters 4 and 5). TYL is downloaded a great deal: so tell me what what you think!  — all comments and advice will, as always, be very gratefully received.

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Elisabeth Brauss at Wigmore Hall

Of the recent streamed performances from Wigmore Hall, Elisabeth Brauss’s lunchtime concert of Beethoven (Op. 10 No. 3),  Mendelssohn (Variations sérieuses), and  Prokofiev (Piano Sonata No. 2) really stood out for me — and not just for me. Astonishingly good playing, without bombast or exaggeration. She is surely on the threshold of a stellar career. To be watched and re-watched. Enjoy!

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And then there is TYL and a Gentle Introduction …

The Teach Yourself Logic Study Guide usually gets the most downloads here — recently, a fairly consistent seventeen hundred or more downloads a month (with occasional upward spikes). The Guide has grown by accretion over the years to the current 93 pages, and to be honest it is by now a bit of an inconsistent mess, in terms of levels of detail and coverage. So given how much it is used and recommended, and given the absence of any obvious alternative, I suppose I really ought to settle down to re-thinking it and re-writing it. Which could be fun in its way but is slightly daunting.

And then, to my considerable surprise and embarrassment, Category Theory: A Gentle Introduction gets an equally consistent six hundred or so downloads a month. Surprise, because there are so many available good sets of lecture notes and freely available books out there (as listed here). Embarrassment, because it is a very rough-and-ready unfinished draft — though it already weighs in at 291 pages; it needs a lot of corrections and a lot of development and expansion to get it into a decent state. And that’s an even more daunting prospect given my pretty amateur and tenuous grasp of category theory! But people have said nice things about the Gentle Introduction even as it stands: and I think it is just different enough from the alternatives in level (more accessible!) and organization (more logical!) to be worth having a good bash at improving it.

I’m really not sure, though, how to juggle thinking about a possible IFL3, re-writing TYL, and diving into a lot of category theory homework. But I suppose it is good not to run out of projects that may be a bit daunting but still seem realistically manageable (at least these are finitely limited projects in a way that more purely philosophical projects tend not to be). I’ll just have to see how the spirit moves me once I’ve really got GWT done and dusted …

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Big Red Logic Books, again

So I now at last have a full draft of the new book-style version of Gödel Without (Too Many) Tears. But I’m going to take my own advice and put this aside for a week or more, before returning with fresh eyes for a last end-to-end read through (and I will need to add some references, some brief suggestions for further reading, and a rough-and-ready index). I may run a final version past a few people for quick comments on some of the newer bits. But hopefully there will soon be another Big Red Logic Book. Big at least in the sense of being large format: but I’ve managed to keep it, as planned, to about a third the length of IGT2. I’m both embarrassed and pleased to say I learnt two or three things in the re-writing too!

How are the other BRLBs faring? I made IGT2 freely available as a PDF nine weeks ago. As I explained before, there was a quite crazy flood of downloads in the first two or three days — about sixty-four thousand — due to a direct link being posted for a couple of days on the front page of Hacker News (which shows, at any rate, the continuing real interest out there in an introduction to Gödel’s theorems even if not in An Introduction to Gödel’s Theorems). After that initial flurry, the rate of downloads has settled down to something much more sensible – amounting to another three thousand or so. Still a surprisingly high number, as there have been readily-accessible unofficial free PDFs floating around the internet for years. There has also been a small uptake for the (more-or-less at cost) Amazon print-on-demand version, more than enough to make it worth having arranged that possibility. I have of course been dipping into a few bits of IGT2 as I revised GWT and inevitably found some passages that I thought could certainly be improved. Still, I don’t think I’ll be rushing to write an IGT3 just yet.

Then I posted the free version of IFL2 a couple of weeks after IGT2, and this has been downloaded over two and a half thousand times. The sales of the Amazon printed copies are level-pegging with the sales of IGT2 — which is a bit disappointing, as I was hoping that people (i.e. you!) would be asking their libraries to order copies. Since this edition of IFL is now not coming from a commercial publisher with a marketing department, librarians will need to be told about it (order details at the end of this earlier post). Now, as we all know, there is a lot of intro logic books out there. This one at least has the considerable merit of being officially zero-cost, as opposed to the crazy pricing of many texts! And having done a bit more homework, I’m a bit surprised to find that — leaving aside forallx and Paul Teller’s old text — there doesn’t seem to much good free competition either. Which makes me think more seriously about putting together an IFL3. 

As I have explained before, the first edition of IFL concentrated on logic by trees. The second edition, as well as significantly revising all the other chapters, replaces the chapters on trees with chapters on a natural deduction proof system, done Fitch-style. Which again won’t please everyone! Chapters on trees are still available; but because of considerations of space in the printed version, those chapters are relegated to the status of online supplements. This was always a second-best solution. Ideally, I would have liked to have covered both trees and natural deduction (while carefully arranging things so that the lecturer/student who only wanted to explore one of these still has a coherent path through the book). With e-publication, the question of length isn’t so vital. And the absence of much by way of freely-available alternatives suggests it wouldn’t be at all a waste of time and effort to put together a third edition. I think it would complement the excellent forallx for those who want a more expansive/discursive (or some would say, more long-winded) introduction but with quite a similar approach. But IFL3 is for the future, and first I need to finish putting answers to the exercises in IFL2 online. And there are two other projects which I must juggle with while working on intro logic. More about them in the next post!

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Les chemins de l’amour

Covid restrictions strike again. One of the Wigmore Hall concerts I was really looking forward to watching online was tonight’s planned programme of French songs from Sabine Devieilhe and Alexandre Tharaud. But it was not to be. As a taster of what we have missed, here they are  performing Poulenc’s ‘Les chemins de l’amour’. This is from their terrific recent CD Chanson d’Amour which I have enjoyed a great deal.

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Free introductions to formal logic?

Browsing through, I notice that  The Logic Book by Bergmann, Moor and Nelson is $51 on Amazon.com. Not exactly cheap for a student.

Oh hold on, that is the price to rent the book for one semester. To buy it, even at Amazon’s discounted price, is $128. Ye gods. That’s simply outrageous, isn’t it?

What about the competition? Hurley and Watson’s Concise Introduction to Logic is $32 to rent for a semester, and $86 to buy (discounted from a ludicrous list price of $182). Copi’s Introduction to Logic apparently marches on to a 15th edition which you can rent for a price-gouging $79 (yes, you read that right: seventy nine dollars to rent the book for one semester): which makes buying it seem quite the bargain at $104 (reduced from an absurd $195).

I could go on. And it isn’t as if those books are (by my lights) particularly good, even if much used and recommended. Nick Smith’s Logic: The Laws of Truth by contrast is excellent; but although it has been out over eight years, it has never been paperbacked by Princeton, and has a list price of $62 ($56 on Amazon). Much better value, but still quite punchy for a student budget.

Which prompts the question: what books are there at this level — intro logic books aimed at philosophy students — which are free (officially free to download), and/or available for at-cost print on demand (for a student who prefers to work from a traditional book).

Here’s what I currently know about. We should probably set aside Neil Tennant’s Natural Logic (here’s a scanned copy from the author’s website), as this is tough going for beginners. So, in chronological order, we have:

Paul Teller, A Modern Formal Logic Primer (originally Prentice Hall, 1989). Now available as scanned PDFs, with exercise solutions too, from this webpage for the book. Old but still admirable, and very clearly written.
Craig DeLancey, A Concise Introduction to Logic (SUNY Open Textbooks, 2017). Webpage for this book. Not to my taste, in either the order of presentation of material or the style of natural deduction system.
P. D. Magnus, Tim Button and others, forallx (The Open Logic Project, frequently updated). Webpage for 2020 Calgary version. Available also from Amazon print on demand. Excellent.
Peter Smith, An Introduction to Formal Logic (2nd edition, originally CUP, 2020) Webpage here. Available also from Amazon print on demand. Doesn’t cover as much and more expansive than forallx, so perhaps more accessible for self-study.

But there must surely be other options. I haven’t done a significant amount of homework on this, so do let me know what’s out there, and I will put together a web-page resource with links and more comments.

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Developing a writing style

There has been for quite a while a short page of notes here at Logic Matters, intended for graduate students (or indeed for anyone) on writing essays, thesis chapters, draft papers. I recently noticed that it is still visited two or three thousand times a year, so I guess there must be links to it out there! So I thought it was worth taking a quick look at it again and revising it just a little. Here’s the not-very-revised version.

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