Logic

GWT2 — a first full draft

Gödel's theorems / 3 Comments /

I now have put together a first complete [updated, now third] draft of the second edition of Gödel Without (Too Many) Tears. You can download it here.

I need to do a careful read-through for typos/thinkos. I also need to update the index, make the typography more uniform between chapters, and e.g. decide on a more consistent policy about when I cross-reference to IGT2. That sort of fun to come over the next month or so. I’ll post updates from time to time, and the link above will keep pointing to the current draft version of GWT2.

It is too late to write a very different book, and after all this is supposed to be just a revised edition of the seemingly quite well-liked GWT1! This is not the moment, then, for radical revisions. But otherwise, all suggestions, comments and corrections, including quick notes of the most trivial typos, will be most welcome! Send to the e-mail address on the first page of PDF, or comment here. (Just note the date of the version you are commenting on.)

Actually, many readers of this blog will have better things to do than spend much time with this sort of intro-level enterprise (though massive thanks are due to a handful who have already been e-mailing comments). But if you aren’t a student yourself, you could well have students who would be interested to take a look and let me know what they find too obscure, and/or give other comments and corrections. (Back in the day, I lined up a whole team of volunteers to look at a couple of chapters each of IGT2: and there turned out to be precisely zero correlation between “status”, from undergraduate to full professor, and the usefulness of comments!) So do please spread the word to any students, undergraduate or graduate, who might be — or ought to be! — interested.

It’s not much of a bribe, I know, but those impoverished students who prompt the biggest corrections/improvements will get a free paperback in due course, as well as having their name in lights in the Preface!

Added: Latest version, linked above, Thursday 12 September.

GWT2 — up to the first incompleteness theorem

Gödel's theorems, Logic / 2 Comments /

I have now revised Gödel Without (Too Many) Tears up to and including the pair of chapters on the first incompleteness theorem. You can download the current version up to Chapter 13 here.

For info: the chapter on quantifier complexity has been revised (adopting a more complex definition of Sigma_1 sentences, so that I don’t have to cheat later in saying that primitive recursive functions can be defined by Sigma_1 sentences). Then the chapter on primitive recursive functions has been slightly revised yet again. I have tried to make the chapter that proves that primitive recursive functions can indeed be defined by Sigma_1 sentences a bit more reader-friendly (the key ideas are elegantly simple: implementing them is unavoidably a bit messy). The chapter on the arithmetization of syntax is little altered. And finally in this instalment, the two chapters on the semantic and syntactic versions of the first incompleteness theorem are more or less untouched.

I’m still on track for getting a second edition out by around the end of October. It goes without saying that all comments and corrections will be gratefully received (and do please alert any students who might be interested in reading through and spotting typos or unclarities). Many thanks once again to David Furcy and Rowsety Moid for corrections and suggestions.

Added 24 Aug Corrected version uploaded.

IFL as a free download, two years on …

IFL, Logic / 2 Comments /

The second edition of my An Introduction to Formal Logic was originally published by CUP. It is now exactly two years ago today that I was able to make the book free to download as a PDF and also make it available as a very cheap paperback, thanks to the Amazon print-on-demand system. How have things gone?

As with the Gödel book I really didn’t know what to expect. But from almost the beginning, IFL2 has been downloaded about 850 times a month, and has sold very steadily over 75 paperbacks a month (with numbers if anything creeping up). Which, on the one hand, isn’t exactly falling stone-dead from the press. But, on the hand, given the very large number of philosophy students who must be taking Logic 101 out there in the Anglophone world, it isn’t an overwhelming endorsement either. However, you can’t please everyone: there isn’t much consensus about what we want from an intro logic book (which is why unwise lecturers like me keep spending an inordinate amount of time writing our own, despite the best advice of our friends …). Modified rapture, then.

So what now? IFL1 was truth-tree based. IFL2 uses a Fitch-style natural deduction system. The intro book I’d ideally write would cover both trees and natural deduction. That wasn’t possible within the CUP page budget. But those constraints are lifted. A PDF can be as long as I want; and in fact the marginal additional printing cost of expanding the paperback by fifty or sixty pages wouldn’t make very much difference to the price. So an expanded IFL3 is certainly a possibility. But do I actually want to write a third edition?

OK, I confess I’m tempted! Not at all because I think the world stands in desperate need of such a book, but because (very sad to relate) I’d actually rather enjoy the exercise of getting things into the best shape I can, before I hang up my expository boots. A plan for 2023? If the gods are willing.

IGT as a free download, two years on …

Gödel's theorems / Leave a Comment /

I’ve just noticed that it is almost exactly two years since I was able to make An Introduction to Gödel’s Theorems, originally published by CUP, freely available to download as a PDF.

After a ridiculously large initial flurry of downloads, the book is now steadily downloaded about 600 times a month. As I’ve said before about such stats, it is very difficult to know how to interpret the absolute numbers: but this looks respectable enough, and  the trend is still upwards.

When I originally announced, without much fanfare, that the PDF was available, I added

I may in due course also make this corrected version of the book available as an inexpensive print-on-demand book via Amazon, for those who want a physical copy. But I doubt that there would be a big demand for that, so one step at a time

Well, of course, I soon did set up the POD paperback (very easy if you already have a publication-quality PDF), and I was proved quite wrong about demand. I thought any sales would be a tiny trickle given the availability of a completely free download; but in fact the paperback of IGT very steadily sells over 50 copies a month, over three times as many it was doing under the auspices of CUP. So I think we can count the experiment as a success!

Back in the day, when I was writing the first edition of IGT, I tried to do the whole thing from memory, reconstructing proofs as I went, on the principle that if an idea or a proof-strategy had stuck in my mind, then it was probably worth including,  and if it hadn’t then maybe not. I did fill in some gaps once I had a complete good draft; but that explains the relative shortage of footnotes to sources. When I look at the book occasionally, it is just a tad depressing to realize that I would struggle to rewrite it from memory now. That struck me forcefully yesterday when, reworking a section in Gödel Without (Too Many) Tears for its second edition, I consulted IGT and came across an important point that, at least for the moment, I’d quite forgotten the intricacies of. Ah well …

GWT2 — another instalment

Gödel's theorems / Leave a Comment /

I have now revised three more segments of Gödel Without (Too Many) Tears. I’m still aiming to get a second edition out by the end of October.

So here is the whole first third of what will be GWT2, now taking us up to and including the first Interlude. The chapter introducing Peano Arithmetic has been quite significantly revised from the current printed version; the chapter on Quantifier Complexity gets some minor tweaks; and the following Interlude is hardly changed at all. Many thanks once again to David Furcy for corrections and suggestions.

I‘m not entirely sure, as I read through, that I’d write the book quite the same way if I were starting all over again from scratch. But no matter: the plan is to improve the current book, not to write a different one!

GWT2 — a third instalment

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I was prompted to start working on a second edition of Gödel Without (Too Many) Tears by discovering that there was a significant muddle in Chapter 5 (I had inconsistently wavered between taking Baby Arithmetic, so called, as having just a negation connective, and having other connectives too). Thanks again to Ben Selfridge for pointing out that embarrassing glitch.

This new instalment of GWT2 corrects that unfortunate mess, and makes a number of other small improvements for clarity/readability, in what is now numbered as Chapter 6. So here it is!

To save readers having to dart between different PDFs, I have included the revised versions of the Preface and earlier chapters (with only trivial changes from the previous posting).

Need I say it? No: but I still will! Corrections and friendly suggestions are immensely welcome at this stage.

Already updated with many thanks to David Furcy and Léon Probst. (You may need to force your browser to reload the file to get the version dated 18 July.)

GWT2 — a first instalment

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I’ve decided to get to work, putting together a revised version of Gödel Without Too Many Tears. 

I’ll obviously correct the known glitches in the first edition. But as I start to read through the opening chapters, I find myself wanting to make quite a lot of little stylistic improvements. And there will be a couple of more significant changes, I think, later in book. So this will be, while not a radically different version, still rather more than a lightly corrected reprint. So we’ll count  it as a second edition, GWT2.

I’ll post revised chapters here in bite-sized instalments, from time to time over the next few weeks. As always, corrections and friendly suggestions are immensely welcome at this stage: now is the time to let me know if you had issues with the first edition.

Here then are the first fourteen pages, based on the old Chapters 1 and 2, now three chapters. Enjoy!

Added 9 July: With many thanks to David Furcy, I’ve already uploaded a corrected version, repairing about ten(!) minor typos.

Logic Works?

Logic / 2 Comments /

I can hardly complain about people adding unnecessarily to the over-supply of introductory logic books, having done it myself. But here’s yet another one, Logic Works: A Rigorous Introduction to Formal Logic by Lorne Falkenstein, Scott Stapleford and Molly Kao (published just six months ago by Routledge). I’ve been asked what I think of it. Having now taken a look at the book, I’ll save you the trouble of doing the same. It’s pretty bad. Not that I’ve struggled through all 645 pages. But you’ll forgive me that: life is short and patience limited.

That’s a strange subtitle, no? As if introductions to formal logic aren’t usually rigorous. Or at least, as rigorous as they need to be — and as they say, “sufficient unto the day is the rigour thereof”. You might be tempted to worry, then, that a book that especially advertises itself as “rigorous” is likely to be unnecessarily laboured. You’d be right. And actually it is worse than that. It’s not just heavy-handed in explaining the technicalities, but quite generally the long-winded prose is depressingly clotted and terminally uninviting. I pity the poor students who have this inflicted on them!

Two sample episodes. Chapter 6 presents a Fitch-style deduction system for propositional logic. Good choice (though the system isn’t as streamlined as it could be). But the authors plod through a turgid presentation, without zip and zest, making very heavy going of things. It is really pretty difficult to imagine a reader coming to appreciate that by doing things Fitch-style we can arrive at a really rather elegant, natural, and highly user-friendly system. Things aren’t helped by the printed pages being a typographical mess. 

The same applies in spades to the grimly laborious chapters introducing the language of predicate logic. Who would ever guess from these longueurs that the beautiful and compelling basic idea of a quantifier/variable notation for expressions of generality is so very neat and attractive once explained that it can be introduced well enough to convey a reading knowledge to any beginning mathematics student in half a lecture? (I was surprised to see that one of the authors does have some mathematical background — yet the writing throughout gives no sense of the aesthetic attractions of rigorous mathematical ideas.)

I could go into more detail, but I won’t. A rather depressing read, then, which I can’t recommend at all. If you want a good introduction to formal logic which also ranges quite widely, I’d stick to Nick Smith’s!

[Added And see Phil’s comment!]

Juliette Kennedy, Gödel’s Incompleteness Theorems

Books, Gödel's theorems / 5 Comments /

I was in the CUP Bookshop the other day, and saw physical copies of the Elements series for the first time. I have to say that the books are suprisingly poorly produced, and very expensive for what they are. I suspect that the Elements are primarily designed for online reading; and I certainly won’t be buying physical copies.

I’ve now read Juliette Kennedy’s contribution on Gödel’s Incompleteness Theorems. Who knows who the reader is supposed to be? It is apparently someone who needs the notion of a primitive recursive function explained on p. 11, while on p. 24 we get a hard-core forcing argument to prove that “There is no Borel function F(s) from infinite sequences of reals to reals such that if ran(s) = ran(s’), then F(s) = F(s’), and moreover F(s) is always outside ran(s)” (‘ran’ isn’t explained). This is just bizarre. What were the editors of this particular series thinking?

Whatever the author’s strengths, they don’t include the knack of attractive exposition. So I can’t recommend this for reading as a book. But if you already know your way around the Gödelian themes, you could perhaps treat this Element as an occasionally useful scrapbook to dip into, to follow up various references (indeed, some new to me). And I’ll leave it at that.

Greg Restall, Proofs and Models in Philosophical Logic

Books, Logic / Leave a Comment /

I notice that Juliette Kennedy’s book on Gödel’s incompleteness theorems in the Cambridge Elements series has now also been published. I’ll no doubt get round to commenting on that in due course, along with John Bell’s short book on type theory. But first, let me say something about Greg Restall’s contribution to the series: as I said, for the coming few days you can freely download a PDF here.

There does seem little consistency in the level/intended audience of the various books in this series. As we will see, Bell’s book is pretty hard-core graduate level, and mathematical in style and approach. Burgess’s book I found to be a bit of a mixed bag: the earlier sections are nicely approachable at an introductory level; but the later overview of topics in higher set theory — though indeed interesting and well done — seems written for a different, significantly more mathematically sophisticated, audience. It is good to report, then, that Greg Restall — as his title promises — does keep philosophers and philosophical issues firmly in mind; he writes with great clarity at a level that should be pretty consistently accessible to someone who has done a first formal logic course.

After a short scene-setting introduction to the context, there are three main sections, titled ‘Proofs’, ‘Models’ and ‘Connections’. So, the first section is predictably on proof-styles — Frege-Hilbert proofs, Gentzen natural deduction, single-conclusion sequent calculi, multi-conclusion sequent calculi — with, along the way, discussions of ‘tonk’, of the role of contraction in deriving certain paradoxes, and more. I enjoyed reading this, and it strikes me as extremely well done (a definite recommendation for motivational reading in the proof-theory chapter of the Beginning Math Logic guide).

I can’t myself muster quite the same enthusiasm for the ‘Models’ section — though it is written with the same enviable clarity and zest. For what we get here is a discussion of variant models (at the level of propositional logic) with three values, with truth-value  gaps, and truth-value gluts, and with (re)-definitions of logical consequence to match, discussed with an eye on the treatment of various paradoxes (the Liar, the Curry paradox, the Sorites). I know there are many philosophers who get really excited by this sort of thing. Not me. However, if you are one, then you’ll find Restall’s discussion a very nicely organized introductory overview.

The shorter ‘Connections’ section, as you’d expect, says something technical about soundness and completeness proofs; but it also makes interesting remarks about the philosophical significance of such proofs, depending on whether you take a truth-first or inferentialist approach to semantics. (And then this is related back to the discussion of the paradoxes.)

If you aren’t a paradox-monger and think that truth-value gluts and the like are the work of the devil, you can skim some bits and still get a lot out of reading Restall’s book. For it is always good to stand back and see an area — even one you know quite well — being organised by an insightful and eminently clear logician. Overall, then, an excellent and very welcome Element.

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