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.
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:
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.
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.
I was intending to post Chapter 17 today — the final chapter, dealing with Löb’s Theorem and related results. But looking again at my draft version yesterday, I thought it was/is rather a mess, and that some of the material is even in the wrong chapter. So some not-quite-trivial rewriting is needed. It will be a day or two before I can get down to doing that.
Meanwhile, many thanks to all those (some here, more by email) who have sent corrections and comments and suggestions about earlier chapters. I’ll try to get a revised version of the whole thing, plus suggestions for further reading, and an index, done by the middle of next month. And then delight the world with another Big Red Logic Book …
Today’s chapter is optimistically entitled ‘Proving the Second Incompleteness Theorem’. Of course we don’t actually do that! But we do say something more about what it takes to prove it (stating the so-called derivability conditions, and saying what it takes to prove them).
As an extra, we say how it can be that there are consistent theories which ‘prove’ their own inconsistency.
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[I’ve retitled the previous post, to keep blog post numbers in sync with chapter numbers!]
We at last move on to the Second Theorem. In Chapter 15, we introduce the theorem, and explain its significance for Hilbert’s programme. This involves a cartoon history trying to bring out the attractions of Hilbert’s programme (surely one of the great ideas in the philosophy of maths — if only it had worked!).
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Today’s short episode is a second ‘Interlude’, separating the chapters on the first incompleteness theorem from the final three chapters on the second theorem. But it mentions (or at least, gestures at) enough interesting points for it to be worth its own post.
[There a reference to a Theorem 51, which is the previously unnumbered result — now indeed to be recognized as a theorem in its own right — at the end of §13.3. which says that if a theory T is p.r. axiomatized and contains Q, the formal predicate ProvT does not capture the property of being a T-theorem.]
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Twenty minutes of delight and balm for the soul: Lea Desandre singing with Thomas Dunford and members of Les Arts Florissants.
In this last (and short!) chapter related to the first incompleteness theorem, we meet ‘Tarski’s Theorem’. And so we arrive at what might be thought of as the Master Argument for incompleteness — for appropriate theories, provability-in-T is expressible in T but truth isn’t, so provability isn’t truth.
Onwards to the second theorem next week!
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We get today to Chapter 13, called ‘The Diagonalization Lemma, and Rosser’s Theorem’. Not that we actually prove Rosser’s theorem in detail, as this is fiddly. But I do establish the Lemma, show how it can be used to derive the syntactic version of the first theorem again, and I indicate the key construction for getting to Rosser’s theorem.
(Looking back at Chapter 12, though, I see that the section which actually proves the first theorem there had become rather oddly arranged: I’ve rearranged a bit, separating out the theorem from a couple of corollaries worth remarking. So I’ve included a revised Chapter 12 too.)
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