Logic Matters -Blog
by Peter Smith
1y ago
Here is the first main chapter of the Study Guide, on First-Order Logic. Nothing much has changed in the recommendations (or the occasional disparaging comments about non-recommended books!). However, the surrounding chat has been tidied up. I have in particular heeded a friendly warning about “mission creep” (the overview sections were getting too long, too detailed — especially about various proof-systems). So I hope the balance is improved. One comment (which I have also now added to Chapter 1 — the section on “Choices, choices” where I say something about how I have decided which texts to ..read more
Logic Matters -Blog
by Peter Smith
1y ago
As I’ve mentioned before, I have started work on revising/updating/extending/cutting-down the much-used Study Guide (Teach Yourself Logic as was, now retitled a bit more helpfully Beginning Mathematical Logic). I’d thought about dropping the three-part structure. But I have decided, after some experimentation, to keep it. So after some preliminaries, Part I is on the core math logic curriculum. Part II (fairly short) looks sideways at some ways of deviating from/extending standard FOL. Part III follows up the topics of Part I at a more advanced level. So, for example, there is an int ..read more
Logic Matters -Blog
by Peter Smith
1y ago
I have just read Stephen Budiansky’s Journey to the Edge of Reason: The Life of Kurt Gödel (OUP 2021). And no, I’m not immediately breaking my self-denying resolution to concentrate on finishing the Study Guide — I wanted to know if this biography should get a recommendation in the historical notes of the relevant chapter! I’ll be brisk. Budiansky’s book comes much praised. But, to be honest, I’m not really sure why. You’ll certainly learn much more about Gödel’s ideas and intellectual circles from John Dawson’s reliable and thoughtful Logical Dilemmas: The Life and Work of Kurt ..read more
Logic Matters -Blog
by Peter Smith
1y ago
So August was the first full month for Logic Matters with its snappy new web host, and with its sparse new look. Everything seems to have settled down to be working pretty satisfactorily (though some further minor tinkering remains to be done when I am in the mood). The stats are pretty much in line with the previous averages — just under 40K unique visitors in the first month. Or so they say. I’m never sure quite what to read into such absolute numbers. Relative numbers are more reliable, no doubt. And one consistency is that — month by month — the Study Guide gets downloaded more than the Th ..read more
Logic Matters -Blog
by Peter Smith
1y ago
At an abstract level of description, the strategy of Gentzen’s consistency proof for PA can be readily described. We map proofs in his sequent calculus version of PA to ordinals less than ?0. We show that there’s an effective reduction procedure which takes any proof in the system which ends with absurdity/the empty sequent and outputs another proof with the same conclusion but a smaller assigned ordinal. So if there is one proof of absurdity in PA, there is an infinite chain of such proofs indexed by an infinite descending chain of ordinals. That’s impossible, so we are done. The devil is in ..read more
Logic Matters -Blog
by Peter Smith
1y ago
As I said in the last post, Chapter 7 of IPT makes a start on Gentzen’s (second) proof of the consistency of arithmetic. Chapter 8 fills in enough of the needed background on ordinal induction. Then Chapter 9 completes the consistency proof. I’ll take things in a different order, commenting briefly on the relatively short ordinals chapter in this post, and then tackling the whole consistency proof (covering some 77 pages!) in the next one. §8.1 introduces well-orders and induction along well-orderings, and §8.2 introduces lexicographical and related well-orderings. Then §§8.3 to 8.5 are a deta ..read more
Logic Matters -Blog
by Peter Smith
1y ago
Let’s pause to take stock. Chapters 3 to 6 of IPT comprise some two hundred pages — a book’s worth in itself — on Gentzen-style natural deduction and normalization theorems, and then on the sequent calculus and cut-elimination theorems. These topics are much-discussed elsewhere, at various levels of sophistication. What’s distinctive about the coverage of IPT is that it is (i) supposed to be accessible to near beginners in logic, while (ii) sticking pretty closely to discussing Gentzen’s own formal systems, and Gentzen’s own proofs about them (and later developments of them): “One of the main ..read more
Logic Matters -Blog
by Peter Smith
1y ago
There used to be a little subscription form in the sidepanel of the old-look Logic Matters where you could subscribe to get email notifications whenever there was an update here. I’ve decided against replicating this. Instead of auto-generated mailings, I’ll just send out occasional short Newsletters to alert people to the more interesting new postings or series of postings. Here’s the first Newsletter, as sent out to previous subscribers. If you want to get future such mailings — and I promise your email box will not be cluttered! — then you can subscribe by following the link you get to by s ..read more
Logic Matters -Blog
by Peter Smith
1y ago
And it is now exactly a year since the self-published version of the second edition of An Introduction to Formal Logic was published as a paperback. This sells about 80 copies a month, very steadily. Again, the figure strikes me as surprisingly high, given that the PDF has also been freely available all along — and that’s downloaded about 850 times a month. Some of the online support materials, like the answers to exercises, are quite well used too. All in all, pretty pleasing. I occasionally get friendly feedback about the book: and it will be interesting to see if sales/downloads jump a ..read more
Logic Matters -Blog
by Peter Smith
1y ago
We are already two hundred pages into IPT: but onwards! We’ve now arrived at Chapter 6, The Cut-Elimination Theorem. Recall, the sequent calculus that Mancosu, Galvan and Zach have introduced is very much Gentzen’s; and now their proof of cut elimination equally closely follows Gentzen. The story goes like this. The chapter begins by introducing the mix rule, which is easily seen to be equivalent to the cut rule. Suppose then we can establish the Main Lemma that a classical proof with a single mix as its final step can be transformed into a mix-free proof of the same end-sequent. Repeatedly ap ..read more

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