Study Guide

The revised Study Guide — elementary proof theory

Here is the draft final chapter of Part I of Beginning Mathematical Logic — so this is the last of the group of chapters on core topics at an elementary level. This one on proof theory is newly written; comments will therefore be particularly welcome.

Having got this far, I am well aware that there are now some mismatches in the level/breadth of the overviews in the various chapters, and also places which call for more cross-references.  For example, I need to go back to the overview on set theory to say just a paragraph or so more about ordinals, in order to make a better connection with the use of small ordinals in the current chapter when waving my arms at Gentzen’s consistency proof. So smoothing out the coverage of Part I of the Study Guide is a next task. But at least there is now a full draft to play with.

The revised Study Guide — intuitionistic logic

After the minor revisions of earlier chapters of the Beginning Mathematical Logic Study Guide, something more exciting! Here is a brand new chapter on intuitionistic logic.

The chapter starts by briskly outlining a standard natural deduction system for intuitionistic logic. Then there are two sections giving rough-and-ready overviews, one on motivation and the BHK interpretation, the other on some additional formal details and giving a sketch of Kripke semantics. There are then the usual sections suggesting main readings, followed by some additional reading options. The chapter ends with some pointers to a few pieces with a more historical/philosophical flavour. The usual chapter format, in other words.

Since this is newly minted, I’d very much welcome comments/suggestions — particularly about alternative/additional readings at the right kind of introductory level. (Also very welcome, advance suggestions for what should appear in the planned later section when we briefly revisit intuitionism at a more sophisticated level in Part III of the Guide.)

The revised Study Guide — model theory

The chapter on model theory in the Beginning Mathematical Logic Study Guide was last updated quite recently, in particular to take account of Roman Kossak’s nice 20221 book Model Theory for Beginners (College Publications). Rather little has changed, then, in this current revision, except some minor tidying (though I have dropped as unnecessary a previously footnoted long proof). But still, here it is, the revised Chapter 5 (as it is now numbered).

The revised Study Guide — second-order logic

What to cover in the Guide straight after standard classical FOL?

Theories expressed in first-order languages with a first-order logic turn out to have their limitations — that’s a theme that will recur when we look at model theory, theories of arithmetic, and set theory. You will find explicit contrasts being drawn with richer theories expressed in second-order languages with a second-order logic. That’s why — although this is of course a judgement call — I do on balance think it is worth knowing just something early on about second-order logic, in order to be in a position to understand something of the contrasts being drawn. Hence this next short chapter.

There are no very substantive changes from the previous version. But it is a little tidier in some respects. So here is Chapter 4: Second-order logic, quite briefly.

The revised Study Guide — first-order logic

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 recommend). If I were choosing a text book around which to shape a lecture course on FOL, or some other topic, I would no doubt be looking at many of the same books that I mention in the Guide; but my preference-rankings could well be rather different. So, to emphasize, the recommendations in this Guide are for books which I think should be particularly good for self-studying logic, without the benefit of classroom introductions or backup.

The revised Study Guide — preliminary instalment

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 introductory chapter on e.g. model theory in Part I, and then some suggestions about more advanced reading on model theory in another chapter in Part III. (Having one long chapter on model theory, one long chapter on arithmetic, etc. made for unwieldy and dauntingly long chapters, so that’s why it is back to the original plan.)

Over the next couple of weeks, I’ll be posting some early revised chapters from Part I, and I’ll very much be welcoming comments and suggestions (and corrections, of course) at this stage. Please, please, don’t hesitate to have your say (either using the comments boxes, or by email to peter_smith at logicmatters.net). A lot of students — possibly including your own students! — are downloading the Guide each month: if you think they are being led astray, now is the time to say!

Here then, for starters, are the Preface and a couple of preliminary chapters. Not terribly exciting, but much snappier than before. They will explain more about the structure and coverage of the Guide to those who don’t already know it. Next up, the long key chapter on FOL.

Back to the Study Guide …

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 Three Big Red Logic Books combined. So really that settles what I need to do next. Namely, eschew all kinds of logical distractions and concentrate on actually finishing rewriting the damned thing: no more procrastination. So that’s my plan for the next ten weeks. I have a time-table. And, if things don’t go too far adrift, I hope to start posting excerpts  from the new version here by the end of the month. Who knows? — I might even get a few useful comments/suggestions from new contributors …

Beginning Mathematical Logic again

I have uploaded a slightly revised version of Part I of the Study Guide, with just a few changes to the arm-waving chat and a couple of additions to the recommendations in the Computation/Arithmetic/Gödel’s Theorem chapter. You can download it here.

I’m working away at Part II, mostly enjoying the (re)reading around. An earlier time-slice of myself might have persisted in reading the less fun books out of a misplaced sense of duty. Now I tend to think that if someone really can’t be bothered to write with transparent clarity and make some honest attempt to take their reader along with them by e.g. providing enough signposts along the way, then maybe I can’t be too bothered about struggling with their ill-written texts. So I move on much more quickly to find something more logically entertaining.

Beginning Mathematical Logic: A Study Guide

I’ve renamed the old Teach Yourself Logic study guide; it is now more aptly called Beginning Mathematical Logic: A Study Guide. And there is now a new version of Part I of the Guide (all 95 pages of it) which you can download from here. It’s taken some time to settle on a style for the expanded Guide (though in the end I have not worried too much about keeping the level of the “overviews” of various topics consistent in level), and also it’s a judgement call where to place e.g. a quick introduction to second-order logic.

If you read the PDF from within a browser (as opposed to downloading it and using a PDF reader) it seems best to use Firefox on a Mac. Because then, if you go back after clicking a link, you are returned to your place in the Guide: Safari returns you unhelpfully to the beginning of the Guide.

All comments/corrections gratefully received as always — but perhaps better to use email until I can sort out the comment handling on the blog. The comments will arrive in my admin dashboard but won’t be visible.

Back to the Study Guide

These have been depressing times, despite good vaccine news, no? Grey winter days do not lift the lockdown spirits. So an unproductive period for me. I don’t think I’m alone in this either.

Regrouping, I realize I’ve been trying to juggle too many balls at the same time recently. So — with apologies to Catarina Dutilh Novaes — I’m going to hang fire on blogging chapter-by-chapter about her interesting The Dialogical Roots of Induction (this is such a wide-ranging book, and it would take me too much time to do the homework to do it detailed justice). I might put together some brisker comments later. I’m also going to back off from the idea of doing some podcasts. I need to focus, and since both are downloaded a lot, I’m going to concentrate over the next few months on completing (i) the new version study guide and (ii) the notes on category theory. Which probably won’t make for many interesting blog posts here!

OK; so I have now uploaded the latest version of the partial Logic: A Study Guide, with a new twelve page chapter on elementary set theory. There is an overview of the topics, and I’ve slightly revised my preference-ordering of recommended texts. It’s been fun (and embarrassingly instructive) to revisit some of those basic set theory book; so I hope that some students will find the results useful!

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