## Richard Zach et al.: Open Logic Text

This is a collaborative, open-source, enterprise, and very much work in progress. The book’s website originally described the project like this:

The Open Logic Text is an open-source, collaborative textbook of formal (meta)logic and formal methods, starting at an intermediate level (i.e., after an introductory formal logic course). It is aimed at a non-mathematical audience (in particular, students of philosophy and computer science), but is completely rigorous.

But in fact (although “textbook” is still used in places) the official description has now changed to “collection of teaching materials” and that’s a much more accurate description. This is a collection of lecture notes by various hands, at varying degrees of sophistication, with varying degrees of polish, various levels of coverage.

The project team lead by Richard Zach is distinguished and I will be very interested to see how things develop. My comments here are on the complete text as retrieved on 2 Oct. 2015.

Part I: First Order Logic (pp. 9-64) What do these opening chapters cover? Ch. 1 is on the syntax and semantics of first-order languages. Ch. 2 talks about theories and their models. Ch. 3 presents the LK sequent calculus and proves its soundness. Ch. 4 proves the completeness theorem by Henkin’s method (and touches on compactness and the downward LS theorem).

That’s a packed program for less than sixty pages. (And a non-mathematical student who tackles it will have to bring more to the party than she is likely to have picked up from a typical introductory formal logic course. Five pages in, it is assumed that the reader knows what an induction hypothesis is and how proofs by induction work. A couple of pages later, we are told that a certain definition is a ‘recursive’ definition of a function, again without explanation. When it comes to talking about first-order theories, we get as first examples the theory of strict linear orders and the theory of groups, with no explanation of what such things are. Actually, a mathematically rather ept audience seems to be presupposed!)

As noted, the proof system considered is the sequent calculus LK. There isn’t a word to link this up to what a student may have encountered in her introductory formal course (very likely a tree-based system, or a Fitch-style proof system), or to explain why we are doing things this way. We get a few examples, and then a three-page soundness proof. Again, tough for the non-mathematical, I’d have thought. So far, in fact, this all reads like slightly souped up lecture hand-outs covering some of the fiddly bits of a course, very respectable if accompanied by a lot of lecture-room chat around and about, but not amounting to something that is likely to work as a stand-alone text for self-study.

Having adjusted to the level of the enterprise, however, the chapter on completeness is pretty clearly done, with the strategy of a Henkin proof (and the pitfalls that have to be negotiated to get the proof-idea to fly) being well explained. This could make useful revision material for students doing courses based on other books as it isn’t tied to LK.

Part I, continued: Beyond First Order Logic (pp. 65-79) After a short overview, the sections of Ch. 5 are titled ‘Many-sorted logic’, ‘Second-order logic’, ‘Higher-order logic’, ‘Intuitionistic logic’, ‘Modal logics’, ‘Other logics’. Yes, in just fifteen pages. The pages on second-order logic are probably useful, enough to give a student a flavour of the issues. Otherwise, the discussions are necessarily pretty perfunctory — place-holders, surely, for later developments.

Part II: Model Theory (pp. 81-103) The website tells us that these twenty-four pages come from the late Aldo Antonelli’s notes. The assumed sophistication of the reader goes up another notch or two. These are pretty terse maths notes in the conventional style (and none the worse for that! — but perhaps again not quite fitting the advertised brief).

Ch. 6 comprises just ten pages on ‘The Basics of Model Theory’ (including the usual proof that countable dense linear orderings without end points are isomorphic, and a brisk discussion of non-standard models of first-order True Arithmetic). Then Ch . 7 proves the interpolation theorem and derives the Beth definability theorem. Ch. 8 proves Lindström’s theorem.

Now I would not have counted full proofs of interpolation and  Lindström’s theorem as entry-level topics on model theory. And that’s not me being idiosyncratic. Cross-checking, I find  that Manzano’s nice elementary text on model theory doesn’t get to either. Even the bible, Hodges’s Shorter Model Theory, never gets to Lindström’s theorem — and we don’t meet interpolation until half way into that book.

So I wouldn’t yet advise tackling these pages as initial reading on model theory. However, taken as stand-alone treatments for those with enough background, Chs. 7 and 8 are actually crisply and clearly done, and so could well be recommended e.g. as supplements to Manzano or other introductions.

Part III: Computability (pp. 105-158) Ch. 9 is on Recursive Functions, and is reasonably clear but brisk (I think that non-mathematicians encountering this material for the first time will not find it easy and will need to read around in more discursive texts). Ch. 10 gives us 11 pages on the Lambda Calculus: this is surely either too much or too little. Almost nothing later depends on this, so the chapter can certainly be skipped. On the other hand, those baffled by the Lambda Calculus (in my experience, quite a few when they first encounter it, even among good maths students) will find this far too quick to be terrifically helpful. An opportunity missed, perhaps.

Ch. 11 is rather different. This is a twenty-five page overview of Computability Theory getting as far as Rice’s Theorem: this is decently well done. There is enough detail for a first pass through this material for self-study: but I could recommend it as making useful revision material.

Part IV: Turing Machines (pp. 160-167). Chs. 12 and 13 are very short, and evidently awaiting considerable expansion.

Part V: Incompleteness (pp. 169-205). I have a horse in this race, of course, so am interesting to see how others canter down the track! Ch. 14 is on the Arithmetization of Syntax. It is difficult to do this stuff with a light touch, and (as elsewhere in OLT) I’d go for adding quite a lot more arm-waving motivation, here along the lines of “Look, consider the relation Pr(p, n) which holds when p codes for a well-formed proof in Q of the wff with number n.  It’s easy to see that you can check whether Pr(pn) without going in for any open-ended searches; here, informally, is how. So Pr is going to be primitive recursive, right?”. And since the chapter takes PA to be formalized in a sequent calculus, the details of arithmetization are inevitably a bit messy.

Ch. 15 is on Representability in Q. This is done the standard way, via the $\beta$-function lemma, and we get out the undecidability of Q and hence of FOL. Then Ch. 16, Theories and Computability,  shows that consistent extensions of Q are undecidable, and complete computably axiomatised theories are decidable — which together give us, in a familiar way, that consistent, computably axiomatized extensions of Q are incomplete. (Oddly, the undecidability of FOL is again proved along the way, without comment.) Ch. 17 then gives a more Gödelian proof of incompleteness via the fixed point lemma: it would perhaps have been good to say more about how this relates to the preceding proof. And then we have something pretty brisk on the Second Theorem and Löb’s Theorem. These chapters are reasonably clear. Stylistically, however, we are looking more at spruced-up (or not so spruced-up) notes rather than a discursive text:  so they are not — I would have thought — as accessible or helpful to the beginner as a number of excellent treatments in the well-known literature.

Part VI: Sets, relations, functions (pp. 207-229). This is really an Appendix, the ‘little bit of set theory you need to know’, done at a very introductory level, getting as far as telling us about Cantor and non-enumerable sets. These pages are routine but clear enough. Naughtily, they tell us that functions are relations (Great Uncle Gottlob sighs, but he is used to this). Oddly, they introduce ordered pairs without tellings us how to represent them by sets, so there is something missing here. But this part is on its way to being a handy stand-alone module.

A general reflection on OLT. Having got to the end, it is clear that the different segments by different hands presuppose quite significantly different levels of sophistication from the reader. This makes me wonder a bit about the wisdom of presenting this resource as one long document rather than as a suite of separate modules. From the point of view of the rather daunted student, splitting things up would make it a lot clearer that you can and should  pick and choose various parts, depending on your background and interests. But also dividing things into modules might encourage the potential contributor.