Book Note: Tony Roy, Symbolic Logic, #1

Tony Roy (Philosophy, California State University, San Bernardino) has generously made available his Symbolic Logic: An Accessible Introduction to Serious Mathematical Logic. I’m commenting here on the version of October 6, 2015. The full main text is no less than 746 pages long, and is followed another 180 pages of worked answers to exercises. So this really is a major endeavour. But despite the great length, it doesn’t range widely: the discussion is “ruthlessly directed at core results” in the hope of thereby indeed making them as accessible as possible. That’s a good goal to have.

The text is divided into four parts. First, The Elements (introducing axiomatic and natural deduction presentations of FOL); second, a brief Transition part (including material on mathematical induction); third, Classical Metalogic, getting as far as touching on compactness and the L-S theorems; fourth, Logic and Arithmetic. I will comment about these four parts in four posts.

Roy isn’t very explicit about his intended audience: but reasonably high-flying mathematicians will, surely, find the 327(!) page Part I much too slow and laboured. Rather, this text at least initially goes at the sort of pace, and has the sort and level of coverage, that we expect in first ‘baby logic’ books aimed at non-mathematical philosophers who may have done a critical reasoning course, but otherwise are new to the subject. So what will such a student reader find: how will he or she cope?

The first chapter gives the usual sort of account of the informal notions of validity and soundness. This does the job, though we can quibble about details. For example, I don’t like to define an argument in such a way that almost no mathematical proof is an argument (Roy’s arguments have just premisses and a conclusion, with no room for intervening steps!). I’m not sure what is gained by defining validity in terms of there being no consistent story in which all the premisses are true and the conclusion false. What makes a story consistent other than its ingredient propositions being possibly true together? But if we have that notion of being possibly-true-together, then we can directly define validity in terms of that. Again, the brief section on validity and form needs to say more: students need to get the idea that a form of argument that is not generally valid can have valid instances. But as I say, these are quibbles.

Chapter 2 is on Formal Languages (or rather the syntax thereof). Roy discussed both sentential and quantificational languages here (but you could easily extract from this and following chapters the bits about sentential logic and read all those through before tackling  any quantificational logic). The official choice of languages is austere, with only $latex \sim, \to, \forall$ as basic, with other connectives and the other quantifier introduced as abbreviations. Not my preference for an introductory book: for one thing, it is better to start with $latex \sim, \land, \lor$ and show how very nicely things go with these, before trying to sell the material conditional to sceptical philosophy students! For another thing, if philosophers are later to meet non-classical logics where the connectives/quantifiers aren’t interdefinable, it’s better to keep things separate at the outset. But those considerations apart, this chapter is routine and clear enough. It finishes with a first glance at a formal syntax for arithmetic, and this language appears in examples over the coming chapters.

Chapter 3 continues the syntactic theme and tackles Axiomatic Deduction. Roy himself notes that the reader might well want to skip this and return to it later. We get a standard Hilbert System (though unadorned with the Deduction Theorem at this stage, so unfriendly). As e.g. in Mendelson, we end up doing logic at one remove, with derivations all metalinguistic: I don’t myself thing that this is a good line to take, but let’s accept that it is one way to do things which can pass muster  if you make it transparently clear what is happening. But I don’t think Roy pulls this off. For example, he defines consequence (syntactic consequence in a deductive system) as holding between formal wffs. But then he slips unannounced (I think) into talking about consequence as a relation between sentence forms or schemata, and his examples of derivations become lists of schemata. This sort of wobbling isn’t a good example for philosophers to follow!

A more minor thing, but again a sign perhaps that Roy is writing hastily (not surprising perhaps given the length of the text): he sloppily states the modus ponens rule  as $latex \mathcal{P}$, $latex \mathcal{P} \to \mathcal{Q}$ $latex \vdash$ $latex \mathcal{Q}$ — which on the standard understanding of $latex \vdash$ is a proposition (or a schema for one) not the articulation of a rule.  More seriously, while in carping mood, the  presentation of the quantification rules and identity rules is, I think, likely to rather too quick for the intended neophyte audience. So overall, not a particularly successful chapter, I think.

Chapter 4 is on Semantics. Wffs of the formal languages are given valuations in terms of assignment of semantic values to elements of the language in the standard way. Though he calls valuations interpretations, Roy doesn’t actually seem at this stage to give the wffs of a formal sentential language, for example, any interpretative content — so the student might indeed wonder whether we (yet) have here any object languages in which we actually can give contentful arguments? Prescinding from general worries of that kind about exactly what is going on, however, I found the details of the treatment of quantificational semantics unhelpfully messy: there are a number of standard textbooks which do things in a more student-friendly way. I’m inclined to say much the same about the 70 page Chapter 5 which, perhaps rather late in the day,  is on Translation. Here we do get introduced to a notion of “interpretation function” (from “ways the world might be” to assignments of semantic values): I think it is questionable how approachable the student reader will find this. Indeed, to my mind, this chapter often makes quite unnecessarily heavy weather of simple things. Students just don’t need to bring to bear the full official apparatus of quantificational semantics in order to learn to work comfortably with translations to and from the formal languages.

Chapter 6 is 120 pp. on Natural Deduction, and presents a Fitch-style system, now with the usual four  connectives and both quantifiers. And Roy also introduces $latex \bot$ into the system as a new symbol treated as an abbreviation for some contradiction, though this addition doesn’t seem to be handled tidily. The chapter, however, provides a great deal of help on proof-strategy for students using a Fitch-style system and could prove useful. A complaint might be, however, that there isn’t a clean enough separation between (i) getting across a basic understanding of  the rules and of how the system works and (ii) the provision of heuristics for proof-discovery. And I suppose that a real worry is that the mathematically ept won’t need anywhere near so much by way of heuristics, while the philosophers who primarily need to get to understand how proofs work (but needn’t fret so much about learning to roll their own proofs) could get lost in all the details.

Which (I  realize) all sounds a bit ungrateful! But, leaving price considerations aside, there is a lot of competition at this level from some very fine introductory logic books for the non-mathematical which are more polished and to my mind better organized — and brisker. There is a balance to be struck between gently-paced accessibility and downright long-windedness. For my money, Roy too often gets the balance wrong.

To be continued.

Teach Yourself Logic 2016 — Last call for suggestions!

As I’ve noted, it is time to update the much-downloaded Teach Yourself Logic Study Guide for 2016, and I’ve recently made a start working through the current version.  So far, the editorial tinkering has been plentiful but minor as far as content is concerned, and even after quite a bit of thought I’m not yet finding myself inclined to make changes to the main recommendations in the early chapters (though this could alter: I’ve still some possible readings to [re]consider). As for form, I’ve decided to keep the one-big-PDF format, rather than go over to a suite of webpages: but I hope the new version will be just a bit easier to find your way around (even such a simple thing as setting off main recommendations in text boxes makes the Guide look less daunting, more navigable).

But now I ask again: any suggestions for additions, improvements? In particular, are there any sets of freely available online lecture notes (your own or by others!) that are especially good and appropriate for self-study? 

To repeat, suggestions from logicians at any stage of their career, whether taking first steps or on their zimmer frame, will be most welcome — either in the comments below, or by email (address at the bottom of my “about” page here).

A moment of cheer

For ever and a day, the old have bemoaned the state of the world and how it is now all going to the dogs. But it is difficult not to feel that, yes, even here in Europe, things really are going badly wrong. Certainly, I’ve found recent events more than usually unsettling and depressing. I offer then something really good coming from France to cheer us up for a moment.

In fact, Sabine Devieilhe’s whole Rameau disc ‘Le Grande Théâtre de l’Amour’ is terrific. Her stunning singing is matched by a very thoughtfully constructed programme: unlike some recital discs which can pall, the way that arias are interspersed with orchestral interludes means you can sit and listen straight through the whole disc with unalloyed delight. Recommended, then, if you too are feeling in need of something life-affirming.

Postcard from Vienna

Ephesos Museum, Vienna
Ephesos Museum, Vienna

A week in Vienna, staying in the Innere Stadt where The Daughter is living for three months. Some unreasonably good weather, warm enough to take coffee sitting outside. The city looking wonderful in the bright sun. And then in a different way, looking wonderful again as we walk delightedly through the centre almost every night after dinner (the pleasure of doing this is of course tinged with shame and indeed anger when we think of the horrible experience of late night English city centres). The cityscapes, street after street, palais after palais, are extraordinary (and yes, we rather liked the caryatids everywhere, despite the Secessionists’ scorn).

The museums and galleries have bowled us over, stunning collections quite beautifully displayed (I’m running out of superlatives here) — and though busy at the weekend, they are not overwhelmed with visitors in the way that the National Gallery and British Museum are. Even modest cafés are as they should be. A very generous son-in-law takes us to Restaurant Steirereck, which  is as good as they say, not to mention the restaurant at Hotel Sacher. Going to the ballet at the Staatsoper has been another delight — seemingly a much more mixed and more relaxed audience than the London equivalent.

It has all been far, far too short. To our surprise, our embarrassing lack of more than a few German phrases has been little hindrance (rather, we’d immediately be addressed in English in shops or cafés if we’d been heard chatting together), and we’ve felt very relaxed here. We must return for longer next year (the joys of retirement); maybe we can arrange a house swap …

Gently into November

I have just put an update for the Gentle Introduction to category theory online. Things have been moving pretty slowly (pressure of other interests, a planned chapter not really working out, spending far too much time revising earlier chapters, etc.); but we’ve inched forward to 157 fun-packed pages. Again, this version is stopping far short of a natural break point. But I’ll not be able to return to this for a few weeks, so I thought I should at least make available the best version of the earlier chapters I now have, and I can add three new chapters. (I note the newly added Theorem 63 which corrects a terrible blunder in some remarks in an earlier version! No promises that there aren’t more blunders to be found.)


Teach Yourself Logic — suggestions? [Repost]

[I posted this back in August: and I’m moving this to the front of the blog to invite more contributions/suggestions!]

I haven’t looked at all at the Teach Yourself Logic Study Guide since the 2015 version came out on January 1st. I earlier had it in mind to do a mid-year update in time for the new (northern hemisphere) academic year: but that bird has long flown. The main Guide continues to be downloaded eighty or more times a month. It certainly seems to serve some need, and I get appreciative emails.  So I will put time aside over the coming months to get a 2016 version ready for next January 1st.

So now’s the time for feedback on both style and content. As far as style goes, while keeping to the spirit of the present Guide, what would make it more user-friendly? Should I keep the one-big-PDF format, or go over to a suite of webpages? [Added: after thinking a bit, I continue to incline strongly to the PDF format — it is easier to maintain, but also easier to read off line, and for students to work with by highlighting, commenting, etc. onscreen. But thoughts on style/layout etc. are still very welcome.]

As to content, any suggestions for additions, improvements? One thing I’ll want to say something about is The Open Logic project [added: I’ve posted some thoughts that  recently] But are there more conventional new(ish) publications, or overlooked older publications,  that could definitely rate a recommendation for student use?

Feedback from logicians at any stage of their career, whether taking first steps or on their zimmer frame, will be most welcome — either in the comments below, or by email (address at the bottom of my “about” page here).

Moving gently on …

Autumn sun, Cambridge on the first day of lectures
Autumn sun, Cambridge on the first day of lectures

I have various time-consuming plans for the next few weeks and  also need to do quite a bit of reading if I’m to put together a decent update for the Teach Yourself Logic guide for 2016. Hence work on revising the category theory notes — already going at, shall we say, a rather gentle pace — will no doubt slow even more. So I thought I would put online the current version of Category Theory: A Gentle Introduction even though it doesn’t reach a natural break point. So far then, the now fourteen chapters (123 pp.), after introducing categories, consider limits and exponentials (constructions within categories) before moving on to start talking about functors (maps between categories). The category theory page here indicates which chapters to then read in the old notes, if you are feeling suitably enthused!

The Aïda of our dreams?

Aïda is perhaps one of those operas where the performance of our dreams is likely to beat any staging: we turn up the headphones, and our imagination does the rest.

The much heralded new studio recording of Aida, conducted by Antonio Pappano and with a stella cast, has now been released. You can listen, e.g., on Apple Music (surely worth a month’s subscription by itself!). I’m not sure this will supersede e.g. the legendary Karajan/Tebaldi/Bergonzi recording in my affections. But oh, it is pretty wonderful. Just listen to Jonas Kaufmann singing Celeste Aïda, and you’ll be hooked …

Git, for the rest of us?

Git is a widely used version control system, much used e.g. by software developers. But others, even writers of one-author paper or book projects,  swear  by it too. Thus Richard Baron writes:

The last time the Open Logic Text was discussed on the Logic Matters blog ( https://logicmatters.net/2015/05/10/the-open-logic-text/ ), there was some discussion of the merits of Git, and Richard Zach put up some material on the OLT blog about the use of Git. Since then I have experimented on GitLab, starting from zero knowledge, and can confirm that it is a wonderful system, even for a one-author project. So I encourage everyone to have a go.

This is from the comment thread here, where a discussion continues. This post is simply to point out the exchange to anyone interested in this sort of thing (but who might not be delving into comments on OLT). And to invite anyone else who has views/experiences about Git or some other version control system (as a paper/book author) to share them —  either in that thread, or [better?] in new comments here.

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