Names and quantifiers

‘Socrates is a philosopher’ gets rendered into an appropriate formal language of predicate logic by the likes of \mathsf{Fn}; ‘Someone is a philosopher’ gets rendered by \mathsf{\exists xFx}. The syntactic difference between a formal name and a quantifier-tied-to-a-variable vividly marks a semantic difference between the rules for interpreting the two resulting formal wffs. And there isn’t the same sort of immediately striking syntactic difference between the vernacular name ‘Socrates’ and the quantifier ‘someone’.

That’s agreed on all sides, I guess. But many have wanted to say more — namely that, in English, there is no syntactic difference between the name and the quantifier. Thus,
Quine writes that ‘one of the misleading things about ordinary language is that the word “something” masquerades as a proper name’. Well, presumably there is no masquerade if you can easily tell them apart by their surface look. So, more carefully, Quine’s idea is presumably that the English quantifier is just like a proper name as far as surface sentence structure is concerned. Or as Michael Dummett explicitly puts it: ‘As far as the sentence-structure of natural language is concerned, signs of generality such as “someone” and “anyone” behave exactly like proper names — they occupy the same positions in sentences and are governed by the same grammatical rules.’

But this is just not true (as Alex Oliver, for one, has had fun pointing out). Thus, contrast ‘Something wicked this way comes’ with the ungrammatical ‘Jack wicked this way comes’, or ‘Someone brave rescued the dog’ with ‘Jill brave rescued the dog’. Or contrast ‘Foolish Donald tweeted’ with the ungrammatical ‘Foolish someone tweeted’. Compare too ‘Senator, you’re no Jack Kennedy’ with ‘Senator, you’re no someone’. Or what about ‘Hey, Siri!’ compared with ‘Hey, someone!’?

Other quantifiers too aren’t interchangeable with names. Consider ‘Nobody’. We get similar failures of substitution: we can’t replace ‘Nobody wise …’ with ‘Jill wise …’, or replace ‘Foolish Donald …’ with ‘Foolish nobody …’, and preserve grammaticality, etc. And ‘Nobody ever finishes War and Peace’ constrasts with the ungrammatical ‘Jack ever finishes War and Peace’, while ‘Jill never finished War and Peace’ contrasts with ungrammatical ‘Nobody never finished War and Peace’.

And so it goes. Quine’s and Dummett’s claims simply overshoot. English grammar doesn’t treat names and quantifiers exactly on a par. But even if it did, ordinary language would only be “misleading” (in Quine’s word) if there was some tendency for us ordinary speakers to get misled. Now, Mark Sainsbury indeed talks of ‘our tendency to regard quantifiers … as names’. But what is the evidence is that we have such a tendency? This is shown, says Sainsbury, ‘by the fact that Lewis Carroll’s jokes are funny’. But that really is hopeless! After all, Carroll’s wordplay (you know the kind of thing: ‘I see nobody on the road,’ said Alice. ‘I only wish I had such eyes … To be able to see Nobody!’ etc. etc.), apart from being wearyingly unfunny, has nothing specifically to do with confusing quantifiers with names (as Alex Oliver notes, it’s pretty much on a par with the likes of ‘What’s your name?’ ‘Watt.’ ‘I said, what’s your name?’ ‘Watt’s my name’ …)

OK: Quine, Dummett are wrong that English names and quantifiers “are governed by the same grammatical rules”, and even if they weren’t, that would give us no reason to suppose that we are misled (Quine, Sainsbury) by the grammatical similarity or that we tend to regard quantifiers as names.

And yet, and yet … Even if the claim that “signs of generality behave exactly like proper names” is false au pied de la lettre, we are left with the feeling that the sort of exceptions we’ve noted are somehow quirks of idiom rather than deeply significant. We are left with the sense that there is something important and true which ought to be  rescuable from the very familiar kind of remarks from Quine Dummett and Sainsbury. But what is it? We say arm-waving things in our intro logic lectures — but what will we happy to put in black and white?

It is tempting to say this:  the syntax of our formal first order language more perspicuously tracks the semantics of the formal language than the syntax of English names vs quantifiers tracks their semantics. But in a way, this is just too easy to say. Of course  the syntax tracks the semantics in our formal language in a way that is  perspicuous even to beginners — we purpose-designed the language to be exactly that way! And who knows how things are in English when it comes to syntactic or semantic theory — of course that’s not perspicuous at all, as half a century of modern linguistic theory has shown?!

Now, Quine, Dummett and Sainsbury aren’t  aiming to contrast the known with the unknown: they make a positive (even if false) claims about English, after all. But then, to repeat the question, just what that is instructive and importantly true can we extract from those incorrect claims about quantifiers “behaving exactly like” (?!) names in English but not in first order languages? I have (or rather had) my lecture patter. But, as I think about the relevant bit of my revised IFL book, I’d be very interested to know what others say to their students!

More Dvorak

If you loved the Pavel Haas Quartet’s recording of the Dvorak Piano Quintet No. 2, Op. 81 and String Quintet op. 97, then you  should also love this new Dvorak recording by the terrific Jerusalem Quartet (with Veronika Hagen and Gary Hoffman). They here play the String Sextet Op 48 and then the String Quintet op. 97 again. The Sextet was (I confess!) new to me, and is quite delightful: the performance is as good as you would expect.

The Quintet is again beautifully played. The Jerusalem’s playing is slightly gentler, slightly more restrained (I suppose) than the Pavel Haas’s: but warmly recommended too.

(You can listen on Apple Music: other streaming services are available …)

Dry January

For my Dry January, I tried to quit reading stuff about Brexit (after all, surely nothing much was going to happen for a month).

Well, I miserably failed to quit outright. But I did cut down a bit, and I did get more Dickens read. But hell’s bloody bells, what an appalling mess on so many levels.

(That’s what you come here for, no? Incisive political analysis like this!)

Category Theory: A Gentle Introduction

At long last, I have updated my  notes Category Theory: A Gentle Introduction (now some x + 291 pages).

A good while ago, I received lists of corrections from a number of people, and just recently I’ve had another tranche of corrections, making over a hundred in all. Mostly these corrections noted typos. But there were also enough mislabelled diagrams, fumbled notation mid-proof,  etc., to have no doubt caused some head-scratching. So I can only apologize for the delay in making the corrections.

I have also added a new early chapter and restored a couple of sections that were in a rather earlier version but got lost in the last one (thereby breaking some cross-references and no doubt producing more head-scratching).

These notes were originally written for my own satisfaction, trying to get some basics clear. But I know some people have found them useful (despite their very obvious shortcomings, unevenness,  and half-finished character). So I hope some others will find the update helpful: you can download it from the category theory page here.

The Language of Category Theory

I’m taking a week or so off from on working the d****d second edition of my logic text (it’s quite fun, if you like that sort of thing, most of the time: but it is good to take a break). I’m instead updating, just a little, my Gentle Intro to Category Theory, about which more when the revised version is ready for prime time (within the week, I hope). So I’ve now had an opportunity to take a quick look at Steven Roman’s An Introduction to the Language of Category Theory (Birkhäuser, 2017) which in fact has been out a whole year.

This book is advertised as one thing, but is actually something rather different. According to the blurb “This textbook provides an introduction to elementary category theory, with the aim of making what can be a confusing and sometimes overwhelming subject more accessible.” We might, then, expect something rather discursive, with a good amount of the kind of informal motivational classroom chat that is woven into a good lecture course and which can be missing from a conventionally structured textbook. But what we get is actually much closer to a brisk set of lecture notes. For the book travels a long way — through the usual introductory menu of categories, functors, natural transformations, universality, adjunctions (as far as Freyd’s Adjoint Functor Theorem) — and all in just 143 pages before we get to answers to exercises. Moreover, these pages are set rather spaciously, with relatively few lines to the typical page. So certainly there isn’t much room for discursive commentary.

And I would have thought that the sequencing of topics would leave floundering some of those who would appreciate a gentler introduction. So we get to the Yoneda Lemma long before we eventually meet e.g. products (and that as part of a general treatment of limit cones). Yet aren’t products a very nice topic to meet quite early on?  — in talking about them, we  explain why it is rather natural not to care about what product-objects are intrinsically (so to speak) but rather natural  to care instead about how the product gadgetry works in terms of maps to and from products. Here then is a rather nice example to meet early to motivate categorial ways of thinking. But not in this book.

Still, look at this for what it is rather than for what it purports to be. In other words,  look at this as a  set of detailed lecture notes which someone could use as back-up reading for perhaps the first half of a hard-core course, to keep things on track by checking/reinforcing definitions and key ideas, with added exercises  (notes which could then later be useful for revision purposes). Then Roman’s book does seem to be  pretty clearly done  and likely to be useful for some students. But if you were wondering what the categorial fuss is about and wanted an introductory book to draw you in, I doubt that this is it.

[Two grumbles. The book is pretty pricey for its length. And why, oh why, in an otherwise nicely produced paperback have the category theory diagrams been drawn in such an ugly way, given the available elegant standard LaTeX packages?]

Georg Kreisel — partial bibliography

Clearing out an old box, I’ve come across a crumpled xerox of a bibliography of some two hundred papers and other pieces by Georg Kreisel, covering up to the early 1990s.

I believe this biblio was passed on to me by Dan Isaacson, though I cannot recall where he got it from. A quick internet search suggests that it isn’t readily available online. But it might well still be of interest to some, so I have scanned it and made it searchable, and here it is.

(Do let me know if there is a more complete biblio anywhere. I’ve always wondered what Kreisel’s reputation would now be had he had the expository facility — or at any rate, the desire to be understood — of e.g. a Putnam or a Feferman.)

Symbol for assignment of a truth-value?

Here’s an odd thing. There seems, browsing along my shelves, to be no really standard symbolic metalinguistic shorthand used in elementary books for assigning a truth-value to a wff (say, in the propositional calculus). You would have expected there to be some.

In the first edition of my Introduction to Formal Logic, I borrowed the symbol ‘$latex \Rightarrow$’
to abbreviate ‘has the value … [on some given valuation]’ and wrote the likes of e.g.

If $latex \mathsf{P} \Rightarrow \textrm{T}$ and $latex \mathsf{Q} \Rightarrow \textrm{F}$ then $latex \mathsf{(P \land Q)} \Rightarrow \textrm{F}$.

But on reflection this was pretty silly, given that the symbol ‘$latex \Rightarrow$’ is already overloaded (not in my book, but elsewhere — like on math.stackexchange! — where, for a start, some use it for the conditional, some use it in place of a turnstile, and some get in a tangle by using it ambiguously for both!). It seems wiser not to add to possible confusion, especially when readers might well simultaneously get to see the double arrow being used in one of these different ways.

A bit of notation that is used, not at all consistently but often enough, is square double-brackets, so ‘$latex [\![\ldots]\!]$’ is used for ‘the value of …’, and we write the likes of ‘$latex [\![\mathsf{P}]\!] = \textrm{T}$’. But this seems to me a bit cluttered for elementary purposes — I’m after readability, rather than portability to more sophisticated contexts. And it misses the dynamism(??) of some type of arrow.

So for the upcoming second edition, I’m tentatively minded to use the \mapsto symbol for value-assignment, and write instead

If $latex \mathsf{P} \mapsto \textrm{T}$ and $latex \mathsf{Q} \mapsto \textrm{F}$ then $latex \mathsf{(P \land Q)} \mapsto \textrm{F}$.

(I suppose a colon could be another possibility, but I’d rather have something more distinctive. And the likes of ‘T($latex \mathsf{P}$)’ isn’t so pretty/easy to read in bulk and is conventionally part of an augmented object language.)

Any objection to the revised arrow? Am I missing some sufficiently  established (or even just nicer) alternative??

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