What is the modern conception of logic? #1

Three and a half years ago, there were a few blogposts here (and also a follow-up document) about whether there is a canonical modern story about how we should conceive of our Ps and Qs, whether we should define validity primarily for schemas or interpreted sentences, and that kind of thing. Just for the fun of discovery (and because I suspect we rush too fast to suppose that there is a uniform ‘contemporary conception’ of such matters) I’m going to return to the issue over some coming blogposts — developing, correcting, adding to, and sometimes retracting what I said before. This kind of nit-picking won’t be to everyone’s taste; but hopefully some might be as intrigued by the variety of views that have been on offer out there in the modern canon of logic texts.

I can’t expect people to remember the previous discussion! —  so I’ll start again from scratch. Here then is Episode #1, even if much of it I have said before.

1. A contemporary conception?

Warren Goldfarb, in his paper ‘Frege’s conception of logic’ in The Cambridge Companion to Frege (2010), announces that his ‘first task is that of delineating the differences between Frege’s conception of logic and the contemporary one’. And it is not a new idea that there are important contrasts to be drawn between Frege’s approach and some modern views of logic. But one thing that immediately catches the eye in Goldfarb’s prospectus is his reference to the contemporary conception of logic. And that should surely give us some pause, even before reading on.

So how does Goldfarb characterize this contemporary conception? It holds, supposedly, that

the subject matter of logic consists of logical properties of sentences and logical relations among sentences. Sentences have such properties and bear such relations to each other by dint of their having the logical forms they do. Hence, logical properties and relations are defined by way of the logical forms; logic deals with what is common to and can be abstracted from different sentences. Logical forms are not mysterious quasi-entities, a la Russell. Rather, they are simply schemata: representations of the composition of the sentences, constructed from the logical signs (quantifiers and truth-functional connectives, in the standard case) using schematic letters of various sorts (predicate, sentence, and function letters). Schemata do not state anything and so are neither true nor false, but they can be interpreted: a universe of discourse is assigned to the quantifiers, predicate letters are replaced by predicates or assigned extensions (of the appropriate arities) over the universe, sentence letters can be replaced by sentences or assigned truth-values. Under interpretation, a schema will receive a truth-value. We may then define: a schema is valid if and only if it is true under every interpretation; one schema implies another, that is, the second schema is a logical consequence of the first, if and only if every interpretation that makes the first true also makes the second true. A more general notion of logical consequence, between sets of schemata and a schema, may be defined similarly. Finally, we may arrive at the logical properties or relations between sentences thus: a sentence is logically true if and only if it can be schematized by a schema that is valid; one sentence implies another if they can be schematized by schemata the first of which implies the second. (pp. 64–65)

Note an initial oddity here (taking up a theme that Timothy Smiley has remarked on in another context). It is said that a ‘logical form’ just is a schema. What is it then for a sentence to have a logical form? Presumably it is for the sentence to be an instance of the schema. But the sentence ‘Either grass is green or grass is not green’ — at least once we pre-process it as ‘Grass is green $\lor$ $\neg$\,grass is green’ — is an instance of both the schema $P \lor \neg P$ and the schema $Q \lor \neg Q$. These are two different schemata (if we indeed think of schemata, as Goldfarb describes them, as expressions ‘constructed from logical signs … using schematic letters’): but surely we don’t want to say that the given sentence, for this reason at any rate, has two different logical forms. So something is amiss.

But let’s not worry about this detail for the moment. Let’s ask: is Goldfarb right that contemporary logic always (or at least usually) proceeds by defining notions like validity as applying in the first instance to schemata?

Some other writers on the history of logic take the same line about modern logic. Here, for example, is David Bostock, in his Russell’s Logical Atomism (2012), seeking to describe what he supposes is the ‘nowadays usual’ understanding of elementary logic, again in order to contrast it with the view of one of the founding fathers:

In logic as it is now conceived we are concerned with what follows from what formally, where this is understood in terms of the formal language just introduced, i.e. one which uses ‘P’, ‘Q’, … as schematic letters for any proposition, ‘a’, ‘b’, … as schematic letters for any reference to a singular subject, and ‘F’, ‘G’, … as schematic letters for any predicate. So we first explain validity for such schemata. An interpretation for the language assigns some particular propositions, subjects or predicates to the schematic letters involved. It also assigns some domain for the quantifiers to range over …. Then a single schematic formula counts as valid if it always comes out true, however its schematic letters are interpreted, and whatever the domain of quantification is taken to be. A series of such formulae representing an argument … counts as a valid sequent if in all interpretations it is truth-preserving, i.e. if all the interpretations which make all the premises true also make the conclusions true. …

We now add that an actual proposition counts as ‘formally valid’ if and only if it has a valid form, i.e. is an instance of some schematic formula that is valid. Similarly, an an actual argument is ‘formally valid’ if and it only if it has a valid form, i.e. is an instance of some schematic sequent that is valid. Rather than ‘formally valid’ it would be more accurate to say ‘valid just in virtue of the truth functors and first-level quantifiers it contains’. This begs no question about what is to count as the ‘logical form’ of a proposition or an argument, but it does indicate just which ‘forms’ are considered in elementary logic.

Finally, the task of logic as nowadays conceived is the task of finding explicit rules of inference which allow one to discover which formulae (or sequents) are the valid ones. … What is required is just a set of rules which is both ‘sound’ and ‘complete’, in the sense (i) that the rules prove only formulae (or sequents) that are valid, and (ii) that they can prove all such formulae (or sequents). (pp. 8–10)

Bostock here evidently takes very much the same line as Goldfarb, except that he avoids the unhappy outright identification of logical forms with schemata. And he goes on to say that not only do we define semantic notions like validity in the first place for schemata but proof-systems too deal in schemata — i.e. are in the business of deriving schematic formulae (or sequents) from other schematic formulae (or sequents).

It isn’t difficult to guess a major influence on Goldfarb. His one-time colleague W.V.O. Quine’s Methods of Logic was first published in 1950, and in that book — much used at least by philosophers — logical notions like consistency, validity and implication are indeed defined in the first instance for schemata. Goldfarb himself takes the same line in his own later book Deductive Logic (2003). Bostock’s own book Intermediate Logic is perhaps a little more nuanced, but again takes basically the same line.

But the obvious question is: are Goldfarb and Bostock right that the conception of logic they describe, and which they themselves subscribe to in their respective logic books, is so widely prevalent? I have certainly heard it said that a view of their kind is ‘canonical’: but what does the canon actually say?

Philosophers being a professionally contentious lot, we wouldn’t usually predict happy consensus about anything much! If we are going to find something like a shared a canonical modern conception, it is more likely to be an unreflective party line of mathematical logicians, who might be disposed to speed past preliminary niceties en route to the more interesting stuff. At any rate, what I propose to do here is to concentrate on the mathematical logicians rather than the philosophers. So let’s take some well-regarded mathematical logic textbooks from the modern canon.

How far, going back, should we cast the net? I start with Mendelson’s classic Introduction to Mathematical Logic (first published in 1964), and some books from the same era. Now, you might reasonably say that — although these books are ‘contemporary’ in the loose sense that they are still used, still recommended — they aren’t sufficiently up-to-date to chime with Goldfarb’s and Bostock’s intentions when they talk about logic as it is ‘nowadays conceived’. Fair enough. It could turn out that, beginning with an initially messy variation in approaches in the ‘early modern’ period (if we can call it that! — I mean the 1960s and 1970s, some seventy and more years after the first volume of Grundgesetze), there does indeed later emerge some convergence on a single party line in the ‘modern modern’ period. Well, that will be interesting if so. And it will be interesting too to try to discern whether any such convergence (if such there has been) is based on principled reasons for settling on one dominant story.

So what we’ll be doing to considering e.g. how various authors have regarded formal languages, what they take logical relations to hold between, how they regard the letters which appear in logical formulas, what accounts they give of logical laws and logical consequence, and how they regard formal proofs. To be sure, we will expect to find recurrent themes running through the different treatments (after all, there is only a limited number of options). But will we eventually find enough commonality to make it appropriate to talk of ‘the’ canonical contemporary conception of logic among working logicians? And if so, will it be as Goldfarb and Bostock describe it?

Let’s look and see …

[To be continued]

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The all-you-can-eat book buffet

One of the fixtures of the Cambridge year is the annual Cambridge University Press booksale. It lasts for a week or ten days in January, with the shelves continually being replenished as they empty. The Press sell off oodles of “damaged” books (where, very often, the only damage is caused by a neat red stamp on the verso of the title page, marking the book as “DAMAGED”). The going rate for a few years has been £3 for any paperback, and £7 for a hardback. The range of titles is extraordinary. And you can pick up some delightful bargains — important but inessential work-related books that it would be really rather nice to have but which you would never have forked out the full price for, or interesting finds that are intriguing enough to take a chance on. So far this year, I’ve picked up a few pleasing purchases, including a copy of Linsky’s The Evolution of Principia Mathematica which I’ll want to dip into (but could never have warranted spending £100 on), and a paperback of David Wyn Jones’s The Life of Haydn which is proving to be fascinating and highly readable.

But — O tempora o mores! — truffling through the sale shelves just isn’t the enjoyable experience it used to be. In the past the rule was that you could only buy ten books at a time (or was it a ten a day? I think so). There were busy times, but it was mostly other readers young and old you were bumping into, and you would have occasional friendly book chats to people as you browsed, swapping recommendations, and (by the sad standard of academics) a quietly Fun Time was had by all.

Now the rules have changed. You can buy as many books as you can cart away. So various second-hand booksellers come with bags and bags, boxes and boxes, and stand around like vultures, pouncing as soon as the staff bring out more stock as shelves empty, immediately grabbing great armfuls, not quite coming to blows but certainly jostling for space. And just as, when at a buffet, people heaping enormous stacks of food onto their plates simply puts you right off your lunch, this too is off-putting — the rapaciousness on display by people who want to turn a quick buck rather than find some interesting (and perhaps previously unaffordable) books to read themselves. Greed is never a good look.

I know that the Press want to get rid of a lot of stock in their sale. But it’s rather sad that in the process of increasing the number of books they get rid of (if that’s what’s happening) the genuinely enjoyable atmosphere seems to have gone.

Update Others have expressed disquiet too: new arrangements have been announced, with early morning until 10.30 and afternoons after 3.30 as “quiet times”. Hopefully this will help.

Another update The new arrangements have indeed made for a much more congenial experience, the couple of times I’ve dropped by since.

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CD choice #2

51F8TdOZldL I have come very late to appreciate Janáček’s piano music — something I largely owe to two discs by Ivana Gavrić. Her fine first CD, titled after that composer’s In the Mists, was a serendipitous find in an Oxfam shop (it also includes, among other things, an excellent performance of Schubert’s A minor Sonata, D. 784). But her second recording from 2011, From the Street, is even better. If you don’t know Janacek’s sequence of ten pieces On an Overgrown Path, then you have a delight awaiting you. Gavrić’s playing seems exceptional here: the Gramophone reviewer rightly wrote of “the intimacy, finely honed nuance, conversational flow and subtle underlining of the composer’s harmonic surprises that Gavrić brings to each of the short pieces”, and other reviewers were equally enthusiastic. There are, I have since discovered, some other terrific recordings available, including one by Marc-André Hamelin. But this still strikes me as outstanding.

Also on the CD are Janáček’s Sonata 1.X.1905 From the Street, Ravel’s Valses Nobles Et Sentimentalise, and not least a wonderful performance of Prokofiev’s Sonata no. 2. (I’m not usually a great one for mixed recital discs, and I usually listen to these performances separately; but actually the programming works very well). So indeed, all very warmly recommended, especially if the Janáček or Prokofiev isn’t familiar.

Footnote Ivana Gavrić won a BBC Music Magazine Award in 2011 for her first CD. It is now time to vote for this year’s Awards. The Pavel Haas Quartet are shortlisted in the Chamber Music section. So you know what you have to do …

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Book Note: Tony Roy, Symbolic Logic, #3

After a longer than intended gap, I return to consider Parts III and IV of Tony Roy’s freely available  Symbolic Logic: An Accessible Introduction to Serious Mathematical LogicThe previous two , rather lukewarm, instalments discussing Parts I and II are here and here (but do please note Roy’s long comment in response).

Part of my beef against this very long text is indeed its length. In teaching maths, often the key task is to engender the kind of understanding that enables the student to see the wood for the trees, to see what are the Big Ideas and what is merely hack-work joining up the dots. The longer you bang on filling in every last detail of a proof, the greater the danger that you will obscure the overall contours of what’s going on (even if you scatter around an amount of signposting). We ask our students in exams to do “bookwork” questions outlining a proof of some major result, and here the name of the game is indeed to highlight the Big Ideas, the key moves, and to confidently know what can be gestured at, or when we can say “rather similarly, we can show …” etc. Where I take issue with Roy’s pedagogic style, then, is in thinking that writing at his length won’t really help foster these skills.

I mention this again because Part III on Classical Metalogic consists mostly of two very dense chapters, one of forty pages, one of fifty pages, going into rather over-the-top detail (by my lights) on relatively few results. So again I wouldn’t recommend these as primary reading for students encountering some metalogic for the first time.

However, on the positive side, the main content of Chapter 9 is unusual in one interesting respect. Roy has earlier introduced both an axiomatic and a natural deduction system for first-order logic. We can of course prove they are equivalent by going via the respective soundness/completeness proofs for the two systems. (That doesn’t really require two lots of proofs as we can point out in particular that what it takes for a Henkin  completeness proof to go through is available in both cases.) But we can also give a syntactic equivalence proof for the two systems by showing how to systematically manipulate in both directions a proof of the one sort into a proof of the other. Now, this tends to be the sort of thing one armwaves about in class, perhaps sketching in a few obvious moves. But I can’t offhand recall any textbook which spells out in any detail, for particular given axiomatic and ND systems, clear routines for moving between proofs in the two systems (any offers here?). Roy however does this at (slightly numbing) length. Good: we can now usefully point any students unsatisfied by classroom armwaving who want to know how such proofs go to Roy’s very careful working-out of detailed routines.

Chapter 10 then tackles soundness and completeness. The completeness proof for the sentential fragment takes eleven pages: another thirty pages labour over the completeness of a first-order calculus with identity. This is, for example, over twice the number of pages needed by the extraordinarily lucid, gently paced, Chiswell and Hodges. I won’t quote chunks, as you can make your own judgement, since the chapter (as with the rest of the Accessible Introduction) is freely available. But I honestly can’t imagine many students finding the extra length going with a doubling in clarity and understanding. Indeed, if a student were stuck on a Henkin completeness proof in one standard text, I’d first suggest looking at another snappy presentation in a different text (before mentioning Roy’s much longer efforts): for the problem would very likely be in seeing the overall strategy, the Big Ideas — and brisker presentations are likely to make those stand out better.

Part III also contains a fragmentary Chapter 11 which belatedly talks about expressive completeness for the sentential connectives, and then says something briefly about compactness and the L-S theorems. But I won’t say anything about this.

To be concluded.

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Easing gently into 2016

There is a new version of the Gentle Introduction to Category Theory. There are no new chapters this time, but there are some significant additions (I now prove a result about Cartesian closed categories with natural numbers objects, which previously was only announced, and prove that free monoids can be thought of as initial objects in a certain comma category). And there are many improvements, both in content and presentation. Note in particular, I correct a mistake about the relation between different notions of diagrams, and clear up what was an unnecessarily messy chapter on the existence of limits. I am very grateful indeed to comments/corrections from Paolo G. Giarrusso and Yufei Cai for prompting some of these improvements.

Although now 178pp., this version is still very incomplete: you can find some rough-and-ready follow-up chapters at the categories page here where there is also an alternative link to the Gentle Intro for those without an academia.edu login.

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A cheering start to the logical New Year?

Just before Christmas, I put a copy of Teach Yourself Logic 2016 on my rather sparse academia.edu page. It has since been browsed there over 45,000 times, and then the whole thing downloaded 2,500 times. I’m not sure how many times TYL has also been downloaded from Logic Matters, as the stats counter here is flakey (though the page it is linked from has been visited over 11,000 times in the last few weeks): but it is hundreds more.

The particular numbers don’t matter. But the trend is good.  At a personal level, this makes the effort I put into TYL continue to seem worthwhile. And more impersonally, this is serious logic we are talking about here: and caring about the future of the subject, it is really  good to find that there is enough interest out there for thousands of people to go to the bother of downloading the Guide, with at least some sense of what they are letting themselves in for. So this is all rather cheering, and puts a spring in my logical step, inspiring me to keep on tinkering with TYL and with related stuff.

But not quite yet! Back to category theory first …

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CD choice #1


Those end-of-year lists of recommended books are really rather depressing, aren’t they? Even setting aside the pretentious, the uninviting, the distinctly esoteric, there remain all those novels, all those biographies, all those histories, and much more, books that do sound so enticing, yet which you know — despite your best resolutions to read more, and idle less on the internet — you are never going to have the time to read.

Lists of the best CDs of the year, however, I find much more cheering. And with a subscription to Apple Music or the like, you can quickly sample a fair selection of the recommendations that you’d earlier missed, and then listen to a goodly virtual pile of the discs that grab you the most, all in the time it would take to get through that six hundred page history book you aren’t going to read ….

Well, I’ve missed the appropriate time to give my own recommendations from the new classical CDs released in 2015 — and anyway, to be honest, it wouldn’t have been that exciting, but mostly just a rather predictable subset of the monthly recommendations in the Gramophone (predictable, at any rate, given the sort of CDs and concerts mentioned here over the years). So let me begin the year by starting something hopefully a bit more interesting, namely a fairly regular series of  ‘CD choice’ posts, mentioning a disc that I’ve been listening to with enjoyment over the previous few days, perhaps emphasizing discs not as well known as they might be. I’ll try, by the way, mostly to mention recordings available on Apple Music (and presumably on other subscription services). It could be a new release, or an old disc that I forgotten that I had, a recent charity-shop find, something caught by chance on internet radio … Who knows? We’ll just see how it goes! [I was thinking of posting weekly, hence the initial title ‘CD of the week’, but I quickly thought better of it — not because I couldn’t recommend a  CD every seven days, but because that many posts on music would unbalance what is still supposed to be mainly a logic-related blog!]

First up, then, a delightful disc first released in 2014, the oboist Albrecht Mayer’s “Lost and Found”. This is subtitled “Oboenkonzerte des 18. Jahrhunderts von Hoffmeister, Lebrun, Fiala und Kozeluh”, which sounds potentially worthy but dull; but in fact, this is simply very, very enjoyable.

So these are concertos for oboe and cor anglais from around the 1780s, contemporary with Mozart and Haydn, from four other composers well known in the day (and not entirely “lost” since!). The music is immediately engaging yet certainly stands up to repeated listenings. The playing on the CD is terrific (as the Gramophone agrees). Try it!

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Sic transit …

The flood of freely available downloads of pre-2005 mathematics and philosophy books from Springer — including many logical classics, for which I posted a couple of very partial “taster menus” here — didn’t last long! Two days on, the free downloads are no longer available. I believe that there may have been issues about Springer making available books for which they didn’t have the full ownership of the copyright, without consulting authors.

It would be cruel to those who missed the party to leave up the previous posts detailing what they’ve missed; so those posts are for the moment deleted. We can only guess at the background story. We will just have to see whether, in due course, Springer do start making older books freely available when they can (I can see why it might be in their overall interests to do so).

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A Christmas card


Gentile da Fabriano, Adoration of the Magi (detail), 1423

With all good wishes for a very happy and peaceful Christmas.

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51QH+wFLTeL._SX323_BO1,204,203,200_We have far too many books for a house this small, despite our best efforts to make room on shelves by passing on the unloved or the never-to-be-reread to Oxfam. So these days, we try not to buy a newly published book if neither of us is going to read it more or less immediately (of course, a different rule applies to happy second-hand finds). That’s why, when Kate Atkinson’s companion piece to her truly brilliant Life after Life came out, despite our both very much looking forward to reading it, we didn’t buy it immediately, both already having tall piles of books waiting for us. And then it got to the point when a paperback was announced for the end of this month, and we decided to hang on and buy it to read over the festive season.

But ah,  a couple of weeks ago, in  the National Trust second-hand bookshop at Wimpole, there it was — an almost pristine copy of A God in Ruins. The beautifully produced hardback for less than half the cost of a paperback. And it’s quite unreasonably cheering — isn’t it? — this kind of serendipitous find. Well ok, not  that serendipitous, when you think about it: Kate Atkinson is a best-selling author, we often drop in to take a quick look at the book shelves of charity shops large and small, so I suppose it was pretty likely that we’d stumble on a copy over the months since it came out. But even that thought doesn’t make the find less cheering. It isn’t the matter of saving a few pounds (or of giving the money to a charity rather than a chain bookstore); that’s nice, to be sure, but it doesn’t account for the pleasure, the happy feeling engendered by the little smidgin of good fortune. And it’s the sort of little thing that sticks in the mind,  “Do you remember finding that by chance when we were on holiday in …?”, it becomes part of your history of your encounter with the book in a way that just marching into Blackwells and picking the volume off a pile never does.

It is, or rather was, the same with CDs, the pleasure of the happy find in charity shops of something well-known that you’ve been wanting for a while, or of something quite obscure but intriguing. I had much more enjoyment over some years acquiring the complete Hyperion Schubert Edition — all 37 volumes of Graham Johnson’s astonishing exploration of all Schubert’s songs with various singers — mostly second-hand than I ever would have done just buying the lot new. I have had more fun again discovering baroque composers you’ve never heard of, or later obscure Bohemians, and the rest. (Of course there are plenty of misses as well as the hits, but the few pounds have gone to charity, so the misses don’t matter at all.)  But now I mostly use Apple Music to stream music, for there is precious little space for more CDs too; and I miss that kind of serendipity. Yes, of course, there is an unending source of music to explore there for the small subscription. Yes, of course, you make happy finds. But, dinosaur that I am, it just doesn’t feel the same.

No way, though, are we swapping real books for e-books. Putting a few on the iPads when travelling is fine: but a well-made real book remains a continuing delight that we are not giving up.  So we’ll carry on the pleasurable truffling through the charity shops: it’s the splendid Oxfam bookshop in Saffron Walden tomorrow, I think …

And how was A God in Ruins? As good as they say. Just wonderful.

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