Peter Smith's Blog, page 84

A Christmas card

From Journey of the Magi to Bethlehem, Benozzo Gozzoli, 1460.

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

Tarski on disjunction

Before going off to Florence, I was reworking chapters on the material conditional for IFL2 (in fact I posted a couple of draft chapters here, which I then thought I could improve on,  and so I rapidly took them down again). While away, it occurred to me that it might be prudent/interesting/useful to take a look at how various standard logic texts over the years have handled the conditional in propositional logic. So here I am, starting to work quickly through a pile of some 25 introductory texts, from  Alfred Tarski’s Introduction to Logic and to the Methodology of the Deductive Sciences (1936/1941) to Jan von Plato’s Elements of Logical Reasoning (2013).

I’m writing some telegraphic notes for myself as I go along, but I’ve quickly realized it would take far too much time (and be far too distracting from what I am supposed to be doing) to work these up to give detailed and fair-minded stand-alone reports here. But let me say something about the first book I turned to. For I was surprised and intrigued when reading Tarski to discover  how rocky his arguments are (and indeed how unclear his position is about the relation of ordinary language and the connectives of the formal logician). But let’s not tangle now with what he says about the conditional; his preceding remarks about disjunction already show some of the problems he gets into. So here are some quick notes about those remarks.

Talking of “or” as used to join two sentences, Tarski very confidently asserts that

… in everyday language, the word “or”  has at least two different meanings,

in particular, inclusive and exclusive meanings. But the only supposed illustration given for this once-popular ambiguity claim is hopelessly weak. Tarski writes

… if a child has asked to be taken on a hike in the morning and to a theater in the afternoon, and we reply:

no, we shall go on a hike or we shall go to the theatre,

then our usage of the word “or” is obviously of the second [exclusive] kind, since we intend to comply with only one of the two requests.

But what’s the “no” doing here? It is denying that we will both go for a hike and go to the theatre. So the envisaged reply is arguably just a verbal variant of

 We shall go on a hike or we shall go to the theatre but not both.

But while everyone agrees that something of the whole form P or Q but not both expresses the exclusive disjunction of P and Q, it certainly doesn’t follow that the clause P or Q taken by itself means exclusive disjunction. So the sole given example doesn’t give us any evidence that there is a distinctive exclusive meaning of “or”.

Tarksi continues

In logic and in mathematics the word “or” is used always in the first, non-exclusive meaning.

But is he entitled to this claim? After all, if what he wrote about the hike/theatre example holds good, surely this would too:

… if a child wonders if the number 49 is divisible by both three and seven, we might reply (as a hint):

no, 49 is divisible by three or 49 is divisible by seven: work out which!

And our usage of the word “or” is obviously of the exclusive kind, since we intend to allow only one of the two disjuncts to be true.

So, by Tarski’s lights, we ought to have here a mathematical use of the exclusive or! What he meant, presumably, is that in the formal logician’s regimented usage, when  “or” is symbolised by ““, the disjunction is inclusive: but this isn’t what he actually says at this point.

Tarski then continues,

Even if we confine ourselves to those cases in which the word “or” occurs in its first meaning, we find quite noticeable differences between its usage in everyday language and that in logic. In common language two sentences are joined by the word “or” only when they are in some way connected in form and content …. It is not altogether clear what kinds of connections would be appropriate here, and any attempt at their detailed analysis and description would lead to considerable difficulties. As we shall see, such connections are disregarded in contemporary logic, where consequently one has to allow some strange examples; and indeed, anybody unfamiliar with its language would presumably be little inclined to consider a phrase such as:

2 x 2 = 5 or New York is a large city

as a meaningful expression, and even less so to accept it as a true sentence.

What does “in some way connected in form and content” mean? We are are given no hint. Yet on any natural reading, the claim that the disjuncts of everyday disjunctions will be so connected seems over-strong. To take a contemporary example: an article about fake news lists ten surprising/bizarre claims, and asks us to spot which five claims are true, and which five are fake. We happen to know four of the claims to be true, and we dismiss another four as false. That leaves us with two claims up for grabs, P and Q. And in the circumstances it is now entirely natural to assert P or Q even though these disjuncts need not be in any obvious sense connected in form and content any more than are 2 x 2 = 5 and New York is a large city (it is just that the newspaper article has made P and Q both salient in the context, and we justifiably think one is true).

Further, whatever the first inclination of the untutored, it is odd for anyone to cast serious doubt on the meaningfulness of  2 x 2 = 5 or New York is a large city. (We’d surely now say that here Tarksi is running together questions of meaning, of semantics, and questions of pragmatics, of conversational appropriateness in a given context.) After all, it is precisely because it is a meaningful bit of English that we understand the sentence here perfectly well, and so realize that it takes a bit of work of describe a situation in which this disjunction would be a conversationally natural thing to assert. Yet of course it takes exactly the same amount of work to describe a situation in which 2 x 2 = 5  New York is a large city would be a natural thing to assert. On this count, at any rate, there’s nothing to distinguish between “” and (inclusive, sentential connective) “or”.

Tarski continues

Sometimes we even take the utterance of a disjunction as an admission by the speaker that he or she does not know which member of the disjunction is true, and which is false. And if we later arrive at the conviction that the speaker knew at the time that one—and, specifically, which—of the members was false, we are inclined to look upon the whole disjunction as a false sentence, even though the other member should be undoubtedly true.

Well, no. We might, in those circumstances, conclude that the speaker was being misleading, even culpably misleading, in asserting (only) the disjunction. But that doesn’t make the disjunction false. It is surely a naive common-place — not a bit of high post-Gricean theory — that we can can, in many ways, be misled as to the truth by the utterance of a perfectly true sentence.

It is intriguing, then, to find Tarski (he of the formal semantic account of truth!) seemingly so at sea over what would strike us as elementary distinctions of issues of semantics and literal truth vs issues of pragmatics. As you would now expect, this doesn’t bode well for his discussion of conditionals!

Logic (etc.) books of the year?

It’s that time again when the weekend papers are full of their lists of books of the year.  I have to say that so many recommendations sound frankly quite unappealing — surely, there’s a lot of literary virtue-signalling going on! —  but that still leaves me wanting to read more  than I will ever have time to get round to. But one book (perhaps not exactly a philosophy book but certainly of philosophical interest) which has been warmly recommended a number of times is Sarah Bakewell’s At the Existentialist Café: Freedom, Being and Apricot Cocktails. I loved Bakewell’s book on Montaigne, How to Live. So, overcoming my analytical prejudices, I’ve just bought her new new book on Sartre and company as a holiday read. I’ll let you know what I think!

But what about logic books this year (or come to that, books on philosophical logic, philosophy of maths, or other topics broadly related to logic matters)? I have bought a number of older books in the last twelve months, but my haul of recent publications in logic, even broadly construed, seems to have been very modest. I’ve mentioned here two collections of essays, the  Cambridge Companion to Medieval Logic, edited by Catarina Dutilh Novaes and Stephen Read (not quite what I’d hoped for) and  Kurt Gödel, Philosopher-Scientist, edited by Gabriella Crocco and Eva-Maria Engelen (a very mixed bag, and pretty disappointing). But I balked at the price of another collection, Gödel’s Disjunction, edited by Leon Horsten and Philip Welch, as the papers again looked likely to be a pretty mixed bag: so I can’t comment on that. The only logic monograph I bought was  the significantly expanded second edition of Alex Oliver and Tim Smiley’s Plural Logic. Otherwise, my purchases seem to have been more skewed towards pure mathematics — the most accessible and fun read being Barry Mazur and William Stein’s Prime Numbers and the Riemann Hypothesis. 

So I’m not really in a position to recommend a logic (etc.) Book of the Year. All suggestions about what I’ve been missing out on will therefore be very welcome!

Another postcard from Florence

Some wonderful sunny days. Flights to Pisa and hotels in Florence both have enough spaces in December to risk last minute bookings when you’ve seen the weather forecast — but that’s not our way: so this has been sheer good luck. So we took advantage of the blue skies to “do” the Roman site at Fiesole — which perhaps in itself isn’t very exciting, but which does have a very attractively presented small archeological museum which is certainly worth the bus-trip up from Florence.

However, it’s not been all galleries, churches and sites. A certain amount of rather terrific food and drink has been consumed (for mid-culture sustenance, there is Eataly: and after a tough day we can recommend again Olio e Convivium, and Il Santo Bevitore, and a new discovery Il Desco). None will break the bank. And it is a tad depressing that a place as rich and cosmopolitan as Cambridge hasn’t anywhere to touch them.

But no, I shouldn’t say that’s “depressing” even in jest. What’s depressing is the evolving ghastly political news from America. And the continuing profoundly damaging mess that Brexit looks certain to be, thoughts of which are bound to nag away as you wander the side streets of old Europe.

Postcard from Florence

Back to Florence for five days. Lovely in the winter sun. It perhaps seems a little busier than this time last year, but of course still nothing like the dire crowds of summer. One high point has been seeing the restructured Botticelli rooms in the Uffizi which were opened in mid October. The improvement on the familiar Room 10-14 is simply stunning. The old large square room  has been partially divided, to make more wall space. So now the Birth of Venus and Primavera are both beautifully isolated and quite wonderfully well lit too. They can never have looked better.

Here though is a painting from the Uffizi that we’d never noticed before, the Madonna of the Well (c. 1510). Not Raphael as you might think at a first glance, but one Francesco Cristofano Guidicis, known as Franciabigio. Lovely though.

 

IFL2 news

I have just heard that CUP are definitely going to offer me a contract for a shiny new edition of my Introduction to Formal Logic.

If the press had never mentioned the idea, I would have probably never taken a hard look at the book again (since I’m no longer teaching from it, and haven’t done for six years) and so I would not have fretted about it.  But once the seed was sown of the idea of a new edition, I of course found myself re-reading the book with a critical eye. Not very happy with what I found! And so then, of course, I did indeed want to try to do a better job. Hence I’m very relieved and pleased that I will get the chance.

The material conditional — analysis or replacement?

Here’s a couple of chapters, just 22 pages, hot off the laptop, from a bit later in the draft second edition of my Intro to Formal Logic:

Ch 15: The material conditional. Ch 16 More on conditionals

This isn’t the whole of the story about conditionals planned in the book. These chapters will have Exercises, and there will some more about biconditionals there, as well as more illustrations of the oddities of identifying ordinary conditionals with material conditionals. Then a few chapters later we will encounter the very natural standard Natural Deduction rules for the conditional, which (against a classical background) give us the material conditional again — so we’ll have occasion to say more about how the material conditional keeps forcing itself on us. But these two are going to be the core chapters, replacing the current Chs 14 and 15.

Everyone has firm views about conditionals, and so no one is going to agree with these chapters! No doubt, that includes my future self. But I’m at the stage where I want to put the chapters in a e-drawer for a while, and return to them fresher in a few weeks. So in the meantime, I’d love to get comments and suggestions — either by email to ps218 at cam dot ac dot uk, or via the comments form below (though new users may have to wait a day or two to get comments approved). Thanks!

Generating truth-tables in LaTeX

I have just added to the “Logicians’ miscellany” page of LaTeX for Logicians a new heading “Help for generating truth-tables”. There is now a link there to a Truth Table Generator webpage by Michael Rieppel. This page contains a JavaScript program which will generate a truth table given one or more well formed formulas of sentential logic, and provide you with LaTeX source for the table.

Thanks to Sara Uckelman for the pointer to this. Any other recommendations for similar or even better resources?

Unsatisfying satisficing

There’s no getting away from it. You know perfectly well you can’t try to optimize a book — that way madness lies (or at least, never finishing). You know perfectly well you have to satisfice. But that is oh so unsatisfying. When I had to finish books for work reasons, I gritted my teeth and let stuff go. Now I’m retired, it’s more difficult not to keep on keeping on editing and (hopefully) improving. I need a contract settled for the second edition of IFL to concentrate the mind. But CUP’s wheels are grinding slowly (fingers crossed that that isn’t a bad sign).

Anyway, for anyone interested, here’s a tolerably polished draft of the first ten chapters of the second edition. Comments as always most welcome — and many thanks to those who have already given me some very useful feedback. And if you want to skip the pre-formal preamble made up of Chapters 1 to 6, and start commenting from Chapter 7 when the formal work gets under way, that would be perfectly welcome. (I’ll send more chapters to anyone who comments on at least some of these first ten.)

Plural Logic, again

A second, paperback, edition of Plural Logic by Alex Oliver and Timothy Smiley is now out from OUP. As the cover says, it is ‘Revised and Enlarged’ – in fact it is almost fifty pages longer, with some new sections and a whole new chapter on Higher-Level Plural Logic. So you should certainly make sure that your library gets a copy.

I did read and comment on a version of the original edition pre-publication. But that was not at a good time for me, and I remember much less detail than I should: so I really want now to re-read the book. One reasons is that, in reworking my Introduction to Formal Logic, I want to excise unnecessary set talk, e.g. when giving the semantics of QL. So I want to remind myself how Oliver and Smiley handle this. And there is also a tenuous potential connection too between thinking about plurals and another interest, my on-the-back-burner introductory discussion of category theory. For I need to think through how far we can get in elementary category theory by conceiving of categories plurally rather than as set-like, thereby avoiding certain problems of ‘size’ hitting us too soon.