Arnie has been on a flying visit to Cambridge. Always very good to see him — and great fun to go out for a long evening of wall-to-wall philosophical gossip at the excellent Riceboat. One thing which I hadn’t registered which is worth passing on: Arnie’s A Structuralist Theory of Logic is now available as a paperback. It’s still not exactly cheap, but I do hope it gives the book another lease of life, as it deserves a higher profile than it has had. At least make sure it is in your university library!
I’ve just started (re)reading John Burgess’ Fixing Frege. It is really full of useful things, but I still think what I thought on a first reading — namely that this is a pretty annoying book, as it surely could have been done so much better. Done better, for a start, by being done more slowly, with some of the technical exposition being handled more carefully and more transparently, with more commentary. For example, which grad students are going to see what’s natural about the Friedman/Simpson hierarchy of subsystems of second-order arithmetic just from the exposition on pp. 67-68? The book is a must-read; but it is also an unnecessarily difficult read given its presumbably intended audience.
Well, I think I’m going to have to admit defeat. I’ve tried reading Fine’s paper for the third time and I’m still stumped by his positive claims about ‘postulational modality’.
The defender of indefinite extensibility thinks that ‘whatever interpretation [of the supposedly absolutely general quantify] our opponent might come up with, it will be possible to come up with an interpretation that extends it’. And supposedly the second modality here, at any rate, is ‘postulational’. Whatever exactly that means. Presumably the thought is that whatever objects you are quantifying over, I can postulate another one — the set-like collection of all the sets you are quantifying over which aren’t members of themselves — which can’t already be in your domain of quantification, on pain of paradox. But how does this differ from there being such a set-like collection? On the one hand, if Fine is to be making a new move here, there better be a difference; on the other hand, it is difficult to understand what the move is without a clear account of the difference — i.e. a treatment of the metaphysics of (some) mathematical entities as postulated entities, which Fine doesn’t give us.
But set aside those worries. Let’s suppose that, while there isn’t any sense in which you can postulate new donkeys into existence (so ‘there are no talking donkeys’ isn’t, so to speak, vulnerable to a legitimate postulated extension of the domain of quantification), you can postulate new sets (or set-like collections). Well, so what? Why can’t the defender of absolute quantification just aver that when he says, e.g. ‘Everything is self-identical’ or ‘Nothing is a talking donkey’ he already means to cover whatever your postulational ingenuity might come up with — and dig his heels in when you insist that you can still find another entity which might comprise all those things at once (so he is vulnerable to the extensibility argument). Rather he takes the argument of Russell’s paradox as showing us that there is no such single entity.
Which is a familiar dialectic of course. So what I’m missing is how talk of ‘postulational modality’ is supposed to move things forward. As I say, I’m stumped — and will be very happy to get comments from anyone whose grip on Fine’s paper is better than mine.
I’ve just noticed that this will be the hundredth post: which is a landmark of sorts! So why do I bother?
Hmmmm, a good question! Here’s what I officially tell myself. It’s a pretty good discipline writing notes on at least some of what I’m reading (otherwise, these days, I forget depressingly much of what I’ve just been thinking about as soon as I move on to the next thing!). And if I am writing notes for myself, I might as well post some of them here, for whatever they are worth. I’ve always really enjoyed reading brief comments and replies, the more relaxed the better — right back from the days of the replies at the end of e.g. Words and Objections and the Schilpp volumes through to, for example, current exchanges on FOM. So hopefully others might similarly find some of my ramblings useful too.
And unofficially? Well, it’s just fun sounding off …
Suppose I think that there is something problematic about absolutely general quantification. So I try to say “You can’t quantify over absolutely everything”. But either that “everything” is absolutely general, and I’ve illustrated how you can quantify over absolutely everything after all. Or else my “everything” is restricted, and I fail to say what I meant to say. Either way, my attempted saying misfires.
So that disposes of the anti-absolutist? Well, no … I just need to be a bit more dialectically supple: I shouldn’t assert a position myself, but rather stand ready to reveal the tempting confusion that the absolutist has fallen into. Faced with a philosopher who stakes out an absolutist position, the enlightened opponent hits him with an extensibility argument (“Ah, take those things you are quantifying over all together as one big domain; now consider the bit of the domain which contains all the non-self-membered things you were quantifying over; then that isn’t one of the things you were quantifying over, on pain of Russell’s paradox”). Then — assuming of course the cogency of such extensibility arguments — the absolutist is in trouble. Which is something the enlightened philosopher, to coin a phrase, shows rather than says.
Kit Fine, at the end of Section 2 of his paper, floats the possibility of taking this rather Wittgensteinian line. But he doesn’t endorse it — rather there are another fifteen pages in which he tries to find the words in which one might cogently state an anti-absolutist position. The idea is to go modal, and talk in particular about “postulational modalities”. This, however, all gets deeply obscure. I’m going to have to read Fine’s paper for a third time and try to make more sense of it. Watch this space …
Good heavens! Amazon UK reports the Gödel book this morning as 3,069 in the sales ranking. That makes me and J.K. Rowling, who lives permanently at number 1, practically neighbours. I’m preparing myself for the inevitable change of life-style.
Googling around to see what their sales rankings really mean, the answer is that a snapshot ranking means diddly squat. Still … in the academic philosophy rankings, currently Gödel (at 7) beats the pomos, so that is — just for the moment! — cheering.
Hooray! A version of my talk at the Isaacson day we had in Cambridge a couple of months ago has been accepted by Analysis, and will appear in January. Michael Clark has kindly agreed to publish it as a preprint on the Analysis website shortly (as soon as I can un-LaTeX it into a W*rd document, arggghhh!).
For the moment, I’ve put a link to a late draft of the paper in the “Other materials” page on the Gödel book website which (at last) I’m starting slowly to build up. I need in particular to put my mind to compiling fun(?) sets of exercises. That’s because IGT does not contain end-of-chapter exercises, for two reasons. First, the book is already long and adding copious exercises would have made it longer still. Secondly, I didn’t want to put off the more philosophically inclined half of my readers by making the book look too forbidding.
I discovered the first misprints today. But fortunately tiny ones — on p. 341 I oddly use “primitively recursive” twice. But as misprints go, these are not going to cause any loss of sleep!
OK, time to make a start on blogviewing Absolute Generality, edited by Augustín Rayo and Gabriel Uzquiano (OUP, 2006).
As in the Church’s Thesis volume, the editors take the easy line of printing the papers in alphabetical order by the authors’ names, and they don’t offer any suggestions as to what might make a sensible reading order. So we’ll just have to dive in and see what happens. First up is a piece by Kit Fine called “Relatively Unrestricted Quantification”.
And it has to be said straight away that this is, presentationally, pretty awful. Length issues aside, no way would something written like this have got into Analysis when I was editing it. This isn’t just me being captious: sitting down with three very bright and knowledgeable graduate students and a recent PhD, we all struggled to make sense of it. There really isn’t any excuse for writing this kind of philosophy with less than absolute clarity and plain speaking directness. It could well be, then, that my comments — such as they are — are based on misunderstandings. But if so, I’m not sure this is entirely my fault!
Fine holds that if there is a good case to be made against absolutely unrestricted quantification, then it will be based on what he calls “the classic argument from indefinite extendibility”. So the paper kicks off by presenting a version of the argument. Suppose the ‘universalist’ purports to use a (first-order) quantifier ∀ that ranges over everything. Then, the argument goes, “we can come to an understanding of a quantifier according to which there is an object … of which every object, in his sense of the quantifier, is a member”. Then, by separation, we can define another object R whose members are all and only the things in the universalist’s domain which are not members of themselves — and on pain of the Russell paradox, this object cannot be in the original domain. So we can introduce a quantifier ∀+ that runs over this too, and hence the universalist’s quantifier wasn’t absolute general.
Well, this general line of argument is of course very familiar. What I initially found a bit baffling is Fine’s claim that it doesn’t involve an appeal to what Cartwright calls the All in One principle. Here’s a statement of the principle at the end of Cartwright’s paper:
Any objects that can be taken to be the values of the variables of a first-order language constitute a domain.
where a domain is something set-like. Which looks to be exactly the principle appealed to in the first step of Fine’s argument. So why does Fine say otherwise?
Well, Fine picks up on Cartwright’s initial statement of the principle:
to quantify over certain objects is to presuppose that those objects constitute a ‘collection’ or a ‘completed collection’ — some one thing of which those objects are members.
And then Fine leans heavily on the word ‘presuppose’, saying that extendibility argument isn’t claiming that an understanding of the universalist’s ∀ already presupposes a conception of the domain-as-object and hence an understanding of ∀+; it’s the other way around — an understanding of ∀+ presupposes an understanding of ∀. Well, sure. But Cartwright was not saying otherwise, but at worst slightly mis-spoke. His idea, as the rest of his paper surely makes clear, is that the extendibility argument relies on the thought that where there is quantification over certain objects then we must be be able to take those objects as a completed collection — but Cartwright isn’t saying that understanding quantification presupposes thinking of the the objects quantified over constitute another object. Anyone persuaded by Cartwright’s paper, then, won’t find Fine’s version of the extendibility argument any more convincing than usual.
[To be continued]
Here is E. T. Jaynes writing in Probability Theory: The Logic of Science (CUP, 2003).
A famous theorem of Kurt Gödel (1931) states that no mathematical system can provide a proof of its own consistency. … To understand the above result, the essential point is the principle of elementary logic that a contradiction implies all propositions. Let A be the system of axioms underlying a mathematical theory and T any proposition, or theorem, deducible from them. Now whatever T may assert, the fact that T can be deduced from the axioms cannot prove that there is no contradiction in them, since if there were a contradiction, T could certainly be deduced from them! This is the essence of the Gödel theorem. [pp 45-46, slightly abbreviated]
This is of course complete bollocks, to use a technical term. The Second Theorem has nothing particularly to do with the claim that in classical systems a contradiction implies anything: for a start, the Theorem applies equally to theories built in a relevant logic which lacks ex falso quodlibet.
How can Jaynes have gone so wrong? Suppose we are dealing with a system with classical logic, and Con encodes ‘A is consistent’. Then, to be sure, we might reflect that, even were A to entail Con, that wouldn’t prove that A is consistent, because it could entail Con by being inconsistent. So someone might say — students sometimes do say — “If A entailed its own consistency, we’d still have no special reason to trust it! So Gödel’s proof that A can’t prove its own consistency doesn’t really tell us anything interesting.” But that is thumpingly point missing. The key thing, of course, is that since a system containing elementary arithmetic can’t prove its own consistency, it can’t prove the consistency of any stronger theory either. So we can’t use arithmetical reasoning to prove the consistency e.g. of set theory — thus sabotaging Hilbert’s hope that we could do exactly that sort of thing.
Jaynes’s ensuing remarks show that he hasn’t understood the First Theorem either. He seems to think it is just the ‘platitude’ that the axioms of a [mathematical] system might not provide enough information to decide a given proposition. Sigh.
How does this stuff get published? I was sent the references by a grad student working in probability theory who was suitably puzzled. Apparently Jaynes is well regarded in his neck of the woods …
A knock on my office door an hour ago, and the porter brought in two boxes, with half a dozen pre-publication copies each of the hardback and the paperback of my Gödel book.
It looks terrific. Even though I did the LaTeX typesetting, I’m happily surprised by the look of the pages (they are symbol-heavy large format pages in small print, yet they don’t seem off-puttingly dense).
As for content, I’ve learnt from experience that it’s best just to glance proudly at a new book and then put it on the shelf for a few months — for if you start reading, you instantly spot things you don’t like, things that could have been put better, not to mention the inevitable typos. But of course, the content is mostly wonderful … so hurry, hurry to your bookshop or to Amazon and order a copy right now.