Hellman's second line of argument against absolutely general quantification rests -- according to the title of Section 4 of his paper -- on the multiplicity of 'factually equivalent ontologies'.
The claim is that 'The same underlying factual situation [can be] described accurately and adequately in ontologically diverse ways. It would be arbitrary and unwarranted to say that just one is "really correct".' What sorts of case does Hellman have in mind? 'Familiar examples cited long ago by Goodman ... and others come from geometry (pure or applied), e.g. a framework with points and lines (say, in the two-dimensional case) vs. a framework with just lines, points being definable as (suitably selected) pairs of intersecting lines.' But hold on, both frameworks agree that there are points and lines. Either way, then, in promiscuously quantifying over absolutely everything, I'll be quantifying over both points and lines. So what's the problem?
Ah, says Hellman, the absolutist must claim that there is one correct answer to the question 'Are there sui generis points, i.e. points which are distinct from pairs of lines or nested volumes, etc., not constructed out of anything else? ... We may not ever know [the answer], but it shows up one way or another in the range of "absolutely everything".' Eh? Why is someone who claims that we can sensibly quantify over everything committed to the quite different claim that issues about what is ontological basic or sui generis have determinate answers? When I say that everything is self-identical (for example), I commit myself inter alia to agreeing that points, whatever they ultimately are, are self-identical and lines, whatever they are, are self-identical. But I just can't see why it is supposed to follow that I'm thereby committing myself to supposing that the ontology game (in the form of raising the question what's really, really fundamental?) is even a game with determinate rules let alone delivers determinate answers.
Hellman offers variants on the points/lines case. He asks us to consider an ontology of space-time regions that doesn't recognize the existence of ordinary objects like books; strictly, instead of saying that there are books over there, we should say that a certain region is "booked", etc. And Hellman's thought is that in this sort of case, 'no entity recognized in the ontology of this theory is literally a book' (contrast the points/lines case, where there are entities available to be identified as points in either ontology). But so what? What has this to do with the question of the possibility of absolutely general quantification over everything?
It has turned out that there are no gods (which was a bit of a surprise in some quarters); so those who thought -- in quantifying over everything that there is -- that they were including gods along with the books and the points and lines would have been mistaken. But the possibility of being mistaken about what exists doesn't in itself undermine the possibility of quantifying over every that exists! Similarly, suppose it turns out that there are wonderfully conclusive arguments for an ontology of space-time regions that makes it false that there are books. That would be more than a bit of surprise! But, playing along with that fantasy, then it would have just turned out that we are as mistaken in thinking -- when we quantify over everything -- that books are included as our ancestors were in thinking that gods were included. That still doesn't undermine the possibility of quantifying over everything.
So, in short, I can't see that Hellman's arguments in Section 4 of his paper have any force as they stand. He just seems to be running together issues about quantifying over 'what exists absolutely, "in Reality"' (his phrase) with the question about whether we can quantify over absolutely everything that exists.
Tuesday, July 31, 2007
Hellman's second line of argument against absolutely general quantification rests -- according to the title of Section 4 of his paper -- on the multiplicity of 'factually equivalent ontologies'.
Monday, July 30, 2007
The temperature is well into the 90s in Tuscany; rather too hot to do very much. So maybe there will be time for a little cooling philosophy. We'll see.
Meanwhile, to the next village and our favourite restaurant, La Bottega del 30, for a late evening meal. Helene the cook in great form. We were greeted by a little plate of a perfect small bruschetta, a mouthful of pecorino fresco with truffle-infused honey, and a tiny carpaccio of salt beef. Then another starter -- a corgette flower stuffed with porcini (amazing) with figs stuffed with goats cheese and baked wrapped in a sort of pancetta (ditto). Next, quite wonderful ravioli stuffed with pigeon. Then very slow cooked duck (an agonizing choice to make). Then deserts to die for. A 2001 Brunello, surprisingly light, perfect for a summer evening; followed by a terrific 2000 Barolo. All, quite, quite absurdly, for the same price as a pretty second-rate chain restaurant meal in Cambridge. So if you are ever near Castelnuovo Berardenga ...
Tuesday, July 24, 2007
In the next section of his paper, Hellman expounds a version of the Dummett indefinite extensibility argument. You know the sort of thing! 'Take some ordinals; then, whatever we start with, there's an operation which gives us a new ordinal (take the successor of the greatest, or if there is no greatest take the limit ordinal) ... Hence there can be no determinate domain containing, once and for all, all ordinals'.
What Dummett actually says (in a passage Hellman quotes) is "Given any precise specification of a totality of ordinal numbers, we can always form a conception of an ordinal number which is the upper bound of that totality, and hence of a more extensive totality." But, as the version above shows, it seems we don't need to lean heavily on the notion of a 'totality' to get the argument going -- a point that Hellman also makes. Oddly, however, Hellman seems to think that the argument presupposes a plenitudinous platonism that involves the thought that "the very possibility of mathematical objects suffices for their actuality". But that doesn't seem right. You could surely be a selective platonist (a sort of Quinean platonist?), who thought that there were kosher mathematical entities that really exist and which are to be contrasted with the mere fictions of mathematical game-playing, and who thought that -- among the kosher entities -- are ordinals, indispensable for theorizing about the order structures we find in the world. But you could still be struck by the thought that, once we countenance do ordinals and the standard ways of getting from old ordinals to new ordinals, then there is no non-arbitrary way of calling a halt which is true to the very concept of an ordinal. And being struck by that thought doesn't require being in thrall to a plenitudinous platonism.
Could it be, though, that we could accept the argument that there is no determinate domain of ordinals (for example) but still countenance quantification over absolutely everything? The thought would be that we can talk determinately about everything that there is, even if we cannot determinately corral off just that portion of what there is that ought to count as an ordinal (any attempt will leave outside it entities with just as much right to be called ordinals). Hellman thinks that this is unpromising for the reason that the "mathematical operations appealed to in connection with pure mathematicalia [as in forming new ordinals] can also be applied to mathematicalia-cum-non-mathematicalia". But this goes too fast. Suppose we have some ordinals plus some other things (making up everything there is!); then we can apply the familiar operation to the given ordinals to give us another thing. But this operation need not extend the tally of everything there is: it could just be that one of those "other things" that made up everything there is has turned out to have as much right to be deemed an ordinal as the ordinals we started with.
Saturday, July 21, 2007
In the last few days, I've got two newly published introductory logic books, both relatively short and aimed at similar audiences.
One is Mathematical Logic by Ian Chiswell and Wilfrid Hodges (OUP). This is notionally targetted at third year maths undergraduates --- which these days, in most UK universities, sadly isn't saying very much. It would also e.g. be a terrific book to put in the hands of philosophy students who have done a first logic course using trees, and who now need to know about natural deduction, understand the formal semantics of quantificational logic, and get as far as the completeness and the LS theorems. As Chiswell and Hodges go along, they also say something about diophantine sets, and mention Matiyasevich's Theorem, which enables them to get out an incompleteness theorem for almost no extra work.
The other book is The Mathematics of Logic by Richard Kaye (CUP) which is aimed perhaps at somewhat more sophisticated students with a wider mathematical background, but it is very good at signalling what are big ideas and what are boring technicalities. It starts off with a few chapters, e.g. on König's Lemma, showing how the sort of ideas that will later turn up in e.g. completeness proofs are mathematically interesting in their own right. Incidentally, Kaye uses, as his way of laying out formal proofs, a Fitch-type system -- which I think is the right choice if you really do want to stick as closely as possible to the 'natural deductions' of the mathematician in the street, though I'm not sure I'd have chosen quite his rules. And the "bonus" in Kaye's book is not an incompleteness theorem but a chapter on non-standard analysis.
The two books (pretty unsurpringly given the authors) seem at least on a rapid glance through to be splendid! Anyone teaching logic will want to "borrow" ideas from both, and any good student at the right level ought to read both.
A comment on our times. Neither book, I imagine, could be entered for RAE purposes [for non-UK readers, the Research Assessment Exercise by which UK departments are ranked, and which determines the level of government funding that the university gets to support that department], since neither book would count as "research". Yet the future of logic as a subject depends much more on having lively and accessible books such as these enthusing the next generation of students than it does on the publication of another research article or two that gets read by nine people ...
Thursday, July 19, 2007
I'll return to the second paper in the Absolute Generality collection, Michael Glanzberg's 'Context and unrestricted quantification', in due course: but as it happens I've just read Geoffrey Hellman's 'Against "Absolutely Everything"', so I'll comment on that while it is tolerably fresh in my mind. There are four main sections in the paper -- an attempt to state a version of anti-absolutist skepticism, an argument for anti-absolutism based on indefinite extensibility, an argument based on the possibility of 'factually equivalent' ontologies, and then a section explaining e.g. how the anti-absolutist makes sense of apparently absolutely general quantifications as in 'there are no talking donkeys'. I'll take these sections in turn.
Hellman's attempt to state a version of anti-absolutist skepticism is actually a bit of a fumble. He starts off by saying that the skeptic (if that's the right word) can state a position 'without self-destruction' by mentioning the purported quantifier 'absolutely everything' and saying, negatively, that in the end he can't give a stable coherent content to it. So far so good. However, Hellman then asks whether there is a defensible positive thesis that the skeptic can articulate. He starts talking about 'the intensional aspects of ontological commitments' in a way which I found a bit baffling (it's hardly a Quinean notion of ontological commitment that's in play). But then in the end, Hellman says that 'essentially the same idea' can be given presented in the negative mode, with the skeptic standing ready to offer e.g. an indefinite extensibility argument whenever the absolutist attempts to use a supposedly absolutely general quantifier, thereby backing up his (the skeptic's) claim not to be able to give coherent content to it. The excursus looking for a 'positive' thesis seems to achieve nothing. So let's pass on.
Wednesday, July 18, 2007
The translation of Eckart Menzler-Trott's Gentzen book as Logic's Lost Genius: The Life of Gerhard Gentzen is announced on the AMS website. I look forward to reading that a lot.
[Added: Shawn comments that it would be good if Gentzen's collected works could be reissued. I wrote to Dover some months ago about this and they said they would "give it our serious consideration". I've just mailed again to see how things have developed, if at all, and will keep you posted.]
Posted by Peter Smith at 9:20 PM
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!
Posted by Peter Smith at 7:07 PM
Sunday, July 15, 2007
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.
Saturday, July 14, 2007
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.
Thursday, July 12, 2007
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 ...
Posted by Peter Smith at 6:43 PM
Wednesday, July 11, 2007
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.
Posted by Peter Smith at 6:38 AM
Monday, July 09, 2007
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!
Posted by Peter Smith at 8:58 PM
Thursday, July 05, 2007
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]
Monday, July 02, 2007
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.