Structuralism in Bristol

The 2nd and 3rd of December there's a conference on structuralism for physics and mathematics in Bristol.

List of speakers and respondents:

Fraser MacBride (Hannes Leitgeb)
Øystein Linnebo (Katherine Hawley)
Stewart Shapiro (Richard Pettigrew)
Steven French (Jeremy Butterfield)
Oliver Pooley (David Wallace)
Simon Saunders (James Ladyman)

Not sure if I'll be down there myself, but I'd like to see my fellow Norwegian Øysten Linnebo talk on this subject (a bit of healthy patriotism). If I'm not mistaken, he wrote a review of Shapiro's book on structuralism some years back. And, as it turns out, I'm not mistaken: here's the review.

Thrid Arché Graduate Conference, etc.

The programme for the upcoming 3rd Arché Graduate Conference is now online. As announced earlier, the keynote speakers are Graham Priest, Diana Raffman and Jason Stanley. The conference is 17th to 19th of November, so for everyone who's nearby at the time, it's well-worth a visit. Thanks go out to the organisers in Arché for putting this together.

I am, however, a bit disappointed that there's no talks on subjects more directly relating to my own work. If pressed on the matter, I think I would have to admit that I look most forward to Priest's talk on the logic of relativism. With the newly started relativism project in Arché, this will hopefully spawn some interesting response from the audience, and, if Priest delivers, loud protests.

Meanwhile, if there's anyone out there who regret not sending in a paper, why not try the upcoming UT Austin 2007 Graduate Conference. More info over at Aidan's the boundaries of language.


PS A digression. After a day of over-the-top system problems with the old version of Blogger, I finally decided to shift to the new Beta version. What can I say: Thank you, Google. The Dashboard is now several times quicker, both for posting and editing - and the new label function is magnificent. Andreas - ding!

Putnam on inferentialism: Preliminary thoughts

Re-reading an old classic, Hacking's 'What is logic?', I came across an interesting passage which relates to the comment thread on an earlier post (see here). Hacking briefly mentions some remarks made by Putnam on inferentialism for logical constants (roughly the idea that the meaning of logical constants can be specified by their rules), concluding with a worrying response of his own. I had to dig out Putnam's Meaning and the Moral Sciences (1978 [this is actually Putnam's Locke lectures held in Oxford 1976]) to find the original context. But here, first of all, is the passage from Hacking:

[Putnam] recalls Gödel's mapping of classical sentential logic into intuitionistic logic, in which, for example, classical conjunction is identified with intuitionist conjunction but classical disjunction is identified with intuitionist -(-p and -q). Then he writes, in italics: "Contrary to what a number of philosophers - including recently Hacking - have asserted, [the operational rules] do not fix the 'meanings' of the logical connectives. Someone could accept all of those rules (and all classical tautologies, as well) and still be using the logical connectives in the non-classical sense just described." Putnam makes this observation in the course of urging that if the logical connectives are given a "quasi-intuitionist" interpretation one can then go on to define 'true', as he puts it, "exactly à la Tarski. Only 'truth' becomes provability...". (Hacking (1979), p. 300)

So, Putnam wants to rule out inferentialism by way of a mapping from CL to IL (this is where the present discussion connects to the comment thread). Putnam's idea seems to be something along the following lines: Take the standard BHK-interpretation of IL constants. That is, an interpretation with provability as the central notion. By mapping the classical constants to the intuitionistic ones, Putnam wants to show that - although we know the interpretations (i.e., BHK for IL and Tarski-sentences for CL) to be different, and, thus, that the meaning of the constants differ - the rules for the constants are the same. As a consequence, Putnam suggests, inferentialism fails.

Before we return to the argument, let us recap the technical apparatus Putnam is employing. Hacking notes that Putnam is recalling Gödel's mapping from CL to IL, but as far as I can see, there is no mention of Gödel in the relevant passage of Putnam's book. However, it doesn't really matter which particular mapping Putnam is referring to, because there are several that will do the trick. Here are the three most well-known embeddings:

K() [Kolmogorov, 1925, aka the negative translation]

1. K(p) := --p (where p is atomic)
2. K(-A) := -K(A)
3. K(A and B) := --(K(A) and K(B))
4. K(A or B) := --(K(A) or K(B))
5. K(A -> B) := --(K(A) -> K(B))


G2() [Gödel, 1933]

1. G2(p) := --p (where p is atomic)
2. G2(-A) := -G2(A)
3. G2(A and B) := G2(A) and G2(B)
4. G2(A or B) := -(-G2(A) and -G2(B))
5. G2(A -> B) := -(G2(A) and -G2(B))


G1() [Gentzen and Bernays, 1933]:

1. G1(p) := --p (where p is atomic)
2. G1(-A) := -G1(A)
3. G1(A and B) := G1(A) and G1(B)
4. G1(A or B) := -(-G1(A) and -G1(B))
5. G1(A -> B) := G1(A) -> G1(B)


Now, initially, all these embeddings do, is to provide a translatory function from CL into IL s.t. formulae are mapped to formulae. However, the careful induction over complexity of formulae above ensures that the translations have certain interesting properties. First, the translations are stable, that is,

(1a) CL |- A <-> K/G1/G2(A)

In other words, A is classically equivalent to any of its three translations. Moreover,

(1b) Any rule

Gamma : Phi Delta : Psi
_____________________
Sigma : Theta

of CL is stable under the translations, i.e.,

Gamma* : Phi* Delta* : Psi*
_____________________
Sigma* : Theta*

where (i) * is K/G1/G2 and (ii) a set Beta* of formulae is defined (A* | A is a member of Beta). Secondly, the translations preserve provability, i.e.,

(2) If a sequent Gamma : Delta is derivable in CL, then Gamma* : Delta* is derivable in IL. (See Mints, 2000. Note that Mints uses a sequent calculus style of natural deduction.)

The converse follows by (1), so if Gamma* : Delta* is derivable in IL, then Gamma : Delta is derivable in CL.

With the results (1) and (2), Putnam seems right to say that both the intuitionist and the classicist could accept the same canonical rules - typically, the introduction-rules of natural deduction - thus being "inferentialistically" equivalent. Putnam takes this as evidence that what we really want is some underlying story like Tarski-semantics or BHK-interpretation to give the "real" meaning to the involved logical constants.

To examplify Putnam's point, take an introduction rule for disjunction: A, therefore A or B. Trivially, this holds in CL. But let us now look at the translation into IL. Notice that disjunction is the only constant which has a complex translation in G1, i.e., -(-A and -B). The question, then, is if this can be deduced in IL as well. It turns out, of course, that it can. Since the method can be expanded to all introduction-rules, Putnam takes it to mean that the intuitionist and the classicist agree on the rules, yet disagree about the meaning of constants. But I suspect that something is being smuggled into the argument: A, therefore -(-A and -B) does not, like its original disjunctive counterpart, satisfy a constraint proposed by Heinrich Wansing: separation.

The idea of meaning as correct use has certain consequences for the format of introduction rules in sequent calculi. First of all, if one wants to avoid a (partially) holistic account of the meaning of the logical operations, the meaning assignment should not make the meaning of an operation f dependent on the meaning of other connectives. The sequent rules for f should give a purely structural account of f's meaning in the sense that they should not exhibit any connective other than f. (Wansing, 2000)

In fact, I seem to remember that Dummett (1991) at some point also proposed this constraint at some point (under the label 'purity'), but I haven't looked it up yet. At any rate, the point is that the new "canonical rules", produced by the translation does not inherit the simplicity of the original rules. This, I submit, is simply saying that they lost their canonical form during translation. I see this point as relating to something I discussed in my dissertation, namely how complexity influences proof-theoretical semantics (in this case, inferentialism). Notice that after the translation, all strings of the form 'A or B' will be transformed to '-(-A and -B). Assuming that we're working with the formulae in a proof-system fashioned after the sub-formula property, such as sequent calculus, the string will now require two rule-applications rather than one.

So what about Hackings own solution? Well, to put it mildly, I find this wildly unsatisfactory. Surprisingly, Hacking only wants to press his inferentialism on the backdrop of already accepted notions of (Tarskian) truth and logical consequence. Needless to say, this makes the further project of expanding inferentialism into proof-theoretic semantics, and giving a definition of validity within this framework, completely redundant. But then again, this leaves us with all the old problems of Tarskian semantics.

Updated.

Good news, bad news, and imcomprehensible news

Over the last two weeks, Leiter Reports has published some previews of the upcoming 2006 PGR Review. Four lists are up so far: one on Metaphysics and epistemology, one on Philosophy of the sciences and mathematics, one on History of Philosophy, and, finally, one on the overall score of UK universities. Note that the three former lists all contain universities across the English speaking contries (US, UK, Canada, Australia).

Now, in the overall list for UK universities, the top 3 is the same as for the 2004 PGR:

1. Oxford University (4.7; 1)
2. Cambridge University (3.7; 2)
3. University of St. Andrews (3.6, 3)
4. Birkbeck College, University of London (3.3, 6)
4. University College London (3.3, 4)
6. King's College, London (3.1, 5)
6. University of Sheffield (3.1, 7)
8. London School of Economics (2.8, 7)
9. University of Bristol (2.6, 14)
9. University of Edinburgh (2.6, 11)
9. University of Leeds (2.6, 11)
9. University of Nottingham (2.6, 13)
9. University of Reading (2.6, 9)
14. University of Warwick (2.3, 10)
15. University of York (2.2, not in top 15)

Although Oxford is still miles ahead, St. Andrews is now breathing Cambridge down the neck. But, alas, it remains uncertain if these scores have incorporated recent changes, for instance that both Daniel Nolan and Carrie Jenkins left for Nottingham. This was reported on Leiter's blog some time ago, but I don't know if it affected the review.

What about the other lists? Well, St. Andrews has a significant shared 6th place on the E&M list:

1. Oxford University (4.3)
2. New York University (3.9)
2. Rutgers University, New Brunswick (3.9)
4. University of Notre Dame (3.8)
5. University of Texas, Austin (3.5)
6. Massachussetts Institute of Technology (3.4)
6. Princeton University (3.4)
6. University of St. Andrews/University of Stirling Joint Program (3.4)
9. University of California, Los Angeles (3.3)
10. Stanford University (3.2)
10. University of Pittsburgh (3.2)
10. University of Southern California (3.2)

Notice that on the lists for certain fields, St. Andrews is listed together with Stirling because of the joint program. I don't know how the contributions split up, but Leiter points out that if the joint program is evaluated on the UK overall list, then the score increases from 3.6 to 3.8, just enough to secure the 2nd place.

On History of Philosophy, St. Andrews/Stirling makes a shared 18th. The Philosophy of sciences and math, however, is a bit frustrating. Here, St. Andrews doesn't even make the Top 20. Why is this? Of course, in philosophy of math, St. Andrews does reasonably well, but in other areas subsumed under the heading, such as Philosophy of Physics, Biology, Social Science, and Cognitive Science, the score can hardly be impressive.

This brings us to the question: what does the lists really tell us? Unless broken up into sub-disciplines, I take it, not much. The way the lists are put together favour the large departments (as noted by Leiter in the description), where the manpower is sufficient to cover all areas. The numbers we are infact given, however, does not reveal whether the ranking is due to an overall good performance or excellent scores in some disciplines. As far as I remember, the 2004 review was divided into smaller categories of specialization, and I hope the new review will provide this as well. If not, I suspect the purpose of the lists (i.e., to help graduate students finding suitable departments) will be obscured.

Read Weatherson's rantings about the same subject here, here and here.

Chess Championship in Elista: The Conclusion

At the moment of writing, Friday the 13th, the first rapid game of the tie-break is being played on what is definitely the last day of the Championship (but perhaps only the beginning of the post-match legal aftermath). Standing analysis on the comment thread on Chessbase indicate that it's a soon-to-be draw. While waiting, let's look back at the events of the last couple of days.

The second part of Elista has proved to be quite hard-fought - surprising perhaps, since most focus seemed to be on the off-board spectacle. However, Topalov must have decided that to compensate for all the bad press he has received because of his manager, what was needed was some real Topalov-style offensive chess. And after equalising, Topalov just kept on going, winning Game 9 to take the lead for the first time in the match. The win came after a blunder by Kramnik, but fortunately for him Topalov repaid the favour the following day: Game 10 saw Kramnik win to equalise, or, in Kramnik-speak, to take back the lead. Game 11 and 12 were both drawn, but not at all bloodless.

Consequently, we ended up where many super-GMs had said we would end up with a 12-games-only-world-championship-match: in the tie-breaks. I can't recite the tie-break rules on my feet, but I'll return with more if that becomes relevant. For now we'll wait for the results of the two rapid games. And, meanwhile, allow yourself to be entertained by a rather nasty piece of history which resurfaced on the chessbase commentary thread here the other day. This is an excerpt from a little showdown between Short and Kamsky, both top 20 GMs. The two of them played a PCA candidate match back in 1994, where allegedly, Short was accused of cheating by the Kamsky team (more or less consisting solely of Kamsky's infamous father). Apparently, Short thought the current scandal in Elista made ampel opportunity to raise the issue with Kamsky live and online.

Update: The first rapid just ended in a draw. More to come.

Update(2): Kramnik has won the second rapid game. In other words, he has a solid position before the last two rapid games. The pressure is definitely on Topalov now, but he's a strong finisher. I notice that many people online are hoping for Kramnik simply because that will give the least complicated legal situation.

Update(3): Topalov goes on to win the third game! The score is now 1.5-1.5 with one game left, rapid with K as white. The game has started, and in the present position, by my lights Kramnik is clearly better. People have no started asking an important question: if Kramnik wins the match, will he play in the Mexico 2007 championship (round robin)?

Update(4): Kramnik is World Champion, holder of the unified title. He puts the K back in king. The final score is 2.5-1.5. I'll come back with more when the dust settles. For now: Congratulations to Kramnik! He won the match, and he won the tie-breaks - that ought to do it.

Quine: Normativity and logic

Starting in two weeks, Arché's two Professorial Fellows, Graham Priest and Stewart Shapiro, will be giving a research seminar on Quine, relating different aspects of his philosophy to the Arché projects: logic and normativity, philosophy of mathematics, vagueness, and modality. There will be two sessions a week, all going through key texts from Quine's bibliography plus important responses to Quine's work.

Must say that I particularly look forward to the logic and normativity part, since this was one of the topics I wrote on during my M.Litt. Famously, Quine argued in his Philosophy of Logic that the classicist and the deviant logician talked past eachother when arguing over the meaning of connectives. On Quine's account, there is no real disagreement, only talk of different logical constants. Incidentally, Stephen Read just stressed earlier today that accepting such an argument has an important impact on the monism/pluralism debate in the philosophy of logic. The question "which logic is the right logic?" (adopted from Tharp's classical 1975 paper, arguing against Quine), is left hanging if the classicist and the deviant talk past eachother when trying to establish the superiority of their logic.

One paper which applies this Quinean insight is McGee and McLaughlin's 'Distinctions Without a Difference' (1994):

An observation of Quine is worth repeating: we cannot hope to determine the correct rules of inference by a semantic investigation, trying to determine which rules are truth-preserving, because the same question 'What are the legitimate rules of inference?' is going to recur as we try to develop the metatheory. The meaning of the logical connectives cannot be given by the metatheory, because the very same connectives are employed in the metalanguage. Instead, what determines what the connectives mean are the inferences in which we employ them; the rules of inference implicitly define the connectives. People who employ different rules of inference mean different things by connectives; so they are inevitably talking at cross purpose. (p. 206)

McGee and McLaughlin conjures this argument up from the hat in the context of vagueness, but I will not go into details about that now (in a word, McGee and McLaughlin wants to justify keeping CL for the sorites argument, and instead denying the sorites premise). Rather, I want to briefly consider the argument in its own right. Because, although McGee and McLaughlin calls it an observations, it certainly looks more like an argument to me. Actually, the manner in which the "observation" is presented by the authors belies the fact that not all of this is entirely uncontroversial. I agree with the inferentialist position, but it does not follow from inferentialism for logical constants that there can be no question of which rules are appropriate (in a given discourse), or which rules are "right" in some normative sense. In fact, Dummett holds precisely this position: inferentialism in conjunction with revisability of use, i.e., revisability of logical laws (see for instance Dummett 1991, ch. 8-9, in particular pp. 209-215).

There will be more on these issues later. For now, read an interesting post by Colin Caret.

Arché Logic Group

As I've mentioned earlier, this semester Arché has started a new logic group (boringly called 'Arché Logic Group', but unofficially named 'The Deviants'). There's now an Arché Twiki page for the seminar, although still quite provisional. The first part of the semester we're working through Graham Priest's "classic" An Introduction to Non-Classical Logic , preparing to go through a forthcoming publication by the same author. The usual (Scandinavian) suspects have been spending late office hours going through all of the exercises. True, there is quite a bit of tedious and repetitious work, but we have encountered a great deal of interesting material in some of the harder parts of the book.

However, this is only the warm-up: In early November we fly in the big guns. Then Priest himself will join in the fun when we proceed to uncharted terrain. By then there will hopefully be some comments on the first book on the Twiki page. I'll come back with more later.

Meanwhile: Today's session reminded me of a theorem that a friend of mine persistently has brought to my attention in another context. This is Glivenko's theorem (Glivenko 1929), which Priest has relegated to a footnote on p. 103.

Theorem. (a) If Gamma |- A in classical logic then ¬¬Gamma |- ¬¬A in intuitionistic logic, assuming that CL negations and implications are replaced by their IL counterparts. (b) If ¬Gamma,Sigma |- ¬A in CL then ¬Gamma,¬¬Sigma |- ¬A in IL.

Priest suggests that this theorem says that although IL is a sublogic of CL, CL is "in a sense" contained within IL. Of course, it is this "in a sense" which makes the theorem philosophically loaded - how are we to understand the resurfacing of classical truths within the intuitionistic system? Has it any bearing on the philosophical projects associated with the logics? Etc. I will not get into that here. Rather, I just want to point to a corollary of Glivenko's result that I was unaware of. Priest directs our attention to what he calls the "unobvious" fact that if |-A in classical logic, and A contains no logical notation except negation and conjunction, then |-A in intuitionistic logic as well (with intuitionistic conj. and neg.). Somewhat surprising perhaps, since intuitionistic and classical negation usually are thought to behave quite differently.

I checked out Priest's reference, which is to the standard Introduction to Metamathematics (1952) by Kleene (see pp. 492-493). The proof is something like this: Consider A as a conj. of n formulae (n is 1 or greater), where each conjunct is itself not a conjunction. Then, all conjuncts are either propositional variables or negated. Furthermore, if A is provable in CL, then all of its conjuncts are provable; but no propositional variables are provable. So, the conjuncts are all negated, and by Glivenko's theorem (b), they are provable in IL.

Any comments on this - philosophical or logical? (Paul Simon, if this isn't an invitation, nothing is.)

Hopefully, the ALG will prosper in the next few years, warranting the 'logic' part of the Arché research description. I'm already conspiring to make it into a proof-theory seminar next semester - one interesting suggestion was to read the manuscript for Greg Restall's forthcoming Proof-theory and Philosophy.

Update: Thanks to Aidan for this link.

Topalov-Kramnik 4:4

After some days of off-the-board intrigues, the players have finally returned to their profession - actually playing chess. Surprisingly, Kramnik decided to continue the match at 4:3 (i.e., with Topalov getting the point from the forfeited Game 5), even though Kramnik's complaint was accepted, and the Appeals Committee subsequently replaced. So, Kramnik is without doubt the moral winner of the psychological battle that's been raging the last week; if he's also the psychological winner, however, is not at all clear.

Game 6, the return to the board, ended in a draw after Kramnik had significant pressure with black for most of the game. And, after the rest day, Game 7 ended with point-sharing as well. Note that because of the Game 5 situation and the mid-match change of colours, this was Kramnik's third consecutive black - and with three draws, he has managed quite well.

Starting his first white since the trouble began, then, Kramnik was expected to put on the pressure, but, oddly, Game 8 appeared to be all Topalov's. Kramnik played too passively with white, allowing Topalov to reach the type of unbalanced position that he loves. Some minor, and uncharacteristic, inaccuracies in Kramnik's end-game won Topalov the full point, thus equalising to 4:4. With four games to go, two with white each, the match is now wide open. The question posed by most commentators, however, is whether the new score situation will lead Kramnik to reevaluate his response to the Appeal Committee. At any rate, he has already stated that he does not rule out taking legal action whatever the outcome of the match.

Here is the Chessbase analysis. Here's the latest from chess commentator Mig.

Welcome to 17.5

Welcome to cell number 17.5 in the Arché Centre, this is where Andreas and I will be spending the next three years in isolation. My fellow monk, highest in the ranks of reflection, has taken some nice pictures of our new work place. Enjoy.

The sequel is here

The new Masses of Formal Philosophy, eds. Vincent F. Hendricks and John Symons, is right around the corner from Automatic Press/VIP. As I mentioned earlier, Restall has promised us his answers to the editors' questions in this volume, and as you can see, quite a few others have joined him.

Even more interesting, at least in my opinion, is the sibling volume Philosophy of Mathematics: 5 Questions. For St. Andrews people, there are some familiar faces: Crispin Wright, Graham Priest, and Stewart Shapiro are all contributing. So is Bob Hale, Jaakko Hintikka, Paul Beacerraf, Charles Chihara, Solomon Feferman, Harvey Friedman, William Tait, and a range of other seminal figures in philosophy of mathematics. Again the volume is edited by Vincent F. Hendricks, but this time together with Hannes Leitgeb, who I wrote about in connection with his excellent talk in Edinburgh.

A very inspiring talk by van Benthem at this year's ESSLLI comes to mind: It reminded me of how important it is for prominent researchers to provide the younger generations with the broad overview of their fields - a perspective which is, I take it, usually lost for PhD-students and young researchers. Reading the interviews from the first book, Formal Philosophy, I think it provided precisely the type of overview that is often omitted in the specialized literature.

Categories: Philosophy, Logic, Mathematics