“Dennis Des Chene”:http://tlonuqbar.typepad.com/phfn/2004/11/modal_glitch.html criticises a comment Timothy Williamson makes in “Must Do Better”:http://www.philosophy.ox.ac.uk/faculty/members/docs/Must%20Do%20Better.pdf.

bq. The principle that every truth is possibly necessary can now be shown to entail that every truth is necessary by a chain of elementary inferences in a perspicuous notation unavailable to Hegel.

Des Chene says this isn’t right, because it’s not a theorem in most modal logics that MLp entails Lp. (Des Chene uses boxes and diamonds, not Hughes + Cresswell notation, but I can’t do that in HTML. M is diamond and L is box.) He also notes that it is a theorem in S5, and wonders whether Williamson is using that logic. In fact Williamson’s argument, which is just a variant on “Fitch’s paradox”:http://plato.stanford.edu/entries/fitch-paradox/ needs only KT, as I’ll show.

Rather than appeal directly to K, I’ll use three consequences of K. (In what follows -> is material implication and > is entailment.)

K1. If A > B then LA > LB
K2. If A > B then MA > MB
K3. It is a theorem that ~M(A & ~A)

Here’s the proof. (Rule R is the rule that whatever is true is possibly necessary.)

Since p is arbitrary it follows that whatever is true is necessarily true, as required.

The proof is obviously not original, but I think it’s so clever that it’s worth posting here, just for fun.

I don’t know whether you can significantly weaken the assumptions below KT and still get the proof to work. There’s an “expert on Fitch’s paradox”:http://www.arts.cornell.edu/phil/faculty/fara.html in the adjacent office who’d probably know, but he’s gone home for the evening. Anyone with suggestions can leave them in the comments.

Having said all that, Professor Des Chane “goes on to say”:http://tlonuqbar.typepad.com/phfn/2004/11/modal_glitch.html#add01 some interesting things about the conception of philosophy Williamson is advocating. He also, I’m pleased to note, includes a shout-out to “Matt Jones’s”:http://www.columbia.edu/cu/history/htm/h_faculty_profile_jones.htm very interesting work on the connection between mathematics and philosophy in the early modern period. There’s some interesting stuff there that you should pop over and read.

I was looking at “Peter King’s website”:http://users.ox.ac.uk/~worc0337/mystuff.html, especially his book “One Hundred Philosophers”:http://shop.abc.net.au/browse/product.asp?productid=160490 and I thought this passage on “David Lewis”:http://users.ox.ac.uk/~worc0337/authors/david.lewis.html was delightful.

bq. Lewis’ philosophical interests were broad, as evidenced by the contents of the five volumes of his collected papers published so far: ethics, politics, metaphysics, epistemology, philosophical logic, language – he wrote on a vast range of subjects, from holes to worlds, from Anselm to Mill, from the mind to time travel. In everything he wrote he was rigorous, committed, and clear, but perhaps the most distinctive thing about him was his attitude to other philosophers, and especially to criticism: _one can scarcely find a book or paper attacking Lewis’ views that doesn’t contain an acknowledgement to him for his help_. What mattered to him – what he loved – were the ideas, the arguments, the philosophy, not winning or being right. He was the ideal, the model philosopher; he’s also (and this is a very different matter) widely regarded as being the best philosopher of his generation – perhaps of the twentieth century. (Emphasis added.)

The model philosopher indeed.

“Matt Carter”:http://braininavat.net/ has made the pictures I was hoping for with the last “Lemmings post”:http://tar.weatherson.org/archives/004008.html. WooHoo! They are mostly below the fold, but who could resist this lemming pic?

p=. !-http://brian.weatherson.org/stanley_lemming.jpg 200×200!
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It might be behind paywalls for many of you, but the latest “Philosophical Perspectives”:http://www.blackwell-synergy.com/servlet/useragent?func=synergy&synergyAction=showTOC&journalCode=phpe&volume=18&issue=1&year=2004&part=null is out. The number of papers by friends (and writers) of TAR is impressively large. I worry a bit that the paper Andy and I wrote for it looks a little slight in such august company. We’re the “Page 2”:http://sports.espn.go.com/espn/page2/index to their “ESPN.com”:http://espn.go.com, perhaps. (Of course in the off season I only read page 2, so take the analogy how you like.) But it’s impressive company to be in. And I think the paper we wrote is basically correct, even if it isn’t quite as deep as some of the other contributors.

There’s been a lot of hubbub, both here and elsewhere in the blogworld, about the Becker-Posner blog. But if it’s intellectual firepower in a group blog you’re after, you should be reading “Left2Right”:http://left2right.typepad.com/. Here’s its “mission statement”:http://left2right.typepad.com/main/2004/11/why_left2right.html, which should be good for setting off a round of debates.

bq. In the aftermath of the 2004 Presidential election, many of us have come to believe that the Left must learn how to speak more effectively to ears attuned to the Right. How can we better express our values? Can we learn from conservative critiques of those values? Are there conservative values that we should be more forthright about sharing? “Left2Right” will be a discussion of these and related questions.

bq. Although we have chosen the subtitle “How can the Left get through to the Right?”, our view is that the way to get through to people is to listen to them and be willing to learn from them. Many of us identify ourselves with the Left, but others are moderates or independents. What we share is an interest in exploring how American political discourse can get beyond the usual talking points.

The contributors so far include “Elizabeth Anderson”:http://left2right.typepad.com/main/2004/11/what_hume_can_t.html, “Kwame Appiah”:http://left2right.typepad.com/main/2004/11/less_contempt.html, “Josh Cohen”:http://left2right.typepad.com/main/2004/11/the_moral_value.html, “Stephen Darwall”:http://left2right.typepad.com/main/2004/11/school_resegreg.html, “Gerald Dworkin”:http://left2right.typepad.com/main/2004/11/less_contempt_m.html, “David Estlund”:http://left2right.typepad.com/main/2004/12/the_first_data_.html, “Don Herzog”:http://left2right.typepad.com/main/2004/12/public_private_.html, “Jeff McMahan”:http://left2right.typepad.com/main/2004/12/supporting_our_.html, “Seana Shiffrin”:http://left2right.typepad.com/main/2004/11/being_forthrigh.html, and “David Velleman”:http://left2right.typepad.com/main/2004/12/debunking_a_dea.html. Wowsa. And many other names you may have heard of, from Peter Railton to Richard Rorty, are listed as being part of the team. This should be worth following.

Kent Bach just reminded me of one of the most interesting philosophy conferences that has happened for a while – the “Philosophy and Wine”:http://www.sas.ac.uk/Philosophy/Wine.htm conference in London next weekend. If you’re in the area, it looks like a lot of fun.

On a more traditional (and local) note, the “Syracuse Grad Conference”:http://web.syr.edu/~degould/philgradconf2005.html has just put out a call for papers.

The latest paper to go up on “Philosophers’ Imprint”:http://www.philosophersimprint.org/ is my “Morality, Fiction and Possibility”:http://www.philosophersimprint.org/004003/.

It’s very exciting to be part of Philosophers’ Imprint, which is the most exciting new journal to appear in many a year. For those left panicking after the comments about journals in “Brian Leiter’s thread”:http://leiterreports.typepad.com/blog/2004/11/philosophy_jour_1.html#comments I can highly recommend Philosophers’ Imprint, whose editors were not only a pleasure to work with, and very helpful with the essay, but were incredibly patient with the time I took to get them the final corrected version.

Dean Zimmerman sent along the following announcement, which should be of interest to many readers. I’d submit a paper if I had, er, a good metaphysical idea to promote.

*Essay Competition: The _Oxford Studies in Metaphysics_ Younger Scholar Prize*

_Oxford Studies in Metaphysics_ is pleased to announce the continuation of its annual essay competition, the Younger Scholar Prize. The contest is administered by the editorial board of _Oxford Studies in Metaphysics_, and supported by the generosity of the A.M.Monius Institute. The competition is open to both current graduate students and scholars who have received their doctoral degree within the past ten years. Essays may be on any topic in metaphysics. Although the essay should not be primarily historical in nature, it may pay careful attention to the metaphysics of important philosophers from any era. The winner receives $2,500, and the winning essay is published in _Oxford Studies in Metaphysics_.

Submissions must be postmarked by January 15, 2005, to be eligible for this year’s prize. (This is a change from the originally announced deadline of November 30, 2004.)
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I’m currently in Atlanta en route to California for the Wettstein conference. (I was meant to fly via Minneapolis but I got rerouted. I imagine there’ll be plenty of chances to get to Minneapolis in the future though.) Posting will be light here and non-existent on OPP until probably the weekend.

In a nice paper in a recent _Philosophical Review_ Alan Hajek argued that Pascal’s argument in the Wager fails because he doesn’t take account of mixed strategies. I’ve been spending too much of today wondering whether the same thing is true in other fields. (Not that I’m entirely convinced by Hajek’s argument, but the response would take another post, and historical research, and that’s for another week.)

For a while I thought mixed strategies could solve some of the problems “Andy Egan discusses”:http://www.geocities.com/eganamit/NoCDT.pdf in his paper on causal decision theory. Maybe they can, but I’m not so sure. For now I just want to discuss what they do to Nick Bostrom’s “Meta-Newcomb Problem”:http://www.nickbostrom.com/papers/newcomb.html.

The first thing to say is that it’s hard to say what they’d do, because Bostrom doesn’t say what his predictors do if they predict you’ll use a mixed strategy. I’ll follow Nozick and say that if they predict a mixed strategy, that’s the same as predicting a 2-box choice. Importantly I make this assumption both for Bostrom’s predictor and his Meta-Predictor. But if the “Predictor” is not predicting, but is in fact reacting to your choice (as is a possibility in Bostrom’s game) then I’ll assume that what matters is what choice you make, not how you make it. So choosing 1 box by a mixed strategy will be the same as choosing 1 box by a pure strategy for purposes of what causal consequences it has.

Given those assumptions, it sort of seems that the “best” thing to do in Bostrom’s case is to adopt a mixed strategy with probability e of choosing 2 boxes, for vanishingly small e. That will mean that if the meta-predictor is “right” your choice will cause the predictor to wait until you’ve made your decision, and with probability 1 less a vanishingly small amount, you’ll get the million. (Scare quotes because I’ve had to put an odd interpretation on the MetaPredictor’s prediction to make it make sense as a prediction. But this is just in keeping with the Nozickian assumptions with which I started.)

Problem solved, at least under one set of assumptions.

Now I had to set up the assumptions about how to deal with mixed strategies in just the right way for this to work. Presumably there are other ways that would be interesting. I’m not interested in games where predictors are assumed to know the outputs of randomising devices used in mixed strategies. That seems too much like backwards causation. But there could be many other assumptions that lead to interesting puzzles.

UPDATE: Be sure to read the many interesting comments below, especially Bob Stalnaker’s very helpful remarks.