The “Arché Scepticism Conference”:http://www.st-andrews.ac.uk/~arche/bk2009/ held over the weekend was a great success. Thanks to all the organisers, presenters, commentators and questioners for a great learning experience. For a real-time recap, see the conferences “Twitter feed”:http://twitter.com/#search?q=%23asc2009.

I’ll be posting my paper as soon as I’ve figured out how to respond to objections Martin Smith and Elia Zardini. That should be sometime in 2011-12. (I suspect a few other paper givers will be in the same position when it comes to dealing with objections Elia raised.)

The best slides from the conference were by “Roger White”:http://twitpic.com/7dinp. In fact it wasn’t very close. The best title was to Elia’s “reply to Anthony Bruekner”:http://twitpic.com/7dd51. Any other participants want to add some more prizes from the conference?

John Holbo and Belle Waring, who many of you will know from Crooked Timber and elsewhere, have a new book up in PDF format.

• Reason and Persuasion

The book contains translations of three Platonic dialogues, detailed commentaries on each of them, and illustrations! John has been using versions of it for teaching purposes for several years, and the book looks like it could be very useful for teaching (and research!) purposes.

• “Kant’s Aesthetics: Overview and Recent Literature”:http://www3.interscience.wiley.com/journal/122410001/abstract, Christian Helmut Wenzel
• “Environmental Ethics: An Overview”:http://www3.interscience.wiley.com/journal/122409997/abstract, Katie McShane
• “Margaret Cavendish on the Relation between God and World”:http://www3.interscience.wiley.com/journal/122409998/abstract, Karen Detlefsen
• “Whatever Became of the Socratic Elenchus? Philosophical Analysis in Plato”:http://www3.interscience.wiley.com/journal/122382395/abstract, Gareth Matthews
• “Demonstratives in Philosophy and Linguistics”:http://www3.interscience.wiley.com/journal/122295940/abstract, Lynsey Wolter
• “Recent Work on Propositions”:http://www3.interscience.wiley.com/journal/122295941/abstract, Peter Hanks
• “Understanding Kripke’s Puzzles about Belief”:http://www3.interscience.wiley.com/journal/122382396/abstract, Michael McGlone
• “Computationalism in the Philosophy of Mind”:http://www3.interscience.wiley.com/journal/122301402/abstract, Gualtiero Piccinini
• “The Problem of Natural Evil I: General Theistic Replies”:http://www3.interscience.wiley.com/journal/122409999/abstract, Luke Gelinas
• “The Problem of Natural Evil II: Hybrid Replies”:http://www3.interscience.wiley.com/journal/122410000/abstract, Luke Gelinas
• “Aristotelian Homonymy”:http://www3.interscience.wiley.com/journal/122301401/abstract, Julie Ward
• “Intelligent Design”:http://www3.interscience.wiley.com/journal/122295943/abstract, William Hasker
• “Incongruent Counterparts and the Reality of Space”:http://www3.interscience.wiley.com/journal/122295942/abstract, Graham Nerlich

Apologies to anyone who couldn’t get at the site for the previous hours. There was a server error. I’ve now fixed it, I think, and the site should be running. There are some details below the fold for anyone interested in that kind of thing.

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Registration is now open for the “Arché Scepticism Conference”:http://www.st-andrews.ac.uk/~arche/bk2009/. This looks like it should be a very good conference; hopefully I’ll post my paper here in a few days.

And the schedule for the “Formal Epistemology Workshop”:http://fitelson.org/few/schedule.html is up, along with links to a bunch of the papers to be presented. I vaguely hope to have something to say about Sarah Moss’s paper “Updating as Communication”:http://www-personal.umich.edu/~ssmoss/updating.pdf (PDF), which looks like an interesting take on _de se_ updating.

In his very interesting paper at the Rutgers Epistemology Conference, “Higher-Order Evidence”:http://philosophy.rutgers.edu/EVENTS/EPISTEMOLOGY/EPIS2009/PAPERS/HigherOrderEvidence.pdf (PDF), David Christensen discusses a lot of cases where, in the process of investigating whether _p_, we learn something about our ability to detect whether _p_. In the primary cases Christensen discusses, we first come to believe _p_, then come to believe that our capacities are impaired in some way. And one of the epistemologically interesting questions is what we should do at this stage. I wanted to consider a slightly different question.

S is investigating a murder. She gets evidence E, and on the basis of that quite reasonably concludes that it is quite likely the butler did it (her credence in that is 2/3), a serious possibility that the gardener did it (her credence in that is 1/4), and very little chance that neither did it (her credence in that is 1/12).

S then is told, by a usually reliable source, that she has taken some drug that leads to people systematically _underestimating_ how strongly their evidence supports various propositions. So if someone’s taken this drug, and believes _p_ to degree 2/3, then _p_ is usually something that’s more or less guaranteed to be true by their evidence.

What should S do?
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Some stuff you might not have seen from around the web.

• Elsevier publishes fake journals. Henry Farrell has some suggestions for what to do about it.
• Alva Noe has an article about drugs in sport, focussing on why it is that we worry about some enhancements but not others. I think Alva is a little too dismissive of the health concerns; I wouldn’t care about steroids at all if they weren’t dangerous. More realistically, I wouldn’t care if a player with cancer took steroids as part of a standard cancer treatment. But it is interesting to think about where we draw the moral lines, and about what that says about what we value in sporting performance. And the terminology here is bizarre – if we cared about all ‘performance enhancing drugs’, presumably we’d want to get rid of the Gatorade in the dugout.
• Jacob Ross has a fascinating paper on Sleeping Beauty and Countable Additivity (PDF).
• This list of cities with the highest quality of life is interesting, but I think they’re not rating things I care about if Perth comes well ahead of New York.

I imagine the following point is well known, but it might be news enough to some people to be worth posting here. The following principle is inconsistent.

bq. Trust Experts. If you know there is someone who is (a) perfectly rational and (b) strictly better informed than you, i.e. they know everything you know and know more as well, whose credence in p is x, then your credence in p should be x.

The reason this is inconsistent is that there can be multiple experts. Here’s one way to generate a problem for Trust Experts.

There are two coins, 1 and 2, to be flipped. The coin flipping procedure is known to be fair, so it is known that for each coin, the chance of it coming up heads is 1/2. And the coins are independent. Let H1 be that the first coin lands heads, and H2 be that the second coin lands heads.

The coins are now flipped, but you can’t see how they are flipped. There are sixteen people, called witnesses, in an adjacent room, plus an experimenter, who knows how the coin lands. Using a randomising device, the experimenter assigns each of the sixteen different propositions that are truth-functions of H1 and H2 to a different witness, and tells them what their assigned proposition is, and what its truth value is. The witnesses know that the experimenter is assigning propositions at random, and that the experimenter always tells the truth about the truth value of propositions. So the witnesses simply conditionalise on the truth of the information that they receive.

Consider first the witnesses who are told the truth values of (H1 v T2) and (H1 v ~T2). One of these will be told that their assigned proposition is true. Whoever that is, call them W1, will have credence 2/3 in H1.

Consider next the witnesses who are told the truth values of (~H1 v T2) and (~H1 v ~T2). Again, one of these will be told that their assigned proposition is true. Whoever that is, call them W2, will have credence 1/3 in H1.

So if you were following Trust Experts, you’d have to have credence 2/3 in H1, because of the existence of W1, and have credence 1/3 in H1, because of the existence of W2. That’s inconsistent, so Trust Experts is inconsistent.

In propositional modal logic, we have a clear distinction between models and frames. A frame is a pair of a set W and a binary relation R on W. A model is a frame plus a valuation function that maps atomic sentence letters to subsets of W. Given this distinction, we can define truth at a point in a model, truth on a model (i.e. truth at all points on a model) and truth on a frame (i.e. truth at all points on all models on the frame.)

In the previous post I was considering logics which, as well as modal operators, had actuality operators and quantifiers. Given that extra expressive capacity, where should we draw the frame/model distinction? There might well be a well established precedent for this, but I wasn’t sure what it was. And it might be that different distinctions are useful for different purposes.

Here are two more specific questions.

First, if we have an actuality operator, we need a way of designating one world as actual in each model. Is this something we do at the level of the frame, or is it something done by something like the valuation function?

Second, is the number of individuals in each world set by the frame, or do different models on the same frame have different individuals in the worlds?

As I said, I imagine there are conventional answers to these questions, and the conventions are probably well motivated. But I wasn’t sure what they were.

I spent last weekend at the Rutgers Semantics Workshop, which was a great success. Thanks to Angela Harper and Jason Stanley for putting on such a good conference. I learned a few things there, some of which quite surprised me. What I’m going to write about here is something I learned from my colleague-to-soon-be Thony Gillies. (I’m translating a little what Thony said, so if there are errors in what follows, they are my responsibility. I also don’t claim originality for any of this – it’s just something I didn’t know, so I’m recording it here in case other people were similarly ignorant.)

In chapter 6 of Counterfactuals, when he is discussing the relation between semantic constraints on the similarity and accessibility relations, and axioms of the counterfactual logic, Lewis writes

bq. There is no special characteristic axiom corresponding to the Limit Assumption. We can therefore say that if any combination of axioms corresponds to a combination of conditions without the Limit Assumption, then then same combination of axioms corresponds also to that combination of conditions with the Limit Assumption added.

Lewis is here talking about characteristic axioms for theories of comparative possibility. But since counterfactual conditionals can be defined in terms of comparative possibility, the result quickly generalises to the counterfactual conditionals. What I never knew was that this result turns crucially on the expressive limitations of the logics Lewis is working in. If we have quantifiers, and an actuality operator, there is an axiom that corresponds to the Limit Assumption. It’s a little hard to state in HTML, but I’ll try. I’ll use > for the counterfactual conditional, -> for the material implication, A for the universal quantifier, and $ for the actuality operator. Then consider the following axiom schema, where F and G are any predicates, and p is a sentence in which x does not occur free.

bq. (L) $(Ax(Fx -> (p > Gx)) -> (p > Ax($Fx -> Gx)))

That is, if it’s actually the case that for any F, it would be G if p were true, then if p were true, all the actual Fs would be G. That sounds pretty plausible to me. Given the limit assumption, I believe it holds fairly trivially. (Assume it doesn’t hold. Then in the actual world the antecedent is true and the consequent false. So if p were true there would be some object that isn’t G despite being actually F. (Wo points out in comment 5 that I should say could here, not would.) By the Limit Assumption, that means that in some of the nearest p-worlds there are some actual Fs that are not Gs. Call some such object b. That means in the actual world, Fb, but it isn’t true in all the nearest p-worlds that Gb. So p > Gb is not actually true. That contradicts Ax(Fx -> (p > Gx)), which we assumed is actually true.) Surprisingly, (L) does not hold given the failure of the Limit Assumption. (Italicised words added due to correction in comment 4.)

Consider a model with a countable infinity of Fs in the actual world, with these Fs numbered 1, 2, 3, etc. And set the similarity ordering to be as follows. For two worlds w1, w2 that differ on the distribution of F-ness, w1 is more similar to actuality than w2 iff the lowest number individual that is F in just one of these two worlds is F in w1, and not F in w2. Now consider a substitution instance of (L) with G = F, and p is Finitely many actual Fs are F.

The first thing to note is that the Limit Assumption fails. For any world where exactly n of the actual Fs are F, the world where the first n+1 actual Fs are F is closer. So for any world that satisfies p, there is a closer world.

Let c be any number, and let o be the c’th actual F. Then consider the world where the first c actual Fs are F, and no other actual Fs are F. That’s a world where p & Fo is true obviously. Moreover, any closer world to actuality must still have Fo true, since the only way we can get closer to actuality is to make some later actual Fs back into Fs. So on Lewis’s semantics for counterfactuals, p > Fo is actually true. Since o was an arbitrary actual F, it follows that Ax(Fx -> (p > Gx)) is actually true.

But clearly it isn’t true that p > Ax($Fx -> Fx). Indeed, p is inconsistent with Ax($Fx -> Fx), since there are infinitely many actual Fs. So (L) fails in this model. Since (L) is true in all models where the Limit Assumption holds, it follows I think that (L) is a nice way to characterise the Limit Assumption.

If we have the Barcan and converse Barcan axioms in our logic, we can simplify (L) considerably. In that case, the axiom (LB) characterises the Limit Assumption, though I won’t prove this.

bq. (LB) Ax(p > Fx) -> (p > AxFx)