(Cross-posting from LWBM)

Some very big news for British philosophy.  From 1st September there will be a new philosophy research centre at Aberdeen founded by Crispin Wright.  Its provisional name is the Northern Institute of Philosophy.  The centre will have the following remit (with areas of envisaged mid-term focus in brackets):

• Epistemology (formal epistemology, entitlement, epistemic externalism, perception)
• Formal Logic, including the history of logic
• Philosophy of Logic (logical consequence, the paradoxes, inferentialism and model-theory, the epistemology of logic and the a priori)
• Philosophy of Language (rule-following, propositions, vagueness, semantics and pragmatics, contextualism and relativism, content externalism)
• Philosophy of Mathematics (foundations, neo-logicism and structuralism)
• Metaphysics (value, taste, meaning, intentionality, time and truth),
• Philosophy of Mind (the metaphysics and epistemology of the self, rationality and rational explanation)
• History of Analytical Philosophy, and its methods, scope and limits

A number of appointments are planned, including 2-3 Institute professors and 6 quarter-time professorial fellows.  There will also be funded PhD places and postdoc positions, as well as funding for networks and workshops.

Exciting times!

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)

There’s an exciting looking new formal epistemology blog, Choice and Inference. They’ve done a lot of good things with getting it set up, including getting LATEX working on the blog. I’ve already added the blog to my RSS reader, and hopefully it will be a success.

What they didn’t do, at least on setting up, was get many female contributors. This was noted over at Feminist Philosophers, where there were also some disturbing reports of critical comments being blocked from the C&I blog. That’s bad – both having a non-diverse list of writers and blocking critical comments. Happily there have been moves to rectify the situation, and the blog owners seem committed to improving the gender balance of the site. So hopefully this story will have a happy ending.

Just what diversity obligations a blog has is a slightly tricky matter. I think anyone is perfectly within their rights to start a solo blog, and if that blog’s authorship is thereby 100% white and male, I don’t see how that’s a problem. I don’t think there’s a problem if they add a second author, even if that still means 100% white and male. A philosophy-oriented group blog that had, say, 10 authors and was 100% white and male, now that I would think was troubling in its lack of diversity. My intuitions about these cases feel fairly strong and robust, and I assume they are tracking something, but I don’t have a good theory about what they are tracking.

The feminist philosophers blog has been doing very good work over recent times keeping track of which events and the like are doing well or badly on gender diversity. That’s not the only kind of diversity we should be worried about though. I’d like to see us, collectively, pay more attention to how white various events (conferences, blogs, etc) are, and how much they are oriented towards people from English-speaking countries. On the latter score at least, Choice and Inference seems to be doing pretty well; which is as things should be given how much of the best work in formal epistemology is being done on the European continent these days. Hopefully this is a kind of development we’ll see more of going forward. There are so many ways in which philosophy could address diversity considerations; having better blogs is a small step in the right direction.

UPDATE: See Jonah’s comments below for information about steps C&I have taken to address the problems being faced. Richard Chappell has further interesting thoughts on the matter.

The concluding paragraph of P.M.S. Hacker’s review of Timothy Williamson’s The Philosophy of Philosophy, taken from the latest Philosophical Quarterly.

bq. The Philosophy of Philosophy fails to characterize the linguistic turn in analytic philosophy. It fails to explain why many of the greatest analytic philosophers thought philosophy to be a conceptual investigation. It does not explain what a conceptual truth is or was taken to be, but mistakenly assimilates conceptual truths to analytic ones. It holds that philosophy can discover truths about reality by reflection alone, but does not explain how. It holds that some philosophical truths are confirmable by experiments, but does not say which. It misrepresents the methodology of the empirical sciences and the differences between the sciences and philosophy. It has nothing whatsoever to say about most branches of philosophy. But it does provide an adequate ‘self-image’ of the way Williamson does philosophy.

I’ll hopefully have a chance to return to some of the substantive points Hacker makes, or tries to make, here. It’s interesting to think about what conceptual truth might be if not analytic truth. (Is the distribution of primes a conceptual truth that’s synthetic? What about geometric facts? What about moral facts? These seem to be interesting questions.) But this post is largely about the rhetoric.

On the one hand, being this over the top in one’s negativity seems counter-productive. On the other, this is a relatively specific bill of charges, certainly relative to most negative reviews. (And each sentence is the conclusion of something that is argued for, with differing degrees of success, in the body of the review.) On the third hand, I don’t think I’ve seen a review this negative by someone at the same university as the reviewee for many a long year.

This week I’ve been running a competition for the best Twitter tweet-length philosophical argument (that’s 140 characters or less), with the prize being kudos, respect and TAR airtime for the top tweet. It’s been so much fun reading the entries that I’m now sad it’s over.  There were 72 entries in total (and of course, I made my decision on idiosyncratic grounds such that nobody should feel offended in any way by not winning). 

I think the winner has to be Mark Steen (@marksteen), for making this great point with 57 characters to spare:

Ordinary objects are mereological sums. Objects can change parts, so sums can too.

I decided early on, given the nature of many of the entries, that I needed a separate category for comedy value, and in fact ended up with a number of other “special awards”.  I’ve put the full list of PSAT awards and the full list of entries online. Enjoy!

And of course, if you think you can do better …

I recently picked up Leonard Nash’s 1973 Elements of Statistical Thermodynamics as some light airplane reading (well, it’s light in the sense of being a 138 page paperback printed on thin paper), because I’ve been interested in figuring out more about the applications of the Principle of Indifference (roughly, that one’s credences in various propositions should be proportional to the number of ways that the propositions can be true). From considering the first example discussed on page 4, I’m already starting to consider connections between this epistemic principle and the underlying metaphysics. In particular, I think that there may be ways in which thermodynamics can be used to give arguments for or against the existence of tropes (or substrata or bundles or other metaphysical posits). Continue reading →

Phlox, the Berlin-based research group, are organizing a conference on modality in Berlin, September 9th-11th, and are calling for papers.  The deadline for submission of abstracts is May 1st.

I’ve thought for a long time that the relation more probable than was not a linear order. That is, it is possible to have propositions A and B such that none of the following three claims hold.

• A is more probable than B.
• B is more probable than A.
• A and B are equally probable.

This isn’t a particularly original idea of mine; it goes back at least as far as Keynes’s Treatise on Probability (which is where I got the idea from).

I’ve also thought for a long time that there was a nice way to model failures of linearity using sets of probability functions. Say there is some special set S of functions each of which satisfy the probability calculus, and then define the relations considered above as follows.

• A is more probable than B =_df For all Pr in S, Pr(A) > Pr(B).
• B is more probable than A =_df For all Pr in S, Pr(B) > Pr(A).
• A and B are equally probable ==_df For all Pr in S, Pr(A) > Pr(B)

This isn’t particularly new either; the idea goes back at least to the 1960s, perhaps earlier. I did have one idea to contribute to this, namely to suggest that this sets of probability functions approach to understanding comparative probability claims was a good way of modelling Keynes’s somewhat hard to model ideas. But now I’m starting to worry that this was a mistake, or at least undermotivated in a key respect.

Note that on the sets of probability functions approach, we can identify probabilities with functions from each Pr in S to a real in [0, 1]. Call such functions X, Y, etc, and we’ll define X(Pr) in the obvious way. Then there is a natural ordering on the functions, namely X >= Y iff for all Pr in S, X(Pr) >= Y(Pr). This ordering will be reflexive and transitive, but not total.

What I hadn’t thought about until today was that there is a natural meet and join on probabilities that we can define. So the meet of X and Y will be the function Z such that Z(Pr) is the maximum of X(Pr), Y(Pr), and the join of X and Y will be the function Z such that Z(Pr) is the minimum of X(Pr), Y(Pr). This isn’t too surprising – it might be a little sad if probabilities didn’t form a lattice.

What’s surprising is that given this definition, they form a distributive lattice. That is, for any X, Y, Z, if we write XMY for the meet of X and Y, and XJY for the join of X and Y, we have (XMY)JC = (XJC)M(YJC). (Or at least I think we do; I might just be making an error here.) That’s surprising because there’s no obvious reason, once you’ve given up the idea that probabilities form a linear order, to believe in distributivity.

Open question: What other interesting lattice properties does _more probable than_ have?

I know that it isn’t a Boolean lattice. There’s no way to define a negation relation N on probabilities such that (a) X > Y iff NY > NX and (b) XMNX is always the minimal element. I think that’s because the only way to define N to satisfy condition (a) is if NX(Pr) = 1 – X(Pr) for all Pr in S, and that relation doesn’t guarantee that XMNX is minimal. But as for other properties, I’m not really sure.

When I was working on _truer_, I spent a lot of time worrying about whether it generated a distributive lattice. Eventually I came up with an argument that it did, but it was very speculative. (Not that it was a particularly original argument; everything about lattice theory in “that paper”:http://brian.weatherson.org/ttt.pdf I borrowed from Greg Restall’s “Introduction to Substructural Logics”:http://www.amazon.com/An-Introduction-to-Substructural-Logics/dp/B000Q66SDG/ref=sr_1_3?ie=UTF8&s=books&qid=1238098686&sr=8-3, which seems to be now out in Kindle version.) It feels bad to simply assume that _more probable than_ generates a distributive lattice simply because the easiest way to model it implies distributivity.

Consider an agent S who doesn’t know what her evidence is. If Williamson is right, then we are in S’s position all of the time. Assuming S is in a position where different evidence would justify different credences in some proposition p, the following three things can’t be true of S.

• Confirmation Awareness: S knows what the rational response is to some class of possible evidential input, and her actual evidence is in that class.
• Credal Awareness: S knows what her credence in p is.
• Rationality Awareness: S knows that she is responding rationally to her evidence.

Williamson thinks that cases like S are common. In any such case, one of the three awareness claims must fail. Which is most likely to fail? That is, for realistic versions of S, which of these three claims is actually false? I think this is relevant for thinking about the possibility of modelling some familiar and interesting cases, such as Sleeping Beauty, in terms of unknown evidence.

Thinking this through is interesting because it affects what we want to say about the applicability of Williamsonian ideas to everyday cases. Consider, for instance, the following toy example, modelled closely on some examples of Williamson.

An agent is in state S1, or state S2, or state S3. Right now they regard each state as equally probable. They are about to get some evidence. When they get that evidence, then whatever state they are in, they won’t know they are not in the adjoining state, and obviously they won’t know they aren’t in the state they are in. But that’s all they won’t know. So if they are in S1, they’ll know they are in S1 or S2. If they are in S3, they will know they are in S2 or S3. And if they are in S2, they won’t get any usable information.

The agent is actually in S1. What should their credence be that they are in S1? Answer: 1/2. Their evidence is that they are in S1 or S2, conditionalising on that leads to a probability of 1/2 that they are in S1. But note, for all the agent knows, they are in S2. And if they are in S2, then their evidence is consistent with S3. In that case, conditionalising on their evidence should lead to a probability of 1/3 that they are in S1.

There is something odd about the case. The agent can’t know (a) that the right thing to do when (and only when) their evidence is S1 or S2 is to have credence 1/2 in S1, (b) that their credence in S1 or S2 is 1/2 and (c) that they are doing the right thing . If they knew (a), (b) and (c), they’d be able to deduce that their evidence was S1 or S2, and from that they’d be able to deduce that they are in S1. But they can’t know any such thing. So one of (a), (b) and (c) fails. In realistic models of this kind of situation, which of them actually fails.

It seems to me easy enough to think of cases where Confirmation Awareness holds. In cases where there are only a few possible evidential inputs, or in cases where the initial credal distribution over possible outcomes is quite straightforward (perhaps because we’re concerned with the behaviour of a chance device with known chances) it can be quite clear how to conditionalise on various pieces of evidence. So while Confirmation Awareness sometimes fails, I think it often holds.

There is a simple argument that Credal Awareness can’t fail, at least for instrumentally rational agents. The agent can just arrange for themselves to be offered bets on p at various odds, and they can look and see which ones they accept. So they’ll know which credence they have. I think that argument is too quick. At best what it shows is that an agent could get extra evidence about what their credence is, not that they already have that evidence. But the argument does show something. At least in cases where there is a big range of possible credences, we don’t have to get new evidence to know whether our credence in p is, say 1/2 or 1/3. So I think in realistic examples of Williamsonian cases, Credal Awareness succeeds.

So that leaves Rationality Awareness. I suspect a lot of the time, when we don’t know what our evidence is, we won’t know that we’re responding rationally to our actual evidence. To the extent that rationality just is a matter of responding rationally to evidence, we won’t know that we are rational. I think if we think of rationality this way though, as a matter of people appropriately in tune with the world through our evidence, it shouldn’t be too surprising that we can’t always tell we are rational. Sometimes responding rationally to evidence requires a little luck.