Distributivity of More Probable Than

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 I borrowed from Greg Restall’s Introduction to Substructural Logics, 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.

Credences and Unknown Evidence

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.

Beall’s Spandrels

As a few bloggers are noting, JC Beall‘s new book Spandrels of Truth is just out with OUP.  The main idea is that semantic paradox is a by-product (or spandrel) created by the introduction into our language of a transparent truth predicate.  Having talked with JC a lot about this material, and read drafts of the book, I think it’s well worth a read if you’re interested in this topic at all. 

(As a bonus, if you order with this discount form you can get 20% off.)

Williamson on Idiolects

In “Conceptual Truth” (Proc Aris Soc Supp 80: 1-41), Timothy Williamson makes the following argument. The context is that he’s attacking Jackson’s argument that there must be some common doctrine held by people who use terms with the same meaning.

Putnam’s insight is relevant far beyond the class of natural kind terms, as Burge observed (1986). Even where we cannot sensibly divide the linguistic community into experts and non-experts, the picture of a natural language as a cluster of causally interrelated but constitutively independent idiolects is still wrong, because it ignores the way in which individual speakers defer to the linguistic community as a whole. They use a word as a word of a public language, allowing its reference in their mouths to be fixed by its use over the whole community. Such verbal interactions between speakers can hold a linguistic practice together even in the absence of a common creed which they are all required to endorse.

Whatever its merits as an argument against Jackson, this seems to me to be a quite bad argument against the view of “natural language as a cluster of causally interrelated but constitutively independent idiolects”. The problem is that individuals may choose to defer to anyone at all. If Williamson’s argument is to work, the ‘linguistic community’ has to be the whole world, and there has to be just one natural language. But that’s crazy, so Williamson’s argument doesn’t work.

We’re all familiar with examples of common loan words like ‘Schadenfreude’. That looks like a case where speakers of English (or other languages) defer, to the extent they defer at all, to experts who are not English speakers. That is, they defer to Germans. But Germans aren’t part of the linguistic community of English speakers.

Now it might be argued that really English speakers are only deferring to other English speakers. After all, ‘Schadenfreude’ is a loanword that has been incorporated into English. But I don’t think this response can be maintained. For one thing, the first English speakers who started using ‘Schadenfreude’ did not defer to other English speakers. For another, the kind of pattern we see here, namely borrowing words from other languages, can happen all the time, and on an ad hoc basis. An individual speaker may choose to defer to English speakers, or Bengali speakers, or Latin speakers, or speakers of any other kind of language, on a moment to moment basis. If Ishani and I find it convenient to adopt some term from Bengali into the language we use to talk to one another, we can, and we are under no obligation to use that term the way that other English speakers do.

If Williamson’s argument against idiolects, and for public languages in a more traditional sense, is going to work, there needs to be a linguistic community that goes with each language. And speakers must be required, in virtue of speaking that language, to defer to it. But this isn’t how language works. We can choose to defer to whoever we want at any time. Or to not defer if we insist on using a term idiosyncratically.

It’s true, and important, that the meanings of terms in my mouth is determined in part by the usage of experts, other language users and so on. But this isn’t inconsistent with the picture of overlapping idiolects. I could well choose to have the meaning of ‘sofa’, or ‘Schadenfreude’, in my language determined by the usage pattern of a broader group. What would be a problem for the idiolect view is if I was required, in virtue of speaking the language of some community, to defer to that very community. But I’m not. And unless Williamson wants to say there is really only one linguistic community, consisting of the whole world, and one public language, which we all speak fragments of, I don’t see how facts about deference can help sustain the traditional picture of public languages.


I just upgraded the version of WordPress that is powering TAR. I hope this doesn’t lead to crashes anywhere, either for viewers or for writers, but if it does, let me know.

Leiter Poll on Journals

Brian Leiter has a poll on what the top general philosophy journals in English are. The “general” condition rules out quite a few top journals (e.g., Ethics, BJPS, etc.) so it’s pretty much the usual suspects at the top.

When I ran a similar poll it was interesting to get demographic data on the respondents. It turned out that Americans who I surveyed ranked Nous more or less top, while British respondents ranked Mind more highly. It would be especially interesting to see the relative prestige of Philosophical Quarterly (one of my favourite journals) and Proceedings of the Aristotelian Society.

Arché Methdology Blog

I don’t know if everyone knows about the blog attached to the Arché Methdology Project. The link to it is

There has been some fascinating posts, and comments threads already on it. (It’s the only blog whose comments thread I subscribe to, because the comments are usually insightful.) And with term starting up again, it should be extremely lively.

Update: link fixed.

Congratulations to Leeds Metaphysicians

Ross Cameron reports,

Some fantastic news for the Leeds metaphysicians: Jason Turner has won the the Younger Scholar Prize in Metaphysics, for his paper ‘Ontological Nihilism‘! This was after a record number of submissions. Well done Jason!

And the joint runners-up are Robbie Williams and Elizabeth Barnes for their paper ‘A Theory of Metaphysical Indeterminacy‘ and me for my ‘Truthmaking for Presentists‘. So a clean sweep for Leeds!

These three papers will all be appearing in a forthcoming volume of Oxford Studies in Metaphysics.

Well done to the Leedsters! And I would be remiss to not note that the winners are all recent graduates of my two employers, Rutgers and St Andrews. So well done to Rutgers and St Andrews as well.