Help! My credences are unmeasurable!

September 29, 2008

This is a brief follow up to the puzzle I posted a few days ago, and Kenny’s very insightful post and the comments to his post, where he answers a lot of the pressing questions to do with the probability and measurability of various events.

What I want to do here is just note a few probabilistic principles that get violated when you have unmeasurable credences (mostly a summary of what Kenny showed in the comments), and then say a few words about the use of the axiom of choice.

Reflection. Bas van Fraassens’ reflection principle states, informally, that if you are certain that your future credence in p will be x, then your current credence in p should be x (ignoring situations where you’re certain you’ll have a cognitive mishap, and the problems to do with self locating propositions.) If pn says “I will guess the n’th coin toss from the end correctly”, then Kenny shows, assuming translation invariance (that Cr(p)=Cr(q) if p can be gotten from q by uniformly flipping the values of tosses indexed by a fixed set of naturals for each sequence in q) that once we have chosen a strategy, but before the coins are flipped, there will be an n such that Cr(pn) will be unmeasurable (so fix n to be as such from now on.) However, given reasonable assumptions, no matter how the coins land before n, once you have learned that the coins have landed in such and such a way, Cr(pn)=1/2. Thus you may be certain that you will have credence 1/2 in pn even though you’re credence in pn is currently unmeasurable.

Conglomerability. This says that if you have some propositions, S, which are pairwise incompatible, but jointly exhaust the space, then if your credence in p conditional on each element of S is in an interval [a, b], then your unconditional credence in p should be in that interval. Kenny points out that conglomerability, as stated, is violated here too. The unconditional probability of pn is unmeasurable, but the conditional probability of pn on the outcome of each possible sequence up to n, is 1/2. (In this case, it is perhaps best to think of the conditional credence as what you’re credence would be after you have learned the outcome of the sequence up to n.) You can generate similar puzzles in more familiar settings. For example what should your credence be that a dart thrown at the real line will hit the Vitali set? Presumably it should be unmeasurable. However, conditional on each of the propositions \mathbb{Q}+\alpha, \alpha \in \mathbb{R}, which partition the reals, the probability should be zero – the probability of hitting exactly one point from countably many.

The Principal Principle. States, informally, that if you’re certain that the objective chance of p is x, then you should set your credence to x (provided you don’t have any ‘inadmissible’ evidence concerning p.) Intuitively, chances of simple physical scenarios like pn shouldn’t be unmeasurable. This turns out to be not so obvious. It is first worth noting that the argument that your credence in pn is unmeasurable doesn’t apply to the chance of pn, because there are physically possible worlds that are doxastically impossible for you (i.e. worlds where you don’t follow the chosen strategy at guess n.) Secondly, although the chance in a proposition can change over time, so it could technically be unmeasurable before any coin tosses, but 1/2 before the nth coin toss, the way that chances evolve is governed by the physics of the situation — the Schrodinger equation, or what have you. In the example we described we said nothing about the physics, but even so, it does seem like we can consistently stipulate that the chance of pn remains constant at 1/2. In such a scenario we would have a violation of the principal principle – before the tosses you can be certain that the chance of pn is 1/2, but your credence in pn is unmeasurable. (Of course, one could just take this to mean you can’t really be certain you’re going to follow a given strategy in a chancy universe – some things are beyond your control.)

Anyway, after telling some people this puzzle, and the related hats puzzle, a lot of people seemed to think that it was the axiom of choice that’s at fault. To evaluate that claim requires a lot of care, I think.

Usually to say the Axiom of Choice is false, is to say that there are sets which cannot be well ordered, or something equivalent. And presumably this depends on which structure accurately fits the extension of sethood and membership, the extension of which is partially determined by the linguistic practices of set theorists (much like ‘arthritis’ and ‘beech’, the extension of ‘membership’ cannot be primarily determined by usage of the ordinary man on the street.) After all there are many structures that satisfy even the relatively sophisticated axioms of first order ZF, only some of which satisfy the axiom of choice.

If it is this question that is being asked, then the answer is almost certainly: yes, the axiom of choice is true. The structure with which set theorists, and more generally mathematicians, are concerned with is one in which choice is true. (It’d be interesting to do a survey, but I think it is common practice in mathematics not to even mention that you’ve used choice in a proof. Note, it is a different question whether mathematicians think the axiom of choice is true – I’ve found often, especially when they realise they’re talking to a “philosophy” student, they’ll be suddenly become formalists.)

But I find it very hard to see how this answer has *any* bearing on the puzzle here. What structure best fits mathematical practice seems to have no implications whatsoever on whether it is possible for an idealised agent to adopt a certain strategy. This has rather to do with the nature of possibility, not sets. What possible scenarios are concretely realisable? For example, can there be a concretely realised agent whose mental state encodes the choice function on the relevant partition of sequences? (Where a choice function here needn’t be a set, but rather, quite literally, a physical arrangement of concrete objects.) Or another example: imagine a world with some number of epochs. In each epoch there is some number of people – all of them wearing green shirts. Is it possible that exactly one person in each epoch wears a red shirt instead? Surely the answer is yes, whether any person wears a red shirt or not is logically independent of whether the other people in the epoch wear a red shirt. A similar possibility can be guaranteed by Lewis’s principle of recombination – it is possible to arbitrarily delete bits of worlds. If so, it should be possible that exactly one of these people exists in each epoch. Or, suppose you have two collection of objects, A and B. Is it possible to physically arrange these objects into pairs such that either every A-thing is in one of the pairs, or every B-thing is in one of the pairs. Providing that there are possible worlds are large enough to contain big sets, it seems the answer again is yes. However, all of these modal claims correspond to some kind of choice principle.

Perhaps you’ll disagree about whether all of these scenarios are metaphysically possible. For example, can there be spacetimes large enough to contain all these objects? I think there is a natural class of spacetimes that can contain arbitrarily many objects – those constructed from ‘long lines’ (if \alpha is an ordinal, a long line is \alpha \times [0, 1) under the lexigraphic ordering, which behaves much like the positive reals, and can be used to construct large equivalents of \mathbb{R}^4.) Another route of justification might be the principle that if a proposition is mathematically consistent, in that it is true in some mathematical structure, that structure should have a metaphysically possible isomorph. Since Choice is certainly regarded to be mathematically consistent, if not true, one might have thought that the modal principles to get the puzzle of the ground should hold.



  1. Hmm… I suppose what I’d want to suggest here is that there might be constraints on possible structures that could count as credences, or epistemic states, or what have you. Although AC certainly seems like it’s true, it might be a conceptual truth that minds big enough to grasp all these sorts of choice sets wouldn’t really be “thinking” in the sense that we understand.

    For a more plausible analogy, minds powerful enough to be logically omniscient (knowing the logical consequences of any set of propositions that they know) couldn’t do mathematics in any sense that we understand of doing mathematics. At least, not the deductive parts of it. Maybe they’d spend all their time figuring out which axioms beyond those of ZFC are true.

    (I also have some responses to the Principal Principle type challenge, suggesting that even in these cases there should be a credence, rather than an unmeasurable set in the mind to go with the unmeasurable set in the world. However, I’m still sharpening up this argument for the paper I’m working on.)

  2. So how would the response go?

    Is it that we can’t have epistemic attitudes at all towards some propositions? For example, that the state you’re in when your credence is unmeasurable isn’t an epistemic attitude at all (so you don’t really have a credence in it after all.) But surely this still violates reflection (and other principles). Reflection entails, among other things, that if you’re certain you’re going to have a particular epistemic attitude towards a proposition, you should have *some* epistemic attitude towards it now.

    Or is it that in circumstances where it seems like your credence is unmeasurable you should actually assign it a value. This means your credences have to violate translation invariance, which seems like a permissible constraint on a credence function (although it may not be rationally required? It would be nice to see an argument one way or the other for that.) Maybe you could appeal to reflection to get the right credences in these scenarios.

    The fact that it was the agent in question that had to grasp weird and complicated strategies isn’t really at the source of the puzzle. Suppose instead I’m watching God about to undergo the coin tosses, and I know he’s going to adopt one of these AC generated stategies. I don’t have to grasp the strategy, but my credence that he’ll get guess n right is unmeasurable. Similar things can be said about the Vitali set on the dartboard. (Actually, I find the analogy with logical omniscience not particularly convincing. After all ‘having credence 1’ is closed under logical consequence, so it seems, by that reasoning, that the ideally rational agents we’re modelling with credence functions aren’t actually thinking either!)

    BTW, I’d be interested in reading that paper you mentioned, when its ready!

  3. I’m interested in ‘the principle that if a proposition is mathematically consistent, in that it is true in some mathematical structure, that structure should have a metaphysically possible isomorph.’ Do you think this is a commonly-held principle? If so, I’d be genuinely glad of some references. Why do you think it’s true? It doesn’t seem to me to follow from, eg, Lewis’ principle of recombination. Is the justification for it some kind of conceivability-possibility inference?

  4. Hi Al,

    I don’t actually know of much discussion of this principle. It’s definitely a principle modal structuralists adhere to anyway. A place to look might be Hellman’s stuff on modal structuralism (but that’s just a guess.)

    Of course, there is Lewis’s discussion in the Plurality of Worlds, where he famously rejects the principle, on the grounds that it will lead to paradox: “Among the mathematical structures that might be offered as isomorphs of possible spacetimes, some would be admitted , and others would be rejected as oversized.” (p103). I don’t think that the principle actually needs to lead to paradox when you let domains overlap (unlike Lewis.) Of course if it did lead to paradox, why isn’t the analogous truism also paradoxical: “if a proposition is mathematically consistent, in that it is true in some mathematical structure, that structure should have a mathematically possible isomorphic structure.”

    I think the link is stronger than the conceivability-possibility link. E.g. I take it that common examples of concievable impossibilities, are also mathematical impossibilities. Also, there are many inconceivable possibilities which look like they can be mathematically possible.

  5. I agree it’s unclear that Lewis’ argument against the principle is sound. What I’m interested in is positive arguments for the principle. I read Hellman a long time ago, but I was left with the impression he wanted the principle to hold because it would give an adequate metaphysics and epistemology for mathematics, and didn’t have particularly good antecedent ground for asserting it.

    What were you thinking of by inconceivable possibilities which can be mathematically possible? The friend of the conceivability-possibility link will want to appeal to something like Chalmers’ ‘ideal positive’ conceivability, and I’d have thought that all mathematical possibilities would be conceivable for an ideal agent.

    You also mention ‘common examples of concievable impossibilities’. I’m a bit confused – what did you mean by this? The kind of conceivable impossibilities I naturally think of are propositions like ‘masses attract by an inverse-cube law’. But what the principle deals with are possible structures – and most people take it that there’s a possible property-law structure which does involve inverse-cube attraction. It’s just that the property wouldn’t be mass, and the law wouldn’t be gravity.

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