Define 'incoherent' credences as ones that don't satisfy the probability axioms. There are familar 'Dutch Book' arguments about what's wrong with having incoherent credences. At a reading group meeting today lead by Al Hajek, I became even more convinced than previously that they leave a lot to be desired (at least as they stand). I thought I'd have a shot at something else.
Ideally, it would be nice to formulate a norm of credence which incoherent credences - or rather, credences which you *know* to be incoherent (there needn't be anything irrational about credences which are *in fact* incoherent if you've no reason to think they are) - are in tension with. I thought of this:
NC: You should try to make your credence in p sensitive only to your evidence concerning p.
Now, suppose you notice that you have different credences in p and q and you have credence 1 in (p iff q). Your certainty that (p iff q) will (at least in lots of standard situations) enable you to (properly) treat all your evidence concerning p as evidence concerning q and vice versa. So that you have the same evidence for and against each of them. This means, since your credences in p and q are different, that at least one of those credences must be sensitive to something other than your evidence concerning the relevant proposition. You can see that you have violated a norm of credence.
This seems to me to be a decent explanation of what is wrong with incoherent credences in this sort of case. Is this kind of explanation sufficiently generalizable? I don't know yet. For one thing, I'm not sure how to get this sort of explanation going in situations where you don't have credence 1 in something. But then, I'm also not sure whether you can get clear cases of irrationality in situations where you don't have credence 1 in something. (In the above case, if your credence in (p iff q) wasn't 1, it would be less clear that it was an epistemic mistake to have different credences in p and q.)
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4 comments:
Hi Carrie,
I find your post extremely interesting, not least because it might help with me a paper that I've been working on. Like you, I don't think that there's anything necessarily irrational about having incoherent credences. Instead of holding that the probability axioms serve as rationality constraints on credences, I proposed, in my paper, the following list of constraints instead:
(1) If Cr(A is true) = 1, then Cr (A) = 1. (When I presented the paper at the RSSS last Novemeber, some people thought that this constraint sounded kind of trivial, but let's ignore the problem for now.)
(2) If Cr (A & B) = 0, then Cr (A ∨ B) = Cr (A) + Cr (B).
(3) If Cr (A ⊃ B) = 1, then Cr (A) ≤ Cr (B).
Ian Hacking and Haim Gaifman have proposed similar constraints on credences, to replace those provided by the probability axioms. One might think that such constraints are too weak, in the sense that they do not penalise people who assign credences of less than 1 to obvious tautologies. Others (e.g. William Talbott, Daniel Garber) think that such constraints lack mathematical structure, and would not allow us to derive certain important results in Bayesian epistemology. I dealt with these, as well as other objections, in my paper. But I'll not bore you with the details. What I'm interested in is how we could provide some direct justification for the above constraints on credences. I think they're rather intuitive, or at least more intuitive than saying that credences ought to satisfy the probability axioms. Still, it'll be nice to have some direct explanation for why credences should satisfy (1)-(3). Your proposed norm of credence (NC) might just provide the kind of explanation I'm looking for. So here are some questions that I have about it.
You wrote that "it would be nice to formulate a norm of credence which incoherent credences - or rather, credences which you *know* to be incoherent [...] are in tension with". And In your follow-up post, you mentioned that it was strange to have "two different credences in (what you know to be) materially equivalent propositions". I'm wondering if you would still say the same if we replace the word "know" by "believe"? So, if I believe mistakenly that A and B are materially equivalent, and assign them different credences, would you say that I'm in tension with NC? What you said elsewhere in your two posts suggests that you would, but I just want to be sure.
Also, could I clarify what notion of evidence you have in mind? For example, is E part of my evidence for A, as long as I believe so, even if in some objective sense, E is not really evidence for A? A related question: if I believe both F and G, which jointly entail A, do I count as having evidence for A even if I fail to put the pieces together?
One last question, at least for now. I can see how NC might explain why (3) is a constraint on rational credences. It's less clear to me how it might explain why (2) should also be a constraint. Do you have an explanation in mind? (Actually, I'm not entirely sure if you think (2) should be a constraint, but I'm assuming that you do, from the first quote in the last but one paragraph.)
Hi Wenghong,
Thanks for your comments. Yes, replacing 'know' with 'believe' looks OK to me (at least, most of the time). I didn't intend to appeal to any specific notion of evidence in NC; I'm just looking for something that sounds intuitively right and is easy to grasp. (I guess I think there are probably different senses of 'evidence', some more external-sounding, some more internal-sounding. Which one or ones best fit into NC isn't something I've thought about yet.)
On your (2), your symbols aren't displaying properly for me but I'm guessing it's meant to be a disjunction in the second set of brackets. I think something like this should be a rule, at least in lots of ordinary situations where the subject is fully aware of what her credences in all the relevant propositions are (and of relevant logical relations etc.). The explanation, I guess, is going to be something like this.
You're sure that if A then AvB. So you see that you can properly treat all your evidence for A as evidence for AvB. And similarly you see that you can treat all your evidence for B as evidence for AvB. Since you're sure that not-(A&B), you can see that there's no risk of double-counting if you take the strength of your evidence for AvB to be just that of your evidence for A added together with that of your evidence for B. So if your credences in A, B and AvB are not related as in (2), that makes it look like one of them is sensitive to something other than the strength of the relevant evidence.
Hi Carrie,
Thanks for the reply. Yes, there's supposed to be a disjunction in the second set of brackets in (2). (I should have made sure that the symbols display properly!)
In explaining how someone who violates (2) might violate NC, you wrote: "Since you're sure that not-(A&B), you can see that there's no risk of double-counting if you take the strength of your evidence for AvB to be just that of your evidence for A added together with that of your evidence for B". I like your explanation for the specific kind of case you considered. But how about a case in which one is sure that not-(A&B), but has evidence for AvB that is neither evidence for A nor B directly? For example, suppose that someone fairly reliable tells me that she knows who'll be promoted, but all she's willing to reveal is it'll be either Anne or Bob. In such a case, I've evidence for "Either Anne or Bob will be promoted", but this evidence is not direct evidence for either "Anne will be promoted" or "Bob will be promoted". Hence, it seems that the explanation you provided for the other case does not work here, although perhaps, a similar explanation can be given, and I'm too obtuse to see it at the moment.
Hmmm ... you're right, that sort of case is harder to account for. I find it correspondingly harder to feel the force of the intuition that one ought to obey (2) in that sort of case, though. It certainly deserves some more thought.
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