In the latest Philosophical Perspectives, Cian Dorr has a very interesting paper about a puzzle about what he calls the Eternal Coin. I hope to write more about the particular puzzle in future posts, but I wanted to mention one thing that comes up in passing about imprecise probabilities. In the course of rejecting a solution to one puzzle in terms of imprecise probabilities, he says
My main worries about this response are worries about the unsharp credence framework itself. In my view, there is no adequate account of the way unsharp credences should be manifested in decision-making. As Adam Elga has recently compellingly argued, the only viable strategies which would allow for someone with an unsharp credential state to maintain a reasonable pattern of behavioural dispositions over time involve, in effect, choosing a particular member of the representor as the one that will guide their actions. (The choice might be made at the outset, or might be made by means of a gradual process of narrowing down over time; the upshot is much the same.) And even though crude behaviourism must be rejected, I think that if this is all we have to say about the decision theory, we lack an acceptable account of what it is to be in a given unsharp credential state—we cannot explain what would constitute the difference between someone in a sharp credential state given by a certain conditional probability function, and someone in an unsharp credential state containing that probability function, who had chosen is as the guide to their actions. Unsharp credential states seem to have simply been postulated as states that get us out of tricky epistemological dilemmas, without an adequate theory of their underlying nature. It is rather as if some ethicist were to respond to some tricky ethical dilemma—say, whether you should join the Resistance or take care of your ailing mother—by simply postulating a new kind of action that is stipulated to be a special new kind of combination of joining the Resistance and taking care of your mother which lacks the objectionable features of obvious compromises (like doing both on a part-time basis or letting the outcome be determined by the roll of a dice). It would be epistemologically very convenient if there was a psychological state we could rationally be in in which we neither regarded P as less likely than HF, regarded HF as less likely than P, nor regarded them as equally likely. But we should be wary of positing psychological states for the sake of epistemological convenience.
I actually don’t think that imprecise (or unsharp) credences are the solution to the particular problem Cian is interested in here; I think the solution is to say the relevant credences are undefined, not imprecise. But I don’t think this is a compelling objection to imprecise credences either.
It is, I think, pretty easy to say what the behavioural difference is between imprecise credences and sharp credences, even if we accept (as I do!) what Adam and Cian have to say about decision making with imprecise credences. The difference comes up in the context of giving advice and evaluating others’ actions. Let’s say that my credence in p is imprecise over a range of about 0.4 to 0.9, and that I make decisions as if my credence is 0.7. Assume also that I have to make a choice between two options, X and Y, where X has a higher expected return iff p is more likely than not. So I choose X. And assume that you have the same evidence as me, and face the same choice.
On the sharp credences framework, I should advise you to do X, and should be critical of you if you don’t do X. On the imprecise credences framework, I should say that you could rationally make either choice (depending on what other choices you had previously made), and shouldn’t criticise you for making either choice (unless it was inconsistent with other choices you’d previously made).
I don’t want to argue here that it makes sense to separate out the role of credences in decision making from the role they play in advice and evaluation. All I do want to argue here is that once we move beyond decision making, and think about advice and evaluation as well, there is a functional difference between sharp and unsharp credences. So the functionalist argument that there is no new state here collapses.
One other note about this argument. I don’t think of sharp and unsharp credences as different kinds of states, or as states that need to be separately postulated and justified. What I think are the fundamental states are comparative credences. The claim that all credences are sharp then becomes the (wildly implausible) claim that all comparative credences satisfy certain structural properties that allow for a sharp representation. The claim that all credences should be sharp becomes the (still implausible, but not crazy) claim that all comparative credences should satisfy those structural properties. Either way, there’s nothing new about unsharp credences that needs to be justified. What needs to be justified is the ruling out of some structural possibilities that look prima facie attractive.