2. E_P s(P) < P(A) E_{P_A} s(P_A) + P(A^c) E_{P_{A^c}} s(P_{A^c}) whenever 0<P(A)<1,

where s(P) is a function from worlds to [-infty,M] for some finite M, P_A is P conditioned on A, and E_P is expectation with respect to P.

Is there such a measure?

Strict propriety entails (2). One might guess that (2) is equivalent to strict propriety, but in fact (2) doesn't even entail propriety. [Let s0(P) be 1 if P is zero on some non-empty set and 0 otherwise. Then (2) holds with non-strict inequality for s0. Let s(P) = s0(P) + a Brier(P) for some small positive a. Then (2) holds with strict inequality for Brier. Hence (2) holds with strict inequality for s. But clearly s isn't proper, at least if a is small. For let P be uniform. Then E_P s0(P) will be zero, but if Q is zero on some non-empty set, E_P s0(Q) will be one, and for small a we will have E_P s(P) < E_P s(Q), contrary to propriety.]

One might throw additivity into the mix, but I am sceptical of additivity.]]>

I don't think the intuition that learning a true proposition improves one's epistemic state survives once we think about (a) misleading evidence and (b) the fact that some propositions are epistemically much more important than others, independently of any formal framework for measuring epistemic utilities.

Let's say that a billion people have received a medication for a dangerous disease over the last year and another billion have received a placebo. I randomly choose a sample of a hundred thousand from each group. Let E be the proposition that in my random sample of the medicated, each person died within a week of reception, and in my random sample of the placeboed, no person died within a week of reception.

Suppose that in fact the drug is safe and highly beneficial, but nonetheless E is true. (Back of envelope calculation says that in any given week, given a billion people, about two hundred thousand will die. So it is nomically possible that the first random sample will consist of only those who die a week after receiving the drug, no matter how safe the drug, and it is nomically possible that the second random sample won't contain anyone who died within a week of the placebo.)

After updating on the truth E, I will rationally believe that the drug is extremely deadly. Result: I am worse off epistemically, because getting right whether the drug is safe is more important than getting right the particular facts reported in E.

The obvious thing about this case is that it is astronomically unlikely that E would be true. The *expected* epistemic value of learning about the death numbers in the drug and placebo samples is positive, and that's what a proper scoring rule yields. But on rare occasions things go badly.

Of course, in my example above, the implicit scoring rule doesn't have uniform weights across propositions like in your examples. But scoring rules with uniform weights seem really unrealistic. In scientific cases, I take it that normally, getting right the particular data gathered from an experiment has much lower epistemic value than getting right the theories that the data is supposed to bear on. That's why years later the details of the data are largely forgotten but the theories live on. And sometimes they live on on false pretences, because the data was misleading. (And sometimes the data was false, of course.) ]]>