If H is true, I am epistemically better off the more confident I am of H, and if H is false, I am epistemically worse off in respect of H the more confident I am of H. Here are three fairly plausible conditions on an epistemic utility assignment (I am not so sure about Symmetry in general, but it should hold in some cases):
- Symmetry: The epistemic utility of assigning credence p to H when H is true is equal to the epistemic utility of assigning credence 1−p to H when H is false.
- Propriety: For any p, if you've assigned a credence p to H, then it is not the case that by your own lights you expect to increase your epistemic utility in respect of H by changing your credence without further evidence.
- Open-mindedness: For any p, if you've assigned a credence p to H, then for every experiment X you do not by your own lights expect to decrease your epistemic utility in respect of H by finding out the outcome of X.
So, here is an interesting question: Are all, some or no symmetric and proper epistemic utility functions open-minded?
I've been doing a bit of calculus over the past couple of days. I might have slipped up, but this morning's symbol-fiddling seems to show that assuming that the utility functions are 2nd-order differentiable at most points (e.g., at all but countably many) there is no symmetric, proper and open-minded epistemic utility function, and for every symmetric, proper and 2nd-differentiable utility function, the only open or closed credences are 0 and 1. But I will have to re-do the proofs to be sure.
If correct, this is paradoxical.