## Sunday, May 6, 2012

### Open-mindedness and propriety

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):

1. 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.
2. 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.
3. 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.
Say that a credence level p is open provided that for every experiment X you do not by your own lights expect to decrease your epistemic utility in respect of H. If a credence level p is open, then when your credence is at p, you are never required, on pain of expecting to lower your epistemic utility in respect of H, to stop up your ears when the result of an experiment is to be announced. A credence level p is closed provided that for every experiment X you expect by your own lights not to increase your epistemic utility in respect of H. (So, a credence level could be both open and closed, if you expect no experiment to make a difference.)

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.