Suppose I have a brother, and I want to ask him to lend me money. When I go to him asking him for a loan, I need to decide whether to wear a blue shirt. How should I decide? An obvious answer is:
- I need to evaluate the conditional epistemic probabilities P(I get loan | I wear blue) and P(I get loan | I don't wear blue), and act according to the higher of these.
The obvious answer is wrong. Here is a case. My brother is completely color blind. But my mother informs me that
- my father has brought me up to have a tendency to wear blue shirts if and only if he brought up my brother to have a tendency to give loans to relatives
This has got to be in the literature. It's reminiscent of standard examples involving probabilistic theories of causation.
So what is the right answer as to how to decide? My strong intuition is that I need to as best I can estimate the integral of P*(I get loan | Q & S=x) dP(x), where P* is objective chance, S is an epistemic random variable representing the complete state of the world (including the laws) just before my choice and P is epistemic probability measure on the set of values of S compatible with my making a choice, for Q = "I choose to wear blue" and Q = "I don't choose to wear blue". Since P*(I get loan | I choose to wear blue & S=x) = P*(I get loan | I don't choose to wear blue & S=x), for every x, given the brother's colorblindness, the two integrals are equal, and so I don't have reason either way with respect to the shirt. And that's the right answer.
But notice an interesting fact. If determinism holds, then for any complete state x of the world just before my choice, either S=x entails that I will choose to wear blue or S=x entails that I won't choose to wear blue. In the former case, P*(I get loan | I don't choose to wear blue & S=x) is undefined, and in the latter case P*(I get loan | I choose to wear blue & S=x) is undefined. Thus, in the two integrals I am supposed to compare, the values of the integrands are never both defined at the same time, if determinism holds. Therefore, if the above is the right way to make decisions—and I think it is—then knowing determinism to hold would make decision theory non-viable.