Followup To: Logic as Probability
If we design a robot that acts as if it's uncertain about mathematical statements, that violates some desiderata for probability. But realistic robots cannot prove all theorems; they have to be uncertain about hard math problems.
In the name of practicality, we want a foundation for decision-making that captures what it means to make a good decision, even with limited resources. "Good" means that even though our real-world robot can't make decisions well enough to satisfy Savage's theorem, we want to approximate that ideal, not throw it out. Although I don't have the one best answer to give you, in this post we'll take some steps forward.
Part of the sequence Logical Uncertainty
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Yes, this is right. Also, http://www.spaceandgames.com/?p=27 :)
If I know all the digits of pi and you think they're evenly distributed past a certain point, I can take your money.
In order to resist this, you need to have a hypothesis for "Manfred will pick the right number" - which, fortunately, is very doable, because the complexity of this hypothesis is only about the complexity of a program that computes the digits of pi.
But nonetheless, until you figure this out, that's the dutch book.
Lol that is a nice story in that link, but it isn't a Dutch book. The bet in it isn't set up to measure subjective probability either, so I don't really see what the lesson in it is for logical probability.
Say that instead of the digits of pi, we were betting on the contents of some boxes. For concreteness let there be three boxes, one of which contains a prize. Say also that you have looked inside the boxes and know exactly where the prize is. For me, I have some subjective probability P( X_i | I_mine ) that the prize is inside box i. For you, all your s... (read more)