So if I ask you if you think I can hit immeasurable set A on a dartboard, you’d say 50%. Same with disjoint immeasurable set B. Same with A U B. I now offer to bet with 1:1 odds that you can hit A, can hit B and can’t hit A U B. If you hit A, you win the first bet, but lose the second two. If you hit B, you win the second, but lose the other two. If you miss both, you lose all three. No matter what, I get money.
Hm. The simplest way around this is to treat the fact that an immeasurable disjoint set B exists as new information to our agent. E.g. if you just tell me to bet on hitting some immeasurable set A, I’ll think the possibilities are just (A) or (not A), and in my state of ignorance will bet at 1:1 odds. But if you then tell me there’s some disjoint set B, now the possibilities are (A), (B), (neither). Maximum entropy dictates that I only assign a 1⁄3 probability to hitting A or B. This handles the dutch book correctly.
If you add knowledge about relationships between a jillion more immeasurable sets, it still produces sensible answers. The biggest trouble I can see is that representing the things we know about relationships between immeasurable sets in this way is tedious.
So if I ask you if you think I can hit immeasurable set A on a dartboard, you’d say 50%. Same with disjoint immeasurable set B. Same with A U B. I now offer to bet with 1:1 odds that you can hit A, can hit B and can’t hit A U B. If you hit A, you win the first bet, but lose the second two. If you hit B, you win the second, but lose the other two. If you miss both, you lose all three. No matter what, I get money.
Hm. The simplest way around this is to treat the fact that an immeasurable disjoint set B exists as new information to our agent. E.g. if you just tell me to bet on hitting some immeasurable set A, I’ll think the possibilities are just (A) or (not A), and in my state of ignorance will bet at 1:1 odds. But if you then tell me there’s some disjoint set B, now the possibilities are (A), (B), (neither). Maximum entropy dictates that I only assign a 1⁄3 probability to hitting A or B. This handles the dutch book correctly.
If you add knowledge about relationships between a jillion more immeasurable sets, it still produces sensible answers. The biggest trouble I can see is that representing the things we know about relationships between immeasurable sets in this way is tedious.