if the vast majority of the measure of possible worlds given Bob’s knowledge is in worlds where he loses, he’s objectively wrong.
That’s a self-consistent system, it just seems to me more useful and intuitive to say that:
“P” is true ⇒ P
“Bob believes P” is true ⇒ Bob believes P
but not
“Bob’s belief in P” is true ⇒ …er, what exactly?
Also, I frequently need to attach probabilities to facts, where probability goes from [0,1] (or, in Eliezer’s formulation, (-inf, inf)). But it’s rare for me to have to any reason to attach probabilities to probabilities. On the flip side, I attach scoring rules in the range [0, -inf] to probability calculations, but not to facts. So in my current worldview, facts and probabilities are tentatively “made of different substances”.
if the vast majority of the measure of possible worlds given Bob’s knowledge is in worlds where he loses, he’s objectively wrong.
That’s a self-consistent system, it just seems to me more useful and intuitive to say that:
“P” is true ⇒ P
“Bob believes P” is true ⇒ Bob believes P
but not
“Bob’s belief in P” is true ⇒ …er, what exactly?
Also, I frequently need to attach probabilities to facts, where probability goes from [0,1] (or, in Eliezer’s formulation, (-inf, inf)). But it’s rare for me to have to any reason to attach probabilities to probabilities. On the flip side, I attach scoring rules in the range [0, -inf] to probability calculations, but not to facts. So in my current worldview, facts and probabilities are tentatively “made of different substances”.