This thought isn’t original to me, but it’s probably worth making. It feels like there are two sorts of axioms. I am following tradition in describing them as “rationality axioms” and “structure axioms”. The rationality axioms (like the transitivity of the order among acts) are norms on action. The structure axioms (like P6) aren’t normative at all. (It’s about structure on the world, how bizarre is it to say “The world ought to be such that P6 holds of it”?)
Given this, and given the necessity of the structure axioms for the proof, it feels like Savage’s theorem can’t serve as a justification of Bayesian epistemolgy as a norm of rational behaviour.
P6 is really both. Structurally, it forces there to be something like a coin that we can flip as many times as we want. But normatively, we can say that if the agent has blah blah blah preference, it shall be able to name a partition such that blah blah blah. See e.g. [rule 4]. This of course doesn’t address why we think such a thing is normative, but that’s another issue.
But why ought the world be such that such a partition exists for us to name? That doesn’t seem normative. I guess there’s a minor normative element in that it demands “If the world conspires to allow us to have partitions like the ones needed in P6, then the agent must be able to know of them and reason about them” but that still seems secondary to the demand that the world is thus and so.
Agreed, the structural component is not normative. But to me, it is the structural part that seems benign.
If we assume the agent lives forever, and there’s always some uncertainty, then surely the world is thus and so. If the agent doesn’t live forever, then we’re into bounded rationality questions, and even transitivity is up in the air.
P6 entails that there are (uncountably) infinitely many events. It is at least compatible with modern physics that the world is fundamentally discrete both spatially and temporally. The visible universe is bounded. So it may be that there are only finitely many possible configurations of the universe. It’s a big number sure, but if it’s finite, then Savage’s theorem is irrelevant. It doesn’t tell us anything about what to believe in our world. This is perhaps a silly point, and there’s probably a nearby theorem that works for “appropriately large finite worlds”, but still. I don’t think you can just uncritically say “surely the world is thus and so”.
If this is supposed to say something normative about how I should structure my beliefs, then the structural premises should be true of the world I have beliefs about.
I don’t think you can just uncritically say “surely the world is thus and so”.
But it was a conditional statement. If the universe is discrete and finite, then obviously there are no immortal agents either.
Basically I don’t see that aspect of P6 as more problematic than the unbounded resource assumption. And when we question that assumption, we’ll be questioning a lot more than P6.
This thought isn’t original to me, but it’s probably worth making. It feels like there are two sorts of axioms. I am following tradition in describing them as “rationality axioms” and “structure axioms”. The rationality axioms (like the transitivity of the order among acts) are norms on action. The structure axioms (like P6) aren’t normative at all. (It’s about structure on the world, how bizarre is it to say “The world ought to be such that P6 holds of it”?)
Given this, and given the necessity of the structure axioms for the proof, it feels like Savage’s theorem can’t serve as a justification of Bayesian epistemolgy as a norm of rational behaviour.
P6 is really both. Structurally, it forces there to be something like a coin that we can flip as many times as we want. But normatively, we can say that if the agent has blah blah blah preference, it shall be able to name a partition such that blah blah blah. See e.g. [rule 4]. This of course doesn’t address why we think such a thing is normative, but that’s another issue.
But why ought the world be such that such a partition exists for us to name? That doesn’t seem normative. I guess there’s a minor normative element in that it demands “If the world conspires to allow us to have partitions like the ones needed in P6, then the agent must be able to know of them and reason about them” but that still seems secondary to the demand that the world is thus and so.
Agreed, the structural component is not normative. But to me, it is the structural part that seems benign.
If we assume the agent lives forever, and there’s always some uncertainty, then surely the world is thus and so. If the agent doesn’t live forever, then we’re into bounded rationality questions, and even transitivity is up in the air.
P6 entails that there are (uncountably) infinitely many events. It is at least compatible with modern physics that the world is fundamentally discrete both spatially and temporally. The visible universe is bounded. So it may be that there are only finitely many possible configurations of the universe. It’s a big number sure, but if it’s finite, then Savage’s theorem is irrelevant. It doesn’t tell us anything about what to believe in our world. This is perhaps a silly point, and there’s probably a nearby theorem that works for “appropriately large finite worlds”, but still. I don’t think you can just uncritically say “surely the world is thus and so”.
If this is supposed to say something normative about how I should structure my beliefs, then the structural premises should be true of the world I have beliefs about.
But it was a conditional statement. If the universe is discrete and finite, then obviously there are no immortal agents either.
Basically I don’t see that aspect of P6 as more problematic than the unbounded resource assumption. And when we question that assumption, we’ll be questioning a lot more than P6.