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torekp comments on The Savage theorem and the Ellsberg paradox - Less Wrong
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It requires more than that, I think.
(I substituted "%" for their symbol, since markdown doesn't translate their symbol.) Let "A %B" represent "I am indifferent between, or prefer, A to B". Then I can A % B and B % C and A % C while violating rule 1 as you've written it.
ETA: To wit, I am indifferent between A and B, and between B and C, but I prefer A to C. This satisfies the total preorder, but violates Rule 1.
I don't think Rule 1 is a requirement of rationality, for basically this very reason. That is: a semiorder may be sufficient for rational preferences.
Your definition of total pre-order:
Looks to me like it's equivalent to what I wrote for rule 1. In particular, you say:
No, this violates total pre-order, as you've written it.
Since you are indifferent between A and B, and between B and C: A%B, B%A, B%C, C%B. By transitivity, A%C and C%A. Therefore, you are indifferent between A and C.
The "other" type of indifference, you have neither A%B nor B%A (I called this incomparability). But it violates totality.
Hope you'll forgive me if I set this aside. I want to grant absolutely every hypothesis to the Bayesian, except the specific thing I intend to challenge.
Oops, good catch. My formulation of "A % B" as "I am indifferent between or prefer A to B" won't work. I think my doubts center on the totality requirement.