Ah, I see - is the idea "if we haven't derived probabilities yet, how can we use probabilistic strategies?"
If we use some non-black-box random process, like rolling a die, then I think the problem resolves itself, since we don't have to use probabilities to specify a die, we can just have a symmetry in our information about the sides of the die, or some knowledge of past rolls, etc. Under this picture, the "mixed" in mixed strategy would be externalized to the random process, and it would have the same format as a pure strategy.
Hmm, no, I was trying to make a different point. Okay, let's back up a little. Can you spell out what you think are the assumptions and conclusions of Savage's theorem with your proposed changes? I have some vague idea of what you might say, and I suspect that the conclusions don't follow from the assumptions because the proof stops working, but by now we seem to misunderstand each other so much that I have to be sure.
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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