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I would reject the completeness axiom. I often face choices where I don't know which option I prefer, but where I would not agree that I am indifferent. And I'm okay with this fact.
I also reject the transitivity axiom -- intransitive preference is an observed fact for real humans in a wide variety of settings. And you might say this is irrational, but my preference are what they are.
Can you give an example of situations A, B, C for which your preferences are A > B, B > C, C > A? What would you do if you need to choose between A, B, C?
Sure. I'll go to the grocery store and have three kinds of tomato sauce and I'll look at A and B, and pick B, then B and C, pick C, and C and A, and pick A. And I'll stare at them indecisively until my preferences shift. It's sort of ridiculous -- it can take something like a minute to decide. This is NOT the same as feeling indifferent, in which case I would just pick one and go.
I have similar experiences when choosing between entertainment options, transport, etc. My impression is that this is an experience that many people have.
If you google "intransitive preference" you get a bunch of references -- this one has cites to the original experiements: http://www.stanford.edu/class/symbsys170/Preference.pdf
It seems to me that what you're describing are not preferences but spur of the moment decisions. A preference should be thought of as in CEV: the thing you would prefer if you thought about it long enough, knew enough, were more the person you want to be etc. The mere fact you somehow decide between the sauces in the end suggests you're not describing a preference. Also I doubt that you have terminal values related to tomato sauce. More likely, your terminal values involve something like "experiencing pleasure" and your problem here is epistemic rather than "moral": you're not sure which sauce would give you more pleasure.
You are using preference to mean something other than I thought you were.
I'm not convinced that the CEV definition of preference is useful. No actual human ever has infinite time or information; we are always making decisions while we are limited computationally and informationally. You can't just define away those limits. And I'm not at all convinced that our preferences would converge even given infinite time. That's an assumption, not a theorem.
When buying pasta sauce, I have multiple incommensurable values: money, health, and taste. And in general, when you have multiple criteria, there's no non-paradoxical way to do rankings. (This is basically Arrow's theorem). And I suspect that's the cause for my lack of preference ordering.
Of course. But rationality means your decisions should be as close as possible to the decisions you would make if you had infinite time and information.
Money is not a terminal value for most people. I suspect you want money because of the things it can buy you, not as a value in itself. I think health is also instrumental. We value health because illness is unpleasant, might lead to death and generally interferes with taking actions to optimize our values. The unpleasant sensations of illness might well be commensurable with the pleasant sensations of taste. For example you would probably pass up a gourmet meal if eating it implies getting cancer.
However you can not know what decisions you would make if you had infinite time and information. You can make guesses based on your ideas of convergence, but that's about it.
A Bayesian never "knows" anything. She can only compute probabilities and expectation values.
Can she compute probabilities and expectation values with respect to decisions she would make if she had infinite time and information?
I think it should be possible to compute probabilities and expectation values of absolutely anything. However to put it on a sound mathematical basis we need a theory of logical uncertainty.