Some people think we should have sets of probability functions to represent our credence in a proposition, rather than a single such function. They are wrong. Here is one reason
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Interesting puzzle. I"m not sure what you mean by "complementarity", but my thought on reflection is that if you know your credence will eventually be higher, then it should already be higher, while if you know it will eventually be lower, then it should already be lower. In this case, you know your "credence" will be different, but it'll be an interesting superposition of higher and lower. In particular, if you observe H1 then it'll range from 0 (anti-correlated) to 1 (correlated), while if you observe ~H1, then it'll range from 1 (anti-correlated) to 0 (correlated).
Yup. And you actually have provided something like the "complementarity" response. Reflection should say "if you know your credence will be so-and-so in the future, then it should be so-and-so now." I see no reason why that should not also apply in the imprecise case. But, as you point out, while your credence in H2 will be [0,1] whether you observe H1 or H2, your credence will be "flipped" in some sense depending on what you observe. So the advocate of imprecise probabilities can say: "look, you don't have the same 'credence' in both cases, because they are 'flipped,' so since you don't know what your future credence will be, there is no violation of reflection."
My thoughts for a response is: you have a different credence only if you take Joyce's "committee" metaphor seriously (I'm mostly responding to Joyce here, and in particular his forthcoming paper here responding to problems like this, and which has the "complementarity" response and the metaphor I am discussing here). Joyce suggests thinking of imprecise probabilities
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I suppose it depends on what sort of support reflection has. On the one hand, you might think that the following principle is just plausible on its face
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Ya, I am definitely hoping for something like your first formulation. I don't think it is unreasonable, even in the context of precise probabilities. If you know that your future (better informed, equally rational, non-messed-with, etc) self will have credence >=x, I see no reason to impose that as a requirement now. And it is a short step from justifying that principle to going to your first formulation of reflection.
I'm not sure I have heard of your way of motivating reflection. Is it true that that alone plus conditionalization gives you reflection for point-valued credences? That is interesting. How does it work if you define ">" in terms of every function in your representor (because that is probably the only good way of making ">" work for intervals)? (I ask because I'm not sure I understand how it even works just for precise probabilities).
Anyways, I agree that understanding imprecise credences is probably best achieved via the committee metaphor. What I'm pushing back against here is that the "relational structure
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My thoughts for a response is: you have a different credence only if you take Joyce's "committee" metaphor seriously (I'm mostly responding to Joyce here, and in particular his forthcoming paper here responding to problems like this, and which has the "complementarity" response and the metaphor I am discussing here). Joyce suggests thinking of imprecise probabilities ( ... )
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I'm not sure I have heard of your way of motivating reflection. Is it true that that alone plus conditionalization gives you reflection for point-valued credences? That is interesting. How does it work if you define ">" in terms of every function in your representor (because that is probably the only good way of making ">" work for intervals)? (I ask because I'm not sure I understand how it even works just for precise probabilities).
Anyways, I agree that understanding imprecise credences is probably best achieved via the committee metaphor. What I'm pushing back against here is that the "relational structure ( ... )
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