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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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 as having a committee of agents, each of who assign precise probabilities to various propositions. If you take this metaphor completely seriously, then you do have a different 'credence,' I suppose. Suppose John thinks the coins are anti-correlated, while Jane thinks the coins are correlated. Then, if you observe H1, John will have P(H2)=0, while Jane will have P(H2)=1. And vice versa if you observe ~H1. So, since the specific members of your "credal committee" will have different probabilities depending on whether you observe H1 or ~H1, you will have a different credence.
But that is silly. What should matter is the structure of your imprecise credence. And the structure of your credence in H2 will be the same whether you observe H1 or ~H1.
And if that is not convincing, you can put forth a version of Reflection which just says: "if you know there is a determinate fact about what your future credence will be, then you should adjust your current credence to reflect that fact." If you are on board with that version of Reflection, then you still get a violation in the above type of case regardless of what you think about the "complementarity" issue.
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: "If you know that your future, better-informed, and equally rational self will have credal state of type T, then you should currently have credal state of type T." That gets the conclusion you want.
Because of my interests, I've mainly thought of reflection through the lens of conglomerability, which says: "If E is a set of mutually exclusive and exhaustive propositions, then it is not the case that for every e in E, P(A|e)>P(A)." If you further assume that all updates are by conditionalization, then this gets you reflection for point-valued credences, but not for intervals, unless perhaps > is defined in some interesting way.
But for me, I've always felt that the only way I can really understand imprecise credences is with some variety of the committee metaphor. It seems that you're saying that an imprecise credence has some sort of intrinsic structure, and that you can ignore the relational structure of how individual "committee members" change their opinions over time. But if I were tempted towards imprecise credences, I wouldn't want to ignore this.
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" of how individual committee members change over time is just an artifact of that metaphor without independent warrant. I can't see any epistemically motivated reason for taking anything other than the "instrinsic structure" of the the imprecise credence seriously, as long as the "instrinsic structure" is appropriately rich.
I mean, say, you can have agent A and agent B, where A and B both have credence (0,1) in propositions C and D, but A might have Credence(C)>Credence(D) while B might have Credence(D)>Credence(C) (because that would be true for each member of their committee). That seems fine, and seems appropriate to say that A is more confident in C than in D, and vice versa for B. But saying that just because you, say, give a name to each member of your committee, and after some conditionalization your named members will have different credences than they would after some other conditionalization, even though the intrinsic structure is identical, seems weird.
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 as having a committee of agents, each of who assign precise probabilities to various propositions. If you take this metaphor completely seriously, then you do have a different 'credence,' I suppose. Suppose John thinks the coins are anti-correlated, while Jane thinks the coins are correlated. Then, if you observe H1, John will have P(H2)=0, while Jane will have P(H2)=1. And vice versa if you observe ~H1. So, since the specific members of your "credal committee" will have different probabilities depending on whether you observe H1 or ~H1, you will have a different credence.
But that is silly. What should matter is the structure of your imprecise credence. And the structure of your credence in H2 will be the same whether you observe H1 or ~H1.
And if that is not convincing, you can put forth a version of Reflection which just says: "if you know there is a determinate fact about what your future credence will be, then you should adjust your current credence to reflect that fact." If you are on board with that version of Reflection, then you still get a violation in the above type of case regardless of what you think about the "complementarity" issue.
Sorry, that went on for too long.
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"If you know that your future, better-informed, and equally rational self will have credal state of type T, then you should currently have credal state of type T."
That gets the conclusion you want.
Because of my interests, I've mainly thought of reflection through the lens of conglomerability, which says:
"If E is a set of mutually exclusive and exhaustive propositions, then it is not the case that for every e in E, P(A|e)>P(A)."
If you further assume that all updates are by conditionalization, then this gets you reflection for point-valued credences, but not for intervals, unless perhaps > is defined in some interesting way.
But for me, I've always felt that the only way I can really understand imprecise credences is with some variety of the committee metaphor. It seems that you're saying that an imprecise credence has some sort of intrinsic structure, and that you can ignore the relational structure of how individual "committee members" change their opinions over time. But if I were tempted towards imprecise credences, I wouldn't want to ignore this.
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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" of how individual committee members change over time is just an artifact of that metaphor without independent warrant. I can't see any epistemically motivated reason for taking anything other than the "instrinsic structure" of the the imprecise credence seriously, as long as the "instrinsic structure" is appropriately rich.
I mean, say, you can have agent A and agent B, where A and B both have credence (0,1) in propositions C and D, but A might have Credence(C)>Credence(D) while B might have Credence(D)>Credence(C) (because that would be true for each member of their committee). That seems fine, and seems appropriate to say that A is more confident in C than in D, and vice versa for B. But saying that just because you, say, give a name to each member of your committee, and after some conditionalization your named members will have different credences than they would after some other conditionalization, even though the intrinsic structure is identical, seems weird.
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and, again, that was kinda long.
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