I’m unsure what “the way it is done” refers to here: is it the general approach loosely illustrated in the first couple of figures, which covers a very broad range of possible graded accounts, or the specific (not firmly endorsed) example indices D or CD?
Sorry, this was pretty vague. I refer to an even larger category: a rule that takes the representor as an input and then wants the output to say something about the comparative justification/arbitrariness/etc. of belief. Most of my comments below are ultimately about reiterating this point.
I think it’s worth questioning the claim that maximality is the only or most plausible decision rule in this context.
I agree it’s worth questioning. However, with the explicit setup in my previous comment (representor via incompleteness+wanting to get action guidance from some method that uses a probability function) maximality seems undeniable. I think any changes to maximality being correct, or the only correct rule, come from changing or adding something to the setup.
Why assume a sharply binary picture?
The situation is that I’m trying to figure out which probability functions an idealized version of me could have. Here it seems clear that anything less plausible than something else available is not worth considering. (Even if we did want to consider these, the narrower representor should still be considered first. Having more functions in the representor can only remove strict preferences, so the secondary consideration done with the larger representor never gives action guidance.) If some function is more plausible than another, that other one should not be considered. Having multiple probability functions comes from incompleteness in the plausibility relation, and incompleteness is not graded. I don’t see this construction having a sense in which we would draw the boundary of the representor.
Just incompleteness would settle every function in or out, so would not allow for vague endpoints. I attribute vague endpoints to indeterminacy in that it can be unsettled whether one function beats another. That is still not a ranking by degree, which is what the graded picture would need.
On G&S (1982), C&F (2009), Hill (2013, 2019). I’m familiar with Hill (2013) and I skimmed and read summaries of the other literature you mention here. Please correct me if I’m wrong, but they all seem to assume a structure beyond a representor and build a decision rule on that without justifying the assumed structure.
This is relevant for outliers in the following sense: extremal outlier distributions are those that are most vulnerable to alternative precisifications, and thus their inclusion or exclusion can depend on arbitrary precisifications (if we insist on the binary picture).
By “extremal outlier distribution” I take it you mean something about how well supported a function is by our reasons. I don’t have an idea what property of a probability function could track that (I’m genuinely interested in finding such properties but pessimistic). In the structural sense extremal means not being a mixture of two other functions in the representor, and that doesn’t seem to relate to plausibility. Also, which structurally extremal functions produce the endpoints/midpoint of an EV interval is specific to the option, so I don’t see a way to identify endpoint/midpoint functions.
The torture example, and to my reading the whole argument, seems to work on an intuition about the size or total authority of different parts of the representor compared to others. Even in the “one” vs “all the rest” case it is not clear to me how to weigh these up in a non ad hoc way. Justifying that would require finding a size or total authority measure on the representor. (For clarity, I’m not saying it needs to be very formal.) The construction of the representor by incompleteness seems to deny the existence of such a measure.
If the plausibility relation is graded in some way and the representor is a coarse version of it, continuity is definitely desirable. On the incompleteness (and possible indeterminacy) view, a rule that doesn’t flip the verdict seems to misrepresent the situation. As stated before, I don’t think incompleteness construction has a sense in which we would draw a boundary.
My takeaway is that the fundamental disagreement is about where the representor comes from. I’m quite convinced that if we have a representor, it is due to incompleteness. What would produce a graded one?
Thank you for the thorough reply!
Sorry, this was pretty vague. I refer to an even larger category: a rule that takes the representor as an input and then wants the output to say something about the comparative justification/arbitrariness/etc. of belief. Most of my comments below are ultimately about reiterating this point.
I agree it’s worth questioning. However, with the explicit setup in my previous comment (representor via incompleteness+wanting to get action guidance from some method that uses a probability function) maximality seems undeniable. I think any changes to maximality being correct, or the only correct rule, come from changing or adding something to the setup.
The situation is that I’m trying to figure out which probability functions an idealized version of me could have. Here it seems clear that anything less plausible than something else available is not worth considering. (Even if we did want to consider these, the narrower representor should still be considered first. Having more functions in the representor can only remove strict preferences, so the secondary consideration done with the larger representor never gives action guidance.) If some function is more plausible than another, that other one should not be considered. Having multiple probability functions comes from incompleteness in the plausibility relation, and incompleteness is not graded. I don’t see this construction having a sense in which we would draw the boundary of the representor.
Just incompleteness would settle every function in or out, so would not allow for vague endpoints. I attribute vague endpoints to indeterminacy in that it can be unsettled whether one function beats another. That is still not a ranking by degree, which is what the graded picture would need.
On G&S (1982), C&F (2009), Hill (2013, 2019). I’m familiar with Hill (2013) and I skimmed and read summaries of the other literature you mention here. Please correct me if I’m wrong, but they all seem to assume a structure beyond a representor and build a decision rule on that without justifying the assumed structure.
By “extremal outlier distribution” I take it you mean something about how well supported a function is by our reasons. I don’t have an idea what property of a probability function could track that (I’m genuinely interested in finding such properties but pessimistic). In the structural sense extremal means not being a mixture of two other functions in the representor, and that doesn’t seem to relate to plausibility. Also, which structurally extremal functions produce the endpoints/midpoint of an EV interval is specific to the option, so I don’t see a way to identify endpoint/midpoint functions.
The torture example, and to my reading the whole argument, seems to work on an intuition about the size or total authority of different parts of the representor compared to others. Even in the “one” vs “all the rest” case it is not clear to me how to weigh these up in a non ad hoc way. Justifying that would require finding a size or total authority measure on the representor. (For clarity, I’m not saying it needs to be very formal.) The construction of the representor by incompleteness seems to deny the existence of such a measure.
If the plausibility relation is graded in some way and the representor is a coarse version of it, continuity is definitely desirable. On the incompleteness (and possible indeterminacy) view, a rule that doesn’t flip the verdict seems to misrepresent the situation. As stated before, I don’t think incompleteness construction has a sense in which we would draw a boundary.
My takeaway is that the fundamental disagreement is about where the representor comes from. I’m quite convinced that if we have a representor, it is due to incompleteness. What would produce a graded one?