Thanks for the reply!
The memory example is interesting. I now think my rejection of (*) (from my previous comment) was too strong but I haven’t figured out what to make of this more precisely. I feel like there is something valuable to be said about different ways that options can exist or be available and how relevant they are to maximality or other decision rules. I’ve always thought of menus as very unambiguous objects, so this is difficult to wrap my head around. I have more specific thoughts about the example below.
I share your intuition for B being impermissible. Having witnessed B being dominated makes it very hard to say that it would be permissible. I am, however, skeptical of my intuition because it needs to make assumptions that I’m not convinced of.
The situation has two conditions: (1) I’m sure that if I remembered what C was, it would be a valid option at the current situation and that it dominates B. Being sure in this way without knowing what C is requires the situation to be the same in every way that could affect these. (2) I’m unable to generate the same options as the other time. I think this can happen in three ways: (i) the inputs to that generation are different, (ii) the generation process is different or (iii) the generation process is just nondeterministic (or possible to run incompletely). Both (i) and (ii) make fully trusting the similarity of the situation implausible so (iii) must be true to satisfy the conditions. As far as I can tell, for the intuition of deferring to a different run of the process to make sense, these assumptions need to be made:
A common reference menu (not necessarily objective/true) must exist, and the process must sample it.
Knowledge of dominance relations on the reference menu is authoritative over relations on samples of it.
I notice that if I consciously deny these, I don’t get the intuitive verdict.
Another example to test the second assumption in isolation: An undeniably trustworthy oracle tells me that an option exists that would dominate B but not A[1]. My intuitive verdict again is that it would be pretty incredible to take B. I feel this intuition is somewhat weaker than in the memory example but still clear.
I read “an option exists” as it being in my menu (because that is at least intuitively what being an option means) and then give it similar privileges as real options. Thinking about it more critically, I have a hard time accepting objects that I cannot choose to play in my decision rule for my objects of choice. In terms of the assumptions, I take the oracle to have access to relations on the reference menu but I’m still unsure if I’m willing to let those be authoritative.
- ^
Or similarly: It is a logical implication of my information. This is the case with mixed options over nowhere-optimal options vs somewhere-uniquely-optimal options.
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?