I think identifying degrees of justification is very compelling. However, I’m sceptical of this being possible to do the way it is done in this post.
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?
I understood this post as working on the assumption that there is something meaningful to be extracted from the representor besides unanimity.
I’d put it differently: Rather than assuming this, I think it’s worth openly exploring the possibility. In particular, I think it’s worth questioning the claim that maximality is the only or most plausible decision rule in this context. I don’t think that’s been established, and I think there are strong arguments and considerations against it (such as those I raise and those raised by others, including Levi (2000, sec. 4), Schervish et al. (2003, sec. 2), and Bradley (2015, sec. 4.1)).
Each probability function carries no information beyond being one admissible way to represent the information we carry.
That’s another premise I would question, or at least I’d question the notion that this is all we can say. As noted in footnote 36: “That we can distinguish plausible probability functions from implausible ones suggests that some level of plausibility-discernment is possible, and it would be surprising if this capacity were restricted to exactly the binary ordering of plausible versus implausible.” Why assume a sharply binary picture?
This does not claim to be an “exhaustive criterion of comparative justification” as you say.
To clarify, what I mean by this in the context of this essay is “an exhaustive criterion for outcome-based choice”, specifically impartial outcome-based choice. That is, my essay is exclusively concerned with c-preference or c-betterness, and I’m arguing that maximality is not an exhaustive criterion of justification in this regard. That we can justify actions in other ways is a point I strongly agree with.
What is the meaningful thing you want to extract from the representor? What information does magnitude or being an outlier carry?
I think the non-sharpness/vagueness of the range of plausible probabilities (as endorsed by Clifton and DiGiovanni) gives a strong clue: this vagueness suggests that there indeed isn’t a clear and sharp cutoff point between plausible and implausible (or admissible and inadmissible) probability functions. There is no sharp cliff between these categories but rather a smooth gradient, or so I’d argue.
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). This is an additional reason why maximality’s verdict in the torture cases seems highly implausible: if we just drew the boundary of P slightly differently for arbitrary reasons, maximality could make its categorical jump from complete indeterminacy to full preference.
This also gives a further reason to prefer rules whose verdicts vary continuously, or at least less abruptly, as the boundary of P changes. Midpoint ordering is one simple candidate (although this consideration does not uniquely privilege it): a small shift in where we draw the boundary shifts the midpoint slightly, whereas it can flip maximality’s verdict categorically. If we must build on vague foundations, it seems more plausible to use a rule whose outputs vary continuously with them.
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.
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?
I’d put it differently: Rather than assuming this, I think it’s worth openly exploring the possibility. In particular, I think it’s worth questioning the claim that maximality is the only or most plausible decision rule in this context. I don’t think that’s been established, and I think there are strong arguments and considerations against it (such as those I raise and those raised by others, including Levi (2000, sec. 4), Schervish et al. (2003, sec. 2), and Bradley (2015, sec. 4.1)).
That’s another premise I would question, or at least I’d question the notion that this is all we can say. As noted in footnote 36: “That we can distinguish plausible probability functions from implausible ones suggests that some level of plausibility-discernment is possible, and it would be surprising if this capacity were restricted to exactly the binary ordering of plausible versus implausible.” Why assume a sharply binary picture?
Some related literature on this includes Gärdenfors & Sahlin (1982), Chateauneuf & Faro (2009), Hill (2013), and Hill (2019).
To clarify, what I mean by this in the context of this essay is “an exhaustive criterion for outcome-based choice”, specifically impartial outcome-based choice. That is, my essay is exclusively concerned with c-preference or c-betterness, and I’m arguing that maximality is not an exhaustive criterion of justification in this regard. That we can justify actions in other ways is a point I strongly agree with.
I think the non-sharpness/vagueness of the range of plausible probabilities (as endorsed by Clifton and DiGiovanni) gives a strong clue: this vagueness suggests that there indeed isn’t a clear and sharp cutoff point between plausible and implausible (or admissible and inadmissible) probability functions. There is no sharp cliff between these categories but rather a smooth gradient, or so I’d argue.
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). This is an additional reason why maximality’s verdict in the torture cases seems highly implausible: if we just drew the boundary of P slightly differently for arbitrary reasons, maximality could make its categorical jump from complete indeterminacy to full preference.
This also gives a further reason to prefer rules whose verdicts vary continuously, or at least less abruptly, as the boundary of P changes. Midpoint ordering is one simple candidate (although this consideration does not uniquely privilege it): a small shift in where we draw the boundary shifts the midpoint slightly, whereas it can flip maximality’s verdict categorically. If we must build on vague foundations, it seems more plausible to use a rule whose outputs vary continuously with them.
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?