3.1. Naturalness/Privileged Language

One might try to maintain that (in some sense) the quantities and ψ are “more natural” (in this context) than α and β or that the “estimation problem” involving and ψ is somehow “privileged” (in comparison to the α/β “estimation problem”). I do not really see how such an argument would go. First, from the point of view of the α/β-language, the quantities and ψ seem just as “gerrymandered” as the quantities α and β might appear from the point of view of Joyce's preferred numerical representation of the truth-values. Moreover, there is a disanalogy to the case of physical magnitudes like velocity, since truth-values do not seem to have numerical properties (per se). That is, there is already something a little artificial about thinking of truth-values as the sort of things that can be “numerically estimated” (where the “estimates” are numerically scored for “accuracy”).

3.2. “Asymmetries” in Accuracy-Dominance in the α/β-Language

One might try to find some (new) accuracy-dominance asymmetry between coherent and incoherent credences in the α/β-language. I see two problems with this strategy. First, in the α/β-language (as opposed to the /ψ-language), some coherent vectors (in Joyce's sense) are Brier dominated by an incoherent vector (witness the example above). Having said that, it is also true that there do exist other coherent credence functions b ^⋆ that Brier dominate b with respect to α and β. As such, we do not have an “utter reversal” of the (full) asymmetry between coherent and incoherent vectors in the /ψ-language. I am not sure that is required here (for our purposes). We have certainly broken the (full) asymmetry between coherent and incoherent vectors. Moreover, we could define up another pair of quantities γ and η (symmetrically interdefinable with respect to α and β—perhaps relative to a new family of intertranslations), which reverses these new (b ^⋆ vs. b) relations of Brier dominance, and so on. So, this response just seems to reiterate the initial problem.

3.3. Disanalogies between “Estimation” and “Prediction.”

I think the most promising (and useful) response to the phenomenon is to argue (i) that there are crucial disanalogies between “estimation” (in Joyce's sense) and “prediction” (in the sense presupposed by Popper and Miller), and (ii) these disanalogies imply that my “reversal argument” is presupposing something incorrect regarding the norms appropriate to “estimation.”

Here, it is important to note that Joyce does not tell us very much about what he means by “estimation.” He does say a few things that are suggestive about what “estimation” is not. Specifically, Joyce clearly thinks:

1. 1. Estimates are not guesses. Joyce (Reference Joyce1998, 587) explicitly distinguishes estimation and guessing. Presumably, guessing (as a cognitive act) does not have the appropriate normative structure to ground the sorts of accuracy norms (for credences) that Joyce has in mind.

2. 2. Estimates are not expectations. Joyce (Reference Joyce1998, 587–88) explicitly disavows thinking of estimates as expectations. Indeed, this is supposed to be one of the novel and distinctive features of Joyce's approach. In fact, it is supposed to be one of the advantages of his argument (over previous, similar arguments). Here, Joyce seems to think that expectations have two sorts of (dialectical) shortcomings, in the present context. First, he seems to think that they have a pragmatic element, which is not suitable for a nonpragmatic vindication of probabilism. Second, expectations seem to build in a nontrivial amount of probabilistic structure (via the definition of expectation, which presupposes that ), and this makes the assumption that estimates are expectations question-begging in the present context.

3. 3. Estimates are not assertions that the values of the parameters are such-and-such. This is clear (just from the nature of these “estimation problems”), since it is not a good idea to assert things that you know (a priori) must be false. And, whenever you offer “estimates” of credences that are nonextreme, you know (a priori) that the parameters ( and ψ) do not take the values you are offering as estimates (an analogous point can be made with respect to what b and b′ “assert” about α and β).

These are the only (definite, precise) commitments about “estimates” that I have been able to extract from Joyce's work (apart from the implicit assumption concerning the appropriateness of “scoring” them in terms of “accuracy” using the Brier score). Unfortunately, these negative claims about what Joyce means by “estimation” do not settle whether my “reversal argument” poses a problem for Joyce. Allow me to briefly explain why.

Let be the claim that S is committed to the values as their “estimates” (in Joyce's sense) of the quantities . What we need to know are the conditions under which the following principle (which is implicit in my “reversal argument”) is acceptable, relative to Joyce's notion of “estimation” :

Presumably, there will be some symmetric intertranslation functions f (in some contexts) such that is acceptable to Joyce. The question is, which translation functions f are acceptable—in my example above?

Since Joyce does not give us a (sufficiently precise) theory of , it is difficult to answer this question definitively. But, if my “reversal argument” is going to be blocked, then I presume that Joyce would want to reject for our intertranslation function f ^⋆ above. It is natural to ask precisely what grounds Joyce might have for such a rejection of our f ^⋆.

It is useful to note that is clearly implausible, under certain interpretations of . Presumably, if involves guessing, then one could argue that should not hold (in general). Perhaps it is just fine for S's guesses about the values of to be utterly independent of S's guesses about (at least, to the extent that I understand the “norms of guessing”). Similarly, if involves expectation, then will demonstrably fail (in general) for nonlinear functions like our f ^⋆. Although, on an expectation reading of “estimate,” will demonstrably hold for all linear intertranslations. Unfortunately for Joyce, neither of these interpretations of is available to him. So, this yields no concrete reasons to reject in our example.

On the other hand, if involved assertion (as in item 3 above), then would be eminently plausible. On an assertion reading of , is tantamount to a simple form of deductive closure for assertoric commitments (in the traditional sense). And this would be very similar to the way Popper (Reference Popper1972) and Miller (Reference Miller1975) were thinking about the predictions of (deterministic, quantitative) scientific theories.Footnote ⁴ It seems clear that is not exactly like that (in this context), but this (alone) does not give us any concrete reasons to reject in this case.

I submit that what we need here is a (sufficiently precise) theory of , which satisfies Joyce's explicit commitments 1–3 above and which is also precise enough to explain why should fail for f ^⋆ (in our example above). At the very least, this article serves as an invitation to Joyce to provide such an (independent) philosophical explication of his .