1. Miller on the Language Dependence of the Accuracy of Predictions
Suppose we have two numerical quantities and ψ. These might be, for instance, the velocities (in some common units) of two objects, at some time (or some other suitable physical quantity of two objects at a time). Suppose further that we have two sets of predictions concerning the values of and ψ, which are entailed by two hypotheses 
and 
, and let us denote the truth about the values of and ψ (or, if you prefer, the true hypothesis about their values)—in our standard units—as T. Let the predictions of 
and 
and the true values T of and ψ be given by table 1. (Ignore the α/β columns of table 1, for now—I will explain the significance of those columns later.)
Table 1 Canonical Example of the Language Dependence of the Accuracy of Predictions
| ψ | α | β | ||
 |
.150 | 1.225 | .925 | 2.000 |
 |
.100 | 1.000 | .800 | 1.700 |
| T | .000 | 1.000 | 1.000 | 2.000 |
It seems clear that the predictions of 
are “closer to the truth T about and ψ” than the predictions of 
are. After all, the predicted values entailed by 
are strictly in between the values predicted by 
and the true values entailed by T. However, as Popper (Reference Popper1972, app. 2) showed (using a recipe invented by David Miller [Reference Miller1975]), there exist quantities α and β (as in table 1) satisfying both of the following conditions:
1. The values α and β are symmetrically interdefinable with respect to and ψ in the following (linear) way:
2. The values for α and β entailed by
are strictly “farther from the truth T about α and β” than the values for α and β entailed by .
are strictly “farther from the truth T about α and β” than the values for α and β entailed by 
. As Miller (Reference Miller1975) explains (see Miller [Reference Miller2006], chap. 11, for a nice historical survey), there is a much more general result in the vicinity. It can be shown that for any pair of false theories 
and 
about parameters and ψ, many comparative relations of “closer to the truth” between 
and 
regarding and ψ can be reversed by looking at what the estimates provided by 
and 
for and ψ entail about quantities α and β, which are symmetrically interdefinable with respect to and ψ, via some (linear) intertranslation of the form
That is, for many cases in which we judge that “
is closer to the truth T about and ψ than 
is” (in many ways of comparing “closeness”), there will exist some member of the above family of symmetric intertranslations such that we will judge that “
is closer to the truth T about α and β than 
is.” In this way, we can often reverse accuracy comparisons of quantitative theories via such redescriptions of prediction problems. As such, many assessments of the accuracy of predictions are language dependent.Footnote 1
2. Joyce on Probabilistic Coherence and the “Accuracy” of Credences
According to Joyce (Reference Joyce1998), if we view credences (of rational agents) as numerical estimates of truth-values of propositions, then we can give an argument for probabilism that is based on considerations having to do with the “accuracy” of such estimates. I will not get into all the details of Joyce's various arguments here. Rather, I will focus on a simple, concrete example that illustrates a (potential) problem of language dependence.
Consider an agent S facing a very simple situation, involving only one atomic sentence P. Suppose that S is logically omniscient (i.e., S assigns the same credences to logically equivalent statements, and he also assigns zero credence to all contradictions and credence one to all tautologies in his toy language). Thus, all that matters concerning S's coherence (in Joyce's sense) is whether S's credences b in P and 
sum to 1 (and are nonnegative). Now, following Joyce, we will associate the truth-value True with the number 1 and the truth-value False with the number 0. Let be the numerical value associated with P's truth-value, and let ψ be the numerical value associated with 
's truth-value (of course, and ψ will vary in the obvious ways across the two salient possible worlds: 
, in which P is false, and 
, in which P is true). We can now state (informally) the sort of theorem(s) that Joyce has been writing about for a number of years.
Theorem (Joyce). If S's credence function b—construed as providing estimates of and ψ—fails to be probabilistic, then there exists a probabilistic b′ that is more accurate than b (according to a suitable “scoring rule”) regarding and ψ—in all possible worlds. And no coherent (probabilistic) credence function is accuracy-dominated in this sense by any incoherent credence function (a key asymmetry).
Joyce makes various assumptions about how to measure “the accuracy of estimates of and ψ—in a possible world.” The various choices of “scoring rule” that one might make in order to render such “accuracy measurements” will not be important for the issue that I am going to raise here. The phenomenon will arise for any such instantiation of Joyce's framework. Rather than describing my “reversal theorem” in such general terms, I will illustrate it via a very simple concrete example, regarding our toy agent S, and assuming the Brier score as our “accuracy measure.” Suppose that S's credence function (b) assigns the following values P and 
(i.e., b entails the following numerical “estimates” of the quantities and ψ; see table 2).
Table 2 The Credence Function (B) of Our Simple Incoherent Agent (S)
| ψ | ||
| b | 1/2 | 1/4 |
Joyce's theorem entails the existence of a coherent set of estimates (b′) of and ψ, which is more accurate than b (under the Brier score) in both of the salient possible worlds. I will say that such a b′ Brier dominates b with respect to and ψ. To make things very concrete, let us look at an example of such a b′ in this case. Table 3 depicts the Euclidean-closest such b′, relative to Joyce's {0, 1}-representation of the truth-values (i.e., and ψ). (Ignore the α/β columns of table 3 for now—I will explain their significance later.)
Table 3 An Example of the Language Dependence of Joycean Brier Domination
| ψ | α | β | ||
| b | 1/2 | 1/4 | 9/16 | 3/16 |
| b′ | 5/8 | 3/8 | 3/4 | 1/4 |
 |
0 | 1 | 7/16 | 9/16 |
 |
1 | 0 | 9/16 | 7/16 |
The estimates entailed by b′ are more accurate—with respect to and ψ—in both 
and 
, according to the Brier score. A natural question to ask (in light of sec. 1, above) is whether there is a Miller-style symmetric intertranslation that can reverse this Brier-dominance relation. Interestingly, it can be shown (proof omitted) that there is no linear Miller-style symmetric intertranslation (of the simple form above) that will do the trick. But there is a slightly more complex (nonlinear) symmetric intertranslation that will yield the desired reversal (as depicted in table 3). Furthermore, it can be shown that this very same numerical intertranslation will yield such a reversal for any coherent function b′ that Brier dominates b for this incoherent agent S (with respect to and ψ). To be more precise, we have the following theorem about our (particular) agent S:
Theorem. For any coherent function b′ that Brier dominates S's credence function b with respect to and ψ, there exist quantities α and β that are symmetrically interdefinable with respect to and ψ, via the following specific symmetric intertranslations.Footnote 2
where b Brier dominates b′ with respect to α and β. It is also noteworthy that the true values of α and β “behave like truth-values,” in the sense that (1) the true value of α (β) in 
(
) is identical to the true value of β (α) in 
(
), and (2) the true values of α and β always sum to one. Indeed, these transformations are guaranteed to preserve coherence of all dominating b′'s, and the “truth-vectors.”Footnote 3
So while it is true that there are some aspects of “the truth” with respect to which S's credence function b is bound to be less accurate than (various) coherent b′'s, it also seems to be the case that (for any such b′) there will be specifiable, symmetrically interdefinable aspects of “the truth” on which the opposite is true (i.e., with respect to which b′ is bound to be less accurate).
In the next section, I consider several possible reactions to this Miller-esque “language dependence of the accuracy of credences” phenomenon. In the end, I think the upshot will be that Joyce needs to tell us more about (precisely) what he means when he says that “credences are (numerical) estimates of (numerical) truth-values.” Specifically, I think the present phenomenon challenges us to get clearer on the precise content of the accuracy norm(s) that are applicable to (or constitutive of) the Joycean cognitive act of “estimation of the (numerical) truth-value of a proposition.”
3. Some Possible Reactions
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. 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. 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. 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).
), and this makes the assumption that estimates are expectations question-begging in the present context.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” 
:
(†) If 
, then 
, where f is a symmetric intertranslation function that maps values of 
to/from values of 
.
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 
.
in which and ψ are primitive parameters, and we treat α and β as defined parameters in 
—as opposed to adopting the language 
, and treating and ψ as defined in 
. Otherwise, we could characterize what is going on here as a dependence of “distances from the truth” on a choice of parameters within a single language 
. I intend this to be a problem of language dependence. So I assume we start with an adopted language + set of primitive parameters. I thank an anonymous referee for pressing this clarification.
