How To Read a Representor

1. Background

Representors arose in response to concerns with the traditional single-function model of belief. Where ${\displaystyle \mathrm{\Omega }}$ is a set of possible worlds, and propositions are subsets of ${\displaystyle \mathrm{\Omega }}$, we let ${\displaystyle \mathbf{P}=\{p,q,r,\dots \}}$ contain all and only those propositions regarding which our subject—Sally—has beliefs to some degree or other. We assume that ${\displaystyle \mathbf{P}}$ is closed under relative complements and binary unions. Then, according to the traditional model, Sally’s beliefs can be represented using a single measure ${\displaystyle P:\mathbf{P}\mapsto [0,1]}$ that satisfies the usual normalisation and additivity constraints—i.e., ${\displaystyle P(\mathrm{\Omega })=1}$ and ${\displaystyle P(p\cup q)=P(p)+P(q)}$ whenever ${\displaystyle p\cap q=\varnothing }$.

There’s something strikingly unrealistic about this. We needn’t go into all of the concerns that have been raised, for they are legion—for discussion, see (Jeffrey 1983), (Seidenfeld 1988), (van Fraassen 1990), (Walley 1991), (Kaplan 1996), (Joyce 2005; 2010), (Sturgeon 2008), (Hájek 2012), (Alon & Lehrer 2014), and (Bradley 2014). But it won’t hurt to briefly consider one example (adapted from Fishburn 1986). Imagine that before you sits an old pack of cards. You’ve been told that some of the cards are missing, but that’s all you’re told—you know neither how many are missing nor which. Now consider:

${\displaystyle p}$ ${\displaystyle =}$ The global population in 2100 will be over 12 billion

${\displaystyle q}$ ${\displaystyle =}$ The next card drawn from this old deck will be a heart

If you’re like most people, then (a) you’ll have some positive degree of confidence regarding each, and (b) you’re unlikely to have exactly as much confidence in ${\displaystyle p}$ as you do in ${\displaystyle q}$. On those assumptions, the traditional model implies that there’s going to be a unique real value ${\displaystyle r}$ such that you’re exactly ${\displaystyle r}$ times as confident in ${\displaystyle p}$ as you are in ${\displaystyle q}$. So, question: exactly how much more or less confident are you in ${\displaystyle p}$ than you are in ${\displaystyle q}$?

You should find that difficult to answer, and not just because there might be some facts about the strengths of your beliefs that are introspectively hard to figure out. That may be the case, but the main problem instead seems to be that such precise values just aren’t very realistic when talking about a squishy psychological quantity like strength of belief—at least not for people like us. Whatever it is about us that grounds the facts about our degrees of belief, it’s not clear that there’s going to be sufficient information down there to determine that we believe ${\displaystyle p}$ down to the ${\displaystyle n^{\text{th}}}$ degree for very large ${\displaystyle n}$. Indeed, it’s not even obvious that ${\displaystyle p}$ and ${\displaystyle q}$ must stand in determinate relations of more, less, or equal confidence. Maybe there’s no fact of the matter as to which one you believe more; or maybe they’re determinately incomparable. You might even be sympathetic to the idea that your strength of belief in ${\displaystyle p}$ can be on a par with your strength of belief in ${\displaystyle q}$, where parity is a special symmetric comparative strength relation holding only if ${\displaystyle p}$ and ${\displaystyle q}$ are not believed to exactly the same degree (à la Chang 2002).

The upshot of the example, in any case, is that there seems to be something about the way our beliefs are, or a way they might be, that the traditional single-function model isn’t able to capture. Worry not just yet what that something is exactly, since that varies from author to author; worry only that it’s missing when we represent beliefs with a probability function. Maybe it’s several things. Either way, a more general model of belief seems to be required.

Representors to the rescue! On this new and improved approach, we represent a system of beliefs by means of a (finite or infinite, but either way non-empty) set of probability functions, ${\displaystyle \mathcal{R}=\{P,P^{\prime },P^{\prime \prime },\dots \}}$, all defined on the same space of propositions ${\displaystyle \mathbf{P}}$. When the representor contains just a single function, then it represents the very same beliefs as would have been represented by that function according to the traditional model. But when the representor contains multiple functions, on the other hand, then it represents … something else.

The credal committee metaphor is frequently used to give the rough idea of what that “something else” is supposed to be. Imagine that every ${\displaystyle P}$ in ${\displaystyle \mathcal{R}}$ gets a vote on what Sally’s beliefs are going to be like, and the vote passes just when the committee is unanimous. If every ${\displaystyle P}$ votes that Sally’s confidence in ${\displaystyle p}$ is greater than her confidence in ${\displaystyle q}$, then Sally’s confidence in ${\displaystyle p}$ really is greater than her confidence in ${\displaystyle q}$—even if there’s no precise value ${\displaystyle r}$ such that all members of the committee agree that Sally’s confidence in ${\displaystyle p}$ is exactly ${\displaystyle r}$ times her confidence in ${\displaystyle q}$. Likewise, if some members of the credal committee vote that Sally is more confident in ${\displaystyle p}$ than she is in ${\displaystyle q}$, while others vote that she’s more confident in ${\displaystyle q}$ than she is in ${\displaystyle p}$, then ${\displaystyle \mathcal{R}}$ as a whole represents neither comparative relation since the committee failed to reach unanimous agreement on the matter.

(Be warned: the metaphor is not an interpretation, and while it can be useful for roughly summarising how an interpretation of the model might go, it can also be misleading. There are some inferences that are naturally suggested by the metaphor that end up being licensed under some interpretations but not others. For example, unreflective application of the metaphor will suggest that Sally has more confidence in ${\displaystyle p}$ than in ${\displaystyle q}$ only if every member of her credal committee votes as such—i.e., only if ${\displaystyle P(p)>P(q)}$ for all ${\displaystyle P\in \mathcal{R}}$. This holds for some interpretations, but not all of them. Likewise, the metaphor suggests that Sally has at least as much confidence in ${\displaystyle p}$ as she does in ${\displaystyle q}$ only if every member of her committee votes as such—i.e., only if ${\displaystyle P(p)\ge P(q)}$ for all ${\displaystyle P\in \mathcal{R}}$. Again: true for some interpretations, not for all. I’ll say more about this below.)

One more thing will be useful before moving on. For any representor ${\displaystyle \mathcal{R}}$, we define its summary function, ${\displaystyle {\mathcal{R}}^s}$, like so:

That is, ${\displaystyle {\mathcal{R}}^s(p)}$ picks out the set of values that the individual measures ${\displaystyle P}$ in ${\displaystyle \mathcal{R}}$ assign to the proposition ${\displaystyle p}$. In some cases, ${\displaystyle {\mathcal{R}}^s(p)}$ may be an interval; in others, ${\displaystyle {\mathcal{R}}^s(p)}$ may be “gappy.” I’ll mostly talk about cases where ${\displaystyle {\mathcal{R}}^s(p)}$ is an interval, but nothing hangs on this. More important to note that while summary functions are useful for describing the spread of values assigned to a proposition by the “credal committee,” a summary function is not just another way of representing a representor. Distinct representors can sometimes determine the very same summary function, so in some cases there’s loss of information when going from the former to the latter. One is a set of real-valued functions, the other is a set-of-reals-valued function, and they shouldn’t be confused.

2. Vagueness Interpretations: The General Idea

Suppose one of us points towards Bruce the cat and says “look at Bruce!” Presumably, there’s some indeterminacy as to what ‘Bruce’ picks out. In the vicinity of the space where we’re pointing there will be many precise cat-like things, Bruce₁, Bruce₂, Bruce₃, …, differing from one another by molecule here or fraction of a whisker there. We’re not really referring to any one of them in particular, though we’re not referring to none of them either. Rather, one might imagine that each serves as a potential referent for ‘Bruce’ and it’s simply undecided which one it should be. Or at least that’s a plausible way to look at things. So say that each of Bruce₁, Bruce₂, Bruce₃, … , is a precisification of what we might mean by ‘Bruce,’ the kind of thing we would be referring to if we were to somehow make our language perfectly precise. Say also, at least to begin, that anything true relative to all such precisifications of our language is true simpliciter, whereas if something is true relative to some precisifications and false on others then it’s indeterminate. Call this the supervaluationist rule.

According to vagueness interpretations, representors represent vagueness in our degrees of belief, and they do so via the same supervaluationist rule (or something very much like it). On the simplest versions of the interpretation, Sally’s representor ${\displaystyle \mathcal{R}}$ contains all and only the probability measures ${\displaystyle P}$ such that, if we were to suitably precisify our language, then ${\displaystyle P}$ would characterise Sally’s beliefs as per the traditional model. For instance, if ${\displaystyle {\mathcal{R}}^s(p)=[0.4,0.5]}$, then Sally’s degree of belief regarding ${\displaystyle p}$ is determinately between ${\displaystyle 0.4}$ and ${\displaystyle 0.5}$ inclusive, but for any more precise degree within that interval it’s going to be indeterminate whether that is the degree to which Sally believes ${\displaystyle p}$. It would be normal in this case for fans of the vagueness interpretation to say that Sally’s beliefs are “vague over the ${\displaystyle [0.4,0.5]}$ interval.” The presumptions, note, are that (i) each of the ${\displaystyle P}$ in ${\displaystyle \mathcal{R}}$ has independent representational import, and (ii) none them determinately misrepresents Sally’s beliefs—in the sense that if ${\displaystyle \mathcal{R}=\{P_1,P_2,\dots \}}$, then according to ${\displaystyle \mathcal{R}}$ it’s indeterminate whether ${\displaystyle P_1}$ represents her doxastic state, or ${\displaystyle P_2}$ does, and so on. Every function in ${\displaystyle \mathcal{R}}$ represents its own thing, and none of them get things determinately wrong. (This will be important.)

Of course there are other ways one could flesh out the details here. Traditional supervaluationism says that truth is truth-under-all-precisifications, and something that’s true on some precisifications but false on others will simply lack a truth-value. Degree-theoretic supervaluationism says that if something is true on all precisifications then it’s 100% true, 0% true if it’s false on all precisifications, and some middling degree of truth otherwise. There can also be variation regarding whether the vagueness is taken to result from semantic indecision, or is instead a feature of the belief system itself and independent of how we talk about it. So there isn’t really one vagueness interpretation, but a family of them. The differences shouldn’t matter for my purposes.

One can find examples of the vagueness interpretation in (van Fraasen 1990; 2006), (Hájek 2003), (Rinard 2015), and (Levinstein 2019). Hájek and Smithson (2012: §3) and Joyce (2010) also present what could be interpreted as instances of the vagueness interpretation, at least under some precisifications. One could also characterise another, strictly broader class of interpretations—the supervaluational interpretations—characterised by their shared application of some supervaluationist logic or other. For example, Williams (2014) interprets sets of probability functions as representing the range of precise attitudes one may rationally take towards a metaphysically indeterminate proposition. Similarly, one might take a representor to represent the rationally permissible precise belief states relative to an agent’s evidence, where the facts about rational permissibility are themselves indeterminate. But these are only superficially similar to what I’m calling the vagueness interpretations. The difference is that vagueness interpretations are concerned with representing vagueness or indeterminacy relating directly to degrees of belief themselves, as opposed to representing vagueness or indeterminacy in connection to which precise degrees of belief are rational.

3. Vagueness Interpretations: The Problem

Let me start with a mention of the problem I’m not going to talk much about. In particular, you might worry that the simple vagueness interpretation is a bit too simple, and applying the supervaluationist rule too liberally will have some absurd consequences. After all, every member of the “credal committee” says that there’s a unique real value ${\displaystyle r}$ such that Sally is exactly ${\displaystyle r}$ times more confident in ${\displaystyle p}$ than she is in ${\displaystyle q}$, provided she has some positive degree of confidence in both. But this was precisely the sort of thing we were trying to avoid saying!

It’s natural to think that such results are mere artefacts of the formalisation, an inevitable consequence of using a set of precise functions to represent an imprecise state and not to be taken seriously. To suppose otherwise smells a bit like what Lewis once called fanatical supervaluationism, “which automatically applies the supervaluationist rule to any statement whatever, never mind that the statement makes no sense that way” (Lewis 1999: 173). A common response, therefore, is to restrict the application of the rule when reading a representor (e.g., Zynda 2000: 49; Rinard 2017: 267). We might say that ${\displaystyle \mathcal{R}}$ represents as true anything that’s true according to every ${\displaystyle P}$ in ${\displaystyle \mathcal{R}}$, with the exception of those existential claims where no instances hold according to every such ${\displaystyle P}$.

You may or may not be convinced by that response—see (Smith forthcoming) for discussion. Either way, since that problem closely relates to a more general and long-standing issue for supervaluationism that has been thoroughly discussed elsewhere, I want to pursue something different. My concern relates to those representors containing functions that, according to the traditional model, represent belief states that are very different from one another. In short, the problem here isn’t so much that the ${\displaystyle P}$ in ${\displaystyle \mathcal{R}}$ are precise when the goal was to represent something imprecise; the problem, rather, it is that the ${\displaystyle P}$ in ${\displaystyle \mathcal{R}}$ have little in common with what they’re supposed to be representing at all.

To get an initial feel for the problem, consider again the precisifications of ‘Bruce.’ Each of Bruce₁, Bruce₂, Bruce₃, and so on, has very precise boundaries, even though one might intuitively think that Bruce does not have precise boundaries. So there’s at least one respect in which what’s true for every precisification is not plausibly true of Bruce. “Not a problem,” some will say, “We’re not committed to saying that Bruce has precise boundaries, because we’re not committed to applying the supervaluationist rule to every statement whatever.” Grant that the response succeeds. Nevertheless—and this is the important part—in all the ways that really matter, every precisification of ‘Bruce’ is still overall very much Bruce-like. Each one walks like Bruce, each one meows like Bruce, and not a one of them, you’ll observe, looks much like a cassowary. If we were to bundle up all the properties we associate with Bruce, then each of the Bruce_i would satisfy the very large majority of them. They all do a good job of playing the Bruce-role, so they all have a good claim to serve as the extension of that name—that is, they all make sense qua precisifications of ‘Bruce.’ Consequently, whatever the precisifications of ‘Bruce’ might be, they cannot be radically unlike one another with respect to their Bruce-y properties.

Keeping that in mind, contrast two representors: ${\displaystyle {\mathcal{R}}_{\text{narrow}}}$ and ${\displaystyle {\mathcal{R}}_{\text{wide}}}$. The first, ${\displaystyle {\mathcal{R}}_{\text{narrow}}}$, determines only a narrow spread of values for any of the propositions in ${\displaystyle \mathcal{R}}$—let’s say, ${\displaystyle {\mathcal{R}}_{\text{narrow}}^s(p)=[0.339,0.341]}$. Now, I think we have a pretty good working idea of what Sally would be like if she were to believe such-and-such propositions to this or that precise degree. Decision theory, for example, gives us a good sense of how Sally’s degrees of belief impact on her choices. Epistemology gives us a good sense of how Sally’s evidence affects changes in her beliefs and hence her decisions conditional on such evidence. We have, in other words, a reasonable grasp of the main functional role associated with the systems of belief represented by probability functions as per the traditional model. And where two such functions assign similar numerical values, they also tend to play overall similar roles. There’s not a great deal of difference in most decision-theoretic or epistemic contexts between believing ${\displaystyle p}$ to degree ${\displaystyle 0.339}$ and believing it to degree ${\displaystyle 0.341}$. As such, it’s entirely plausible that Sally could be in a state such that her behaviour and behavioural dispositions conditional on evidence are similar to but not quite how we’d expect if she believed ${\displaystyle p}$ to degree 0.339, and also similar to but not quite how we’d expect if she believed ${\displaystyle p}$ to degree 0.341, and likewise for the many values between. The functions in ${\displaystyle {\mathcal{R}}_{\text{narrow}}}$ are all alike to one another, and so I can imagine that they all might represent precisified versions of a state that’s simultaneously similar to all of them. Given this, the vagueness interpretation is clear enough (pun intended) for the case of ${\displaystyle {\mathcal{R}}_{\text{narrow}}}$.

Not so for ${\displaystyle {\mathcal{R}}_{\text{wide}}}$, which this time determines a much wider spread of values for many of the propositions in ${\displaystyle \mathcal{R}}$—say, ${\displaystyle {\mathcal{R}}_{\text{wide}}^s(p)=[0,1]}$. This is often described as having beliefs that are “vague over the entire unit interval”—but what could that mean? As above, I have a good working idea of what Sally’s behaviour (and behaviour conditional on evidence) would be like if she were absolutely certain that ${\displaystyle p}$. And I have a good working idea of what Sally would be like if she were absolutely certain that ${\displaystyle \mathrm{\neg }p}$. There isn’t much similarity between them. And neither is very similar to the case where Sally has 50% confidence towards ${\displaystyle p}$, or 25% confidence. It’s hard to imagine how Sally could be in a state such that she behaves similar to but not quite how we’d expect if she believed ${\displaystyle p}$ to degree ${\displaystyle 0}$, and also similar to but not quite how we’d expect if she believed ${\displaystyle p}$ to degree 1, and likewise for the many values between. It seems, rather, that to be in any state such that ${\displaystyle P(p)=1}$ provides a reasonable precisification thereof is ipso facto to be in a state such that ${\displaystyle P(p)=0}$, or ${\displaystyle P(p)=0.5}$, or ${\displaystyle P(p)=0.25}$, doesn’t.

Don’t say that where Sally’s beliefs are represented by ${\displaystyle {\mathcal{R}}_{\text{wide}}}$, then she’ll be in a state that causes her to be indeterminately disposed between behaving in the ${\displaystyle P(p)=1}$ way, the ${\displaystyle P(p)=0}$ way, the ${\displaystyle P(p)=0.5}$ way, and so on. For what could that really mean other than that Sally isn’t really disposed to behave in any of those ways at all? Imagine that Sally is considering prices for a dollar bet on ${\displaystyle p}$. We could meaningfully say that she’s equally disposed to accept any price between $0 and $1 as fair, or we could say that she lacks a disposition one way or the other, but in either case she’ll be determinately unlike what we’d expect if she had 0% confidence that ${\displaystyle p}$—in that scenario she wouldn’t be willing to pay any price for the bet! And she’ll be determinately unlike what we’d expect if she had 100% confidence in ${\displaystyle p}$. Or 50% confidence. Or 25%.

Clarification one: the problem isn’t that we have no account of the functional role associated with a representor like ${\displaystyle {\mathcal{R}}_{\text{wide}}}$. The decision-theoretic role of ${\displaystyle {\mathcal{R}}_{\text{wide}}}$, or its epistemic role, will be implicit in those theories in which it figures. The problem is that, no matter what the role ends up being, it’s hard to make sense of how all the ${\displaystyle P}$ in ${\displaystyle {\mathcal{R}}_{\text{wide}}}$ can each serve as sensible precisifications of whatever it is that ${\displaystyle {\mathcal{R}}_{\text{wide}}}$ supposedly represents. Those functions are associated with states that are very different from one another in all the ways that matter vis-à-vis beliefs. So at least some of the ${\displaystyle P}$ in ${\displaystyle {\mathcal{R}}_{\text{wide}}}$ will seem to determinately misrepresent Sally’s beliefs, given that each has divergent implications regarding the functional role of that state that cannot all be close to the truth. Better, surely, to say that ${\displaystyle {\mathcal{R}}_{\text{wide}}}$ represents something determinately distinct from anything that might be represented by its members separately.

Clarification two: there are of course many ways we might conceivably make sense of a radically indeterminate doxastic state. Functionalists will sometimes say that an agent could be in a state that occupies the functional role of pain for her even while that same state occupies the role of pleasure for her population, and thus there’s simply no fact of the matter as to whether she’s really in a state of pain or in a state of pleasure. One might imagine saying something like this about believing ${\displaystyle p}$ to degree 0 and believing ${\displaystyle p}$ to degree 1. Or if you buy into quantum indeterminacy, then we could perhaps construct a Schrödinger’s believer scenario where Sally is in a superposition of radically different belief states. No doubt there are other imaginary cases involving broken teletransporters and omnipotent demons and whatnot. But the point here isn’t that there’s no way to make sense of extreme indeterminacy in strength of belief. Rather, the point is that it’s unclear how to make sense of extreme indeterminacy in the cases of interest to advocates of the vagueness interpretation—and they’re typically interested in doxastic indeterminacy as a normal response to incomplete or non-specific evidence, not indeterminacy as a result of this one weird quirk of functionalism and hypothetical quantum mechanics experiments.

Ramsey pointed out long ago that excessive precision in the measurement of belief feels a lot like “working out to seven places of decimals a result only valid to two” (Ramsey 1931: 76). Representors like ${\displaystyle {\mathcal{R}}_{\text{narrow}}}$ can capture this thought nicely. There isn’t much difference, functionally, between believing ${\displaystyle p}$ to degree 0.339 or to degree 0.341, and any plausible epistemology or decision theory is going to treat those states of belief as being generally similar to one another in most respects—and likewise the interval ${\displaystyle [0.339,0.341]}$. So it makes sense to say that Sally’s beliefs are “vague over the ${\displaystyle [0.339,0.341]}$ interval,” in much the same way it makes easy sense to speak of the boundary for ‘tall’ being vague over the interval from about 5^′11^′′ to 6^′1^′′. But talk of beliefs that are vague over the entire unit interval sounds a lot like saying the fuzziness of ‘tall’ extends from the tiniest infants right up to the tallest basketballers. In the case of ${\displaystyle {\mathcal{R}}_{\text{wide}}}$, it’s a lot harder to make sense of indeterminacy as to whether this or that member of the representor represents Sally’s beliefs, given that what’s represented by those precisifications are all very dissimilar from one another—and, consequently, at least some of those precisifications must also represent something determinately unlike whatever they’re supposedly precisifications of.

4. Comparativist Interpretations: The General Idea

According to vagueness interpretations, it’s indeterminate which of the ${\displaystyle P}$ in ${\displaystyle \mathcal{R}}$ are supposed to represent Sally’s beliefs. According to comparativist interpretations, by contrast, in most cases when there’s more than one ${\displaystyle P}$ in ${\displaystyle \mathcal{R}}$, then each of them will determinately misrepresent Sally’s beliefs in some way or another. That’s not a problem, because according to this kind of interpretation it’s the entire set ${\displaystyle \mathcal{R}}$ which does the representing, and no individual ${\displaystyle P}$ within ${\displaystyle \mathcal{R}}$ has any representational import independent of the whole.

But I’m getting ahead of myself. I should start with comparativism, the idea that numerical degrees of belief are really just a way of representing what are ultimately nothing more than relations of relative confidence.² To discuss this, we’ll need some more notation:

${\displaystyle p\succsim q}$ iff Sally is at least as confident that ${\displaystyle p}$ as she is that ${\displaystyle q}$

${\displaystyle p\succ q}$ iff Sally is more confident that ${\displaystyle p}$ than she is that ${\displaystyle q}$

${\displaystyle p\sim q}$ iff Sally is just as confident in ${\displaystyle p}$ as she is in ${\displaystyle q}$

${\displaystyle p▽q}$ iff Sally’s confidence in ${\displaystyle p}$ is incomparable to her confidence in ${\displaystyle q}$

I assume that ${\displaystyle p▽q}$ holds whenever ${\displaystyle p}$ and ${\displaystyle q}$ are not related by ${\displaystyle \succsim }$, ${\displaystyle \succ }$, or ${\displaystyle \sim }$ in either direction, provided of course they both belong to the relevant algebra ${\displaystyle \mathbf{P}}$. I am therefore ignoring the possibility that there may be other non-conventional forms of comparability, such as parity. I also take it for granted that if either ${\displaystyle p\succ q}$ or ${\displaystyle p\sim q}$, then ${\displaystyle p\succsim q}$; that seems analytically true if anything is, and seems to be common ground among comparativists. (The other direction is not so obvious.) Given this, ${\displaystyle p\succsim ̸q}$ implies ${\displaystyle p\nsucc q}$ and ${\displaystyle p\nsim q}$, and so it suffices from now on to say:

${\displaystyle p▽q}$ iff ${\displaystyle p\succsim ̸q}$ and ${\displaystyle q\succsim ̸p}$

Given that, according to the traditional comparativist interpretation of a probability function ${\displaystyle P}$, that function represents the facts about Sally’s beliefs by virtue of representing her comparative confidences—specifically:

${\displaystyle p\succsim q}$ iff ${\displaystyle P(p)\ge P(q)}$

${\displaystyle p\succ q}$ iff ${\displaystyle p\succsim q}$ and ${\displaystyle q\succsim ̸p}$

${\displaystyle p\sim q}$ iff ${\displaystyle p\succsim q}$ and ${\displaystyle q\succsim p}$

The important thing to note here is that there’s no possibility of incomparability. The relation ${\displaystyle \ge }$ over the real numbers is complete in the sense that for any two reals ${\displaystyle x,y}$, either ${\displaystyle x\ge y}$ or ${\displaystyle y\ge x}$; hence, any function ${\displaystyle P:\mathbf{P}\mapsto [0,1]}$ interpreted as above automatically represents ${\displaystyle \succsim }$ as being likewise complete over ${\displaystyle \mathbf{P}}$.

Representors provide an alternative means of representing comparative confidence relations, with the benefit of allowing for incompleteness and hence for representing incomparability. Or rather: representors provide several distinct ways of representing potentially incomplete confidence relations, corresponding to several varieties of comparativist interpretation. Again, there’s not really a single interpretation here, but a family of them. One way to capture the differences is in terms of which of ${\displaystyle \succsim }$, ${\displaystyle \succ }$ and/or ${\displaystyle \sim }$ are treated as definitional primitives. On the traditional single-function model, it’s typical to let ${\displaystyle \succsim }$ be the uniquely primitive confidence relation, and simply define ${\displaystyle \succ }$ and ${\displaystyle \sim }$ as its asymmetric and symmetric parts respectively. This is what I did above, but it’s not the only way I could have done it. I could just as easily have let ${\displaystyle \sim }$ and ${\displaystyle \succ }$ be the primitive relations, and then defined ${\displaystyle \succsim }$ as the disjunction of the two (i.e., ${\displaystyle p\succsim q}$ iff ${\displaystyle p\succ q}$ or ${\displaystyle p\sim q}$). Or I could have treated ${\displaystyle \succsim }$ and ${\displaystyle \succ }$ as the primitives and used them to define ${\displaystyle \sim }$ (e.g., ${\displaystyle p\sim q}$ iff ${\displaystyle p\succsim q}$ and ${\displaystyle q\succsim p}$, or iff ${\displaystyle p\nsucc q}$ and ${\displaystyle q\nsucc p}$). Or I could have let all three be considered independently primitive. The point is that it doesn’t matter—it’ll make no difference at all when it comes to reading any real-valued function ${\displaystyle P}$ as a representation of Sally’s comparative confidences. But these choices do make a difference when we shift over to the representor model.

One comparativist interpretation of a representor treats ${\displaystyle \succsim }$ as the unique primitive. On this interpretation we say that ${\displaystyle \mathcal{R}}$ represents that ${\displaystyle p\succsim q}$ just in case every function in ${\displaystyle \mathcal{R}}$ agrees that ${\displaystyle p}$ is at least as probable than ${\displaystyle q}$, and then we let ${\displaystyle \sim }$ and ${\displaystyle \succ }$ be defined as the symmetric and asymmetric parts of ${\displaystyle \succsim }$ as usual. Call this the ${\displaystyle \succsim }$-interpretation:

${\displaystyle p\succsim q}$ iff ${\displaystyle \mathrm{\forall }P\in \mathcal{R}:P(p)\ge P(q)}$

${\displaystyle p\succ q}$ iff ${\displaystyle p\succsim q}$ and ${\displaystyle q\succsim ̸p}$

${\displaystyle p\sim q}$ iff ${\displaystyle p\succsim q}$ and ${\displaystyle q\succsim p}$

Consequence: ${\displaystyle p\succ q}$ just in case ${\displaystyle P(p)\ge P(q)}$ for all ${\displaystyle P}$ in ${\displaystyle \mathcal{R}}$, with ${\displaystyle P(p)>P(q)}$ for at least some but not necessarily all of them. This is probably the most common way of reading a set of probability functions as a representation of comparative probability relations. Or, at least, it’s the way that comes up most often in the literature, to the extent that the intended interpretation is ever explicitly and unambiguously characterised. See, for example, (Nehring 2009), (Alon & Lehrer 2014), (Miranda & Destercke 2015) (Harrison-Trainor & Holliday 2016), (Harrison-Trainor et al. 2018), (Konek 2019), (Ding et al. 2021), and (Eva & Stern 2023). We can also find a version of the ${\displaystyle \succsim }$-interpretation in Kaplan’s ‘Modest Probabilism” (1996; 2002; 2010).³

By contrast, Eva (2019: 394–395) puts forward a distinct (though obviously similar) comparativist interpretation, according to which ${\displaystyle \succ }$ and ${\displaystyle \sim }$ are definitionally primitive and ${\displaystyle \succsim }$ is their disjunction. Call this the ${\displaystyle \succ }$/${\displaystyle \sim }$-interpretation:

${\displaystyle p\succsim q}$ iff ${\displaystyle p\succ q}$ or ${\displaystyle p\sim q}$

${\displaystyle p\succ q}$ iff ${\displaystyle \mathrm{\forall }P\in \mathcal{R}:P(p)>P(q)}$

${\displaystyle p\sim q}$ iff ${\displaystyle \mathrm{\forall }P\in \mathcal{R}:P(p)=P(q)}$

But wait—there’s more! Builes et al. (2022) seem to put forward what we can call the ${\displaystyle \succsim }$/${\displaystyle \succ }$-interpretation:⁴

${\displaystyle p\succsim q}$ iff ${\displaystyle \mathrm{\forall }P\in \mathcal{R}:P(p)\ge P(q)}$

${\displaystyle p\succ q}$ iff ${\displaystyle \mathrm{\forall }P\in \mathcal{R}:P(p)>P(q)}$

${\displaystyle p\sim q}$ iff ${\displaystyle p\succsim q}$ and ${\displaystyle q\succsim p}$

It likely won’t be immediately obvious what the impact of these differences will be, but an example will help. Imagine that Sally has been given a coin by a magician, and has been asked to toss it twice. She knows that magicians’ coins are often biased, though not always, and if it is biased then it’ll be highly variable in which direction and to what extent. As far as she knows, it could be completely biased towards heads, or completely biased towards tails, or anything between.

Given this, we might decide to represent Sally’s beliefs by means of a representor, call it ${\displaystyle {\mathcal{R}}_{\text{coin}}}$, such that if

${\displaystyle p}$ ${\displaystyle =}$ The coin will land heads on the next toss

${\displaystyle q}$ ${\displaystyle =}$ The coin will land heads on both of the next two tosses

then for all ${\displaystyle P}$ in ${\displaystyle {\mathcal{R}}_{\text{coin}}}$,

and

Don’t worry about whether you think this is the right way to represent Sally’s beliefs in this situation; the important point for the example is that ${\displaystyle P(p)=P(q)}$ only where ${\displaystyle P(p)=1}$ or ${\displaystyle P(p)=0}$, and otherwise ${\displaystyle P(p)>P(q)}$. Now, since every ${\displaystyle P}$ in ${\displaystyle {\mathcal{R}}_{\text{coin}}}$ agrees on ${\displaystyle P(p)\ge P(q)}$, but they don’t all agree on ${\displaystyle P(q)\ge P(p)}$, then according to the ${\displaystyle \succsim }$-interpretation we should read ${\displaystyle {\mathcal{R}}_{\text{coin}}}$ as saying:

On the other hand, since neither ${\displaystyle P(p)>P(q)}$ nor ${\displaystyle P(p)=P(q)}$ for all ${\displaystyle P}$ in ${\displaystyle {\mathcal{R}}_{\text{coin}}}$, on the ${\displaystyle \succ }$/${\displaystyle \sim }$-interpretation we read ${\displaystyle {\mathcal{R}}_{\text{coin}}}$ as saying that ${\displaystyle p}$ and ${\displaystyle q}$ are incomparable:

And on the other other hand, the ${\displaystyle \succsim }$/${\displaystyle \succ }$-interpretation reads ${\displaystyle {\mathcal{R}}_{\text{coin}}}$ as saying:

The foregoing is useful for highlighting the dangers arising from unreflective reliance on the credal committee metaphor. According to the ${\displaystyle \succ }$/${\displaystyle \sim }$-interpretation and the ${\displaystyle \succsim }$/${\displaystyle \succ }$-interpretation, every voter on the committee needs to agree that ${\displaystyle p\succ q}$ in order for ${\displaystyle p\succ q}$ to be true, in line with what the metaphor suggests, but not so for the ${\displaystyle \succsim }$-interpretation. Likewise, if every voter agrees that ${\displaystyle p\succsim q}$, then ${\displaystyle p\succsim q}$ according to the ${\displaystyle \succsim }$-interpretation and the ${\displaystyle \succsim }$/${\displaystyle \succ }$-interpretation, but not always according to the ${\displaystyle \succ }$/${\displaystyle \sim }$-interpretation. And while all three comparativist interpretations agree that ${\displaystyle \mathcal{R}}$ represents ${\displaystyle p\sim q}$ just in case everyone on the committee votes ${\displaystyle p\sim q}$, they also all imply that ${\displaystyle p\nsim q}$ inasmuch as a single voter puts their hand up for either ${\displaystyle p\succ q}$ or ${\displaystyle q\succ p}$—which contrasts with the vagueness interpretations, according to which ${\displaystyle p\nsim q}$ only when every committee member votes for ${\displaystyle p\succ q}$ or ${\displaystyle q\succ p}$. Everyone can agree that the metaphor gets some things right and some things wrong, but good luck getting them to agree on what.

5. Comparativist Interpretations: The Problem

Distinguish comparativist interpretations from comparativism. The former are just a way of reading representors, and it’s uncontroversial that representors can be used to represent incomplete confidence orderings. Comparativists advocate something stronger: that’s all a representor needs to represent, because those states of comparative confidence are what ultimately comprise our systems of belief. What matters is just the ordering: ordinal-equivalence is meaning-equivalence. That, as they say, is what’s real; aught else is just an artefact of the numerical representation. The question for us is whether we might reasonably want representors to represent something more.

One of the most frequently cited motivations for comparativism is the plausible idea that there seems to be nothing about our beliefs that calls for a unique numerical representation, or any numerical representation at all for that matter. Comparativism is in a position to explain this, and that is an explanatory virtue of the view (see, e.g., Koopman 1940a: 269; Fine 1973: 15; Zynda 2000: 64–67; Stefánsson 2017). Builes et al. summarise this idea nicely:

Comparativism is based on the intuitive thought that while numerical probabilities represent belief states, there’s nothing about our belief states that mandates a unique numerical representation. In other words, there’s nothing “0.69-ish” about my degree of confidence in ${\displaystyle p}$, beyond the fact that 0.69 can serve as an adequate representation of my degree of confidence within a particular representational system. But 69, for example, or 732.6 for that matter, would work just as well, provided the system was structured in the right way. (2022: 7)

A similar motivation is that comparativism is able to capture and explain certain intuitive possibilities that don’t play nicely with traditional real-valued representations of belief—for instance, (a) that Sally might have more confidence in ${\displaystyle p}$ than she does in ${\displaystyle q}$, without there being any particular degree to which she has more confidence in ${\displaystyle p}$; and (b) that Sally’s confidence in ${\displaystyle p}$ need not be more, less, or equal to her confidence in ${\displaystyle q}$, since ${\displaystyle p}$ and ${\displaystyle q}$ may be incomparable. Fine briefly mentions something along these lines as a reason for adopting purely relational models of belief:

(2) [Comparative probability] provides a wider class of models of random phenomena than does the usual [i.e., precise, real-valued] quantitative theory… Point (2) refers to the curious phenomenon that there exist relatively simple examples of what we consider to be valid [comparative probability] statements that are incompatible with any representation in the usual quantitative theory. (Fine 1973: 15–16)

Now all this would provide a very compelling motivation for comparativism indeed, if comparativism were in a unique position to capture these intuitively plausible thoughts. But it isn’t. Non-comparativists can say these sorts of things too—and indeed they do! The presupposition behind the proposed motivation seems to be that the only theoretical option besides comparativism is the view that degrees of belief correspond to unique real numbers literally inscribed somewhere inside the head. In reality, though, non-comparativists typically just agree that the particular numbers we use are just one way among many for numerically representing a qualitative psychological system, presumably by virtue of some structural similarity, and that the psychological systems being represented need not always have the kind of structure that allows for representation with precise real numbers. All this is common ground. What isn’t common ground is whether the qualitative psychological systems being represented by a (precise or imprecise) numerical model of belief can be fully characterised in terms of comparative confidence relations. That’s the debate.⁵

But there’s an argument for comparativism in the nearby vicinity that’s worth considering in more detail. Let absolute degrees of belief be the kinds of doxastic attitudes that relate an agent to a proposition and a degree (which may or may not be represented numerically); for instance, those attributed when we say that Sally is very confident that ${\displaystyle p}$, is certain that ${\displaystyle p}$, or believes ${\displaystyle p}$ to degree 0.69, and so on. Premise one: the very notion that there are degrees of belief presupposes a minimal relational structure—there must at least be a transitive and reflexive ordering over them, as anything less and we’d be stretching the concept of degrees beyond recognition. Premise two: any system of absolute degrees of belief immediately determines a corresponding system of comparative confidences—for instance, if Sally’s degree of belief in ${\displaystyle p}$ is greater than her degree of belief in ${\displaystyle q}$, then ${\displaystyle p\succ q}$. Premise three: a non-transitive system of comparative confidences seems to be possible. So there seem to be possible systems of comparative confidence that correspond to no possible system of absolute degrees of belief. So absolute degrees of belief are not plausibly more fundamental than comparative confidence; facts about the latter do not supervene on facts about the former.

I’m inclined to accept the conclusion. But that’s not enough yet to conclude that comparative confidences are more fundamental than absolute degrees of belief. For one thing, it may be that neither is more fundamental than the other—perhaps the facts about both fall out simultaneously from the facts about some third species of doxastic state, such as outright beliefs. Or maybe, as Lewis often suggested (e.g., 1986: 36–37; 1994: 430), we can see the system of beliefs as a whole as comprising the fundamental doxastic unit. On this picture—which I’ll advocate in §7—comparative confidences and absolute degrees of belief both are just ways of describing salient aspects of a total doxastic state characterised by its functional role in connection to evidence and behaviour. Or, instead of positing a third state from which both comparative confidence and absolute degrees of belief derive, you might suppose that comparative confidence is merely one among a plurality of primitive kinds of doxastic state. The other primitives may include, e.g., judgements as to when propositions are evidentially independent of one another, or states of certainty, or full belief, and so on. Konek (2019: 308–314) refers to this as the pluralist view—that there’s more than one species of primitive doxastic state, such that the facts about our beliefs cannot all be reduced just to facts about comparative confidence. Many have recommended adding at least a primitive qualitative independence relation alongside comparative confidence as one of the basic doxastic relations that are normally represented by our precise probabilistic models of belief,⁶ and Joyce (2010) suggests an interpretation of representors like this. (I’ll say more about pluralism in the next section.)

So the explanatory motivations that are usually put forward for accepting comparativism are not very compelling—there are some intuitive possibilities that comparativism is able to explain, that’s true, but it’s not uniquely positioned to explain those possibilities. Moreover, a wide range of contemporary theories that employ (precise or imprecise) numerical representations of belief make regular appeal to extra-ordinal properties of those representations that cannot be taken to represent anything expressible wholly in terms of comparative confidence. Generally speaking, the role played by our numerical representations of belief in the bulk of current theory requires those representations to carry more than merely ordinal information. Given this, we have at least some reason for wanting more from an interpretation than the comparativist interpretations give us.

I cannot discuss every example in detail, but I can look at one very simple case in decision theory. Let ${\displaystyle \mathbf{P}=\{\mathrm{\Omega },p,\mathrm{\neg }p,\varnothing \}}$, with ${\displaystyle \mathrm{\Omega }\succ p\succ \mathrm{\neg }p\succ \varnothing }$. According to the interpretations I’ve discussed, a representor ${\displaystyle \mathcal{R}}$ will determine these comparative confidences if (but not always only if), for all ${\displaystyle P}$ in ${\displaystyle \mathcal{R}}$, ${\displaystyle 1>P(p)>P(\mathrm{\neg }p)>0}$. This will include any ${\displaystyle \mathcal{R}}$ such that ${\displaystyle {\mathcal{R}}^s(p)\subseteq (0.5,1)}$. So according to comparativism, these representors all represent the same system of beliefs.⁷ But now take any decision theory for imprecise probabilities that generalises expected utility theory, in the sense that it includes that theory as a special case whenever ${\displaystyle \mathcal{R}}$ is singleton. This covers all the well-known theories—${\displaystyle \mathrm{\Gamma }}$-maximin, E-admissibility, maximality or interval dominance. (See Troffaes 2007 for an overview.) That theory will entail that there is a decision-theoretically relevant difference between at least some, if not all, of these “ordinally equivalent” representors. For instance, imagine that Sally is choosing between two gambles:

${\displaystyle \alpha :}$ receive $1 if ${\displaystyle p}$ is true, nothing otherwise

${\displaystyle \beta :}$ receive $2 if ${\displaystyle p}$ is false, nothing otherwise

Which should she choose? Case 1: if ${\displaystyle \mathit{\text{min}}[{\mathcal{R}}^s(p)]>2/3}$, then Sally should prefer gamble ${\displaystyle \alpha }$. Case 2: if , then Sally should prefer gamble ${\displaystyle \beta }$. Case 3: if ${\displaystyle \mathit{\text{max}}[{\mathcal{R}}^s(p)]>2/3>\mathit{\text{min}}[{\mathcal{R}}^s(p)]}$, then depending on the theory she might either be indifferent between the two gambles, prefer ${\displaystyle \alpha }$ to ${\displaystyle \beta }$, prefer ${\displaystyle \beta }$ to ${\displaystyle \alpha }$, or lack a preference—either way it’ll differ from either Case 1 or Case 2.

There’s nothing controversial about this. It’s well-known that decision theories for “precise” probabilities usually attribute differential import to ordinally equivalent representations of belief. Preferences licensed by some probability function when conjoined with a utility function need not be licensed by another probability function when conjoined with that same utility function, even if the two probability functions determine the same confidence ordering. This is true for normative theories (e.g., expected utility theory, or risk-weighted utility theory) and descriptive theories (e.g., cumulative prospect theory). The same extends to decision theories for “imprecise” probabilities. That is: pairs of representors that determine the same confidence orderings can and often do carry differential import for some decision situations, according to these theories.

And it’s no different outside of decision theory. Epistemology supplies more examples—to say nothing yet of game theory, information theory, or linguistics. Probabilistic independence is centrally important for our theories of evidence and learning, but it’s long been known that independence cannot generally be defined in terms of binary comparative confidence.⁸ Theories of peer disagreement require interpersonal comparisons of confidence, which seem especially tricky for comparativists to explain. (See Elliott 2022b for the feasibility of interpersonal confidence comparisons within a comparativist framework.) The Principal Principle presupposes extra-ordinal distinctions between rational belief states, given that there are meaningfully distinct but ordinally equivalent objective chance functions. And so on, and on, and on. (And on.) In short: across a wide range of contexts, in decision theory and elsewhere, numerical representations of belief are attributed theoretical roles that require them to carry meaningful extra-ordinal information.

“So what?,” the inevitable interjection goes, before the argument is yet complete. “Aren’t you just presupposing that all these theories are correct in appealing to this extra-ordinal information, and therefore begging the question against comparativism? And don’t we have reason already to suppose these theories often help themselves to more information than they’re entitled, as for instance when standard decision theories represent decision-makers as having complete awareness of their state-space? Our current theories are rife with idealisation—so what’s to stop comparativists from simply saying that inasmuch as these theories make use of extra-ordinal information, then this is yet another idealisation not to be taken too seriously?”⁹

In response, we can focus on the forest or on the trees. Start with the trees. While it’s clearly true that current theories of belief and decision-making are over-idealising in many respects when applied to ordinary agents—as when they presume full awareness, or very precise gradations in degree of belief—those idealisations nevertheless seem to have a sensible interpretation in scenarios involving idealised but still possible agents. While we don’t have full awareness of our state-space, it’s not impossible that someone could. While we might not have very precise degrees of belief, it’s not impossible that someone could. The “extra information” isn’t meaningless—it’s not a mere artefact of the representation, but something with a legitimate role to play in conceivable scenarios. The situation with comparativism is quite different. Comparativists aren’t saying that there’s information encoded in a probability function, or in a representor, which makes sense for ideal agents but not for us. Rather, comparativists are saying that if a theory treats two numerical models of belief differently even though they determine the same confidence orderings, then the theory is appealing to meaningless information that doesn’t have a proper interpretation for any scenario. And while I’ll happily concede that contemporary theories of decision-making and belief update and whatnot are unrealistic for agents like us, that fact doesn’t yet give me a reason to suppose that extra-ordinal information has no meaningful role to play.

Now the forest: the argument begs no questions against comparativism, since it neither concludes with nor is premised on anything implying the falsity of that view. Here is a summary of the whole thing, which should help:

Across a wide range of theoretical contexts, ordinally equivalent representations of belief are typically attributed differential import. It is (at best) unclear whether our current theories can be revised so to fit with comparativism without significant loss in terms of, inter alia, fit with empirical data, fit with reflective intuitions about rationality, explanatory power and capacity for integration with adjacent theories—the usual theoretical virtues.¹⁰ So, while it’s consistent with these facts that comparativism might still be correct, given the present state of theorising it is ceteris paribus reasonable to doubt that numerical representations of belief represent only comparative confidence orderings.

I said this earlier, and now I’ll say it again: my goal is not to convince you that comparativism is mistaken. It’s not clear it is mistaken. However, it’s not clear that comparativism is correct either. That’s the point. The goal is to provide reasons in favour of an interpretation that’s capable of saying more than comparativism allows. The general schema of functional interpretation that I propose below is consistent with the possibility that comparativism is correct, but it’s also consistent with the opposite possibility—and that’s a good thing.

The fact that our current theories overwhelmingly tend to presuppose the meaningfulness of extra-ordinal information is a reason to think that such information is meaningful. It’s not a conclusive reason by any means, but it’s certainly not negligible either. By way of analogy, consider John Wheeler’s (1964; 1980) pregeometry programme. According to Wheeler, our theories of space and time should be reconstructed in such a manner as “breaks loose at the start from all mention of geometry and distance” (1980: 3–4)—that is, without presupposing the meaningfulness of any essentially geometric structures and concepts (even to the point of giving up the concept of distance), and weakening common assumptions about the nature of spacetime (such as continuity). There have been some limited attempts in this direction, such as replacing continuous spacetime in special relativity theory with a weaker discrete spacetime, with some partial successes. (See Meschini et al. 2004 for a user-friendly overview.) However, we’re far from having anything approaching a general theory of spacetime that doesn’t presuppose the meaningfulness of a very good deal of classical geometric concepts, and at this stage it’s not at all clear whether such a thing is really feasible. Those facts seem to suffice for taking very seriously the possibility that certain basic geometric concepts and assumptions really are essential to our physical theories, and for adopting such assumptions as reasonable working hypotheses.

Back to comparativism. I’ve argued that, as things stand, there don’t seem to be any interesting possibilities that comparativism is in a unique position to explain, and moreover comparativism sits ill-at-ease with the fact that numerically distinct but ordinally equivalent representations of belief are typically afforded differential import across a very wide range of contemporary theoretical contexts. These facts may change. Maybe better theories will be developed, which do not appeal to extra-ordinal information in the numerical representation of belief. Until such time, I’m going to take the widely presumed meaningfulness of extra-ordinal information implicit in the bulk of present theorising at face value, as telling us something important about the nature of belief.

6. Beyond Doxastic Structure

Numerical models of belief are representations of some qualitative psychological system, presumably by virtue of their possessing some similarity of structure. On this I agree with the comparativists. The interesting debate is not between those who do and do not think that the numbers we happen to employ when ascribing degrees of belief are representations of some underlying psychological structure; rather, it concerns what that structure is. What, in other words, are the qualitative psychological properties and relations captured by our numerical representations of belief, and what, therefore, are the properties of those representations which must be shared among any alternatives with an equal claim to representational adequacy?

There’s two main ways we might think about these questions. On the one hand, we might think that what’s being represented can be fully characterised in terms of qualitative doxastic concepts and relations. Let’s refer to that as a qualitative doxastic structure. When comparativists say that probability functions and representors represent comparative confidence orderings, they’re referring to a qualitative doxastic structure in this sense. When pluralists say that our numerical representations of belief represent some other primitive doxastic states as well, such as qualitative independence relations, they’re positing a richer qualitative structure but still an essentially doxastic structure. A rather different approach is to suppose that the numbers represent not so much the internal structure of the belief system itself, or not only that, but also something about how our beliefs relate to certain other psychological phenomena—preferences, actions, or evidence, for example. What’s real may in part be a matter of the role the system of beliefs plays in our broader psychological economy.

That’s all very abstract and not a little vague, and to explain it fully I’ll need to take a detour through some measurement theory. Consider first the familiar story of length. Lengths are standardly measured on a ratio scale, which is to say that transformations between all the normal measures of length (meters, feet, miles, parsecs, etc.) always preserve ratios. This is not idle stipulation: ratios of lengths on these measures have genuine physical meaning, and that meaning can be appreciated simply by considering how lengths relate to one another—that is, without considering how lengths relate to other quantities. If Spot the dog is twice as long as Bruce the cat, then if we were to have two copies of Bruce and line them up them head-to-tail, their combined length would be as long as Spot. And if Harry the hamster is two-thirds as long as Bruce, then three copies of Harry should be as long as two copies of Bruce.

A bit more formally, let ${\displaystyle \langle \mathbf{O},\succsim ,\circ \rangle }$ be the qualitative length structure, where ${\displaystyle \mathbf{O}}$ is the set of physical objects, ${\displaystyle \succsim }$ is the at least as long relation, and ${\displaystyle \circ }$ is a concatenation operation with ${\displaystyle a\circ b\sim c}$ meaning that if ${\displaystyle a}$ and ${\displaystyle b}$ are lined up end-to-end then the result will be as long as ${\displaystyle c}$. The standard measures of length all correspond to structure-preserving mappings from the qualitative length system ${\displaystyle \langle \mathbf{O},\succsim ,\circ \rangle }$ into the numerical system ${\displaystyle \langle {\mathbb{R}}^{\ge 0},\ge ,+\rangle }$, where ${\displaystyle {\mathbb{R}}^{\ge 0}}$ is the non-negative reals and ${\displaystyle \ge }$ and ${\displaystyle +}$ have their usual interpretations. That is, ${\displaystyle L:\mathbf{O}\mapsto {\mathbb{R}}^{\ge 0}}$ is a structure-preserving mapping from ${\displaystyle \langle \mathbf{O},\succsim ,\circ \rangle }$ into ${\displaystyle \langle {\mathbb{R}}^{\ge 0},\ge ,+\rangle }$ when, for all ${\displaystyle a,b,c}$ in ${\displaystyle \mathbf{O}}$,

${\displaystyle a\succsim b}$   iff ${\displaystyle L(a)\ge L(b)}$

${\displaystyle a\circ b\sim c}$ iff ${\displaystyle L(a)+L(b)=L(c)}$

Call this an additive representation, since it maps the concatenation operation into addition. Ratios are meaningful relative to additive representations of length, and that meaning is reflected directly in what’s invariant across all such representations: if ${\displaystyle L}$ maps ${\displaystyle \langle \mathbf{O},\succsim ,\circ \rangle }$ into ${\displaystyle \langle {\mathbb{R}}^{\ge 0},\ge ,+\rangle }$, then so too does ${\displaystyle L^{\prime }}$ just in case ${\displaystyle L^{\prime }}$ and ${\displaystyle L}$ are related by a ratio-preserving transformation.

The important thing to note about the example is that the underlying qualitative structure is characterised in terms of relations between lengths, without reference to any other quantities. As a result, and to put the point roughly, it is possible to explain the qualitative meaning of length ratios wholly in terms of how lengths relate to other lengths. So, if Spot is twice as long as Bruce, then that’s because ${\displaystyle \text{Bruce}\circ \text{Bruce}\sim \text{Spot}}$; and if Harry is two-thirds as long as Bruce, then that’s because ${\displaystyle \text{Bruce}\circ \text{Bruce}\sim \text{Harry}\circ \text{Harry}\circ \text{Harry}}$.

Comparativists and pluralists alike suppose that the numerical representation of belief is much like the measurement of length in this respect. On the simplest versions of traditional comparativism, the idea is that a real-valued measure ${\displaystyle P}$ maps ${\displaystyle \langle \mathbf{P},\succsim \rangle }$ into ${\displaystyle \langle {\mathbb{R}}^{\ge 0},\ge \rangle }$, such that ${\displaystyle p\succsim q}$ iff ${\displaystyle P(p)\ge P(q)}$. More sophisticated versions of the view add that the union of disjoint propositions behaves a lot like the concatenation operation in the case of length, and should therefore be mapped into addition: if ${\displaystyle p\cap q=\varnothing }$, then ${\displaystyle P(p\cup q)=P(p)+P(q)}$. Pluralists enrich the underlying qualitative system still further with additional doxastic relations to be captured by the numerical representation—e.g., if ${\displaystyle \mathrm{\perp }}$ is a qualitative independence relation, then ${\displaystyle p\mathrm{\perp }q}$ should imply ${\displaystyle P(p\cap q)=P(p)\cdot P(q)}$. The “imprecise” versions of these views replace the single measure with a set of measures, and then say that the qualitative relations represented are those determined in common by all measures in the set. In all these cases, what’s supposedly being represented is a qualitative doxastic structure, characterised in doxastic terms and without reference to other non-doxastic parts of our psychology.

But things don’t have to work this way. The case of length is only one model by which the measurement of belief might be understood, and it is not always possible to appreciate what a numerical model of some phenomenon represents without understanding how that phenomenon systematically interacts with others as part of a broader system. The theory of conjoint measurement was developed to explain how relations between quantities can give rise to meaningful information that’s not apparent when each is considered in isolation (Debreu 1960; Luce & Tukey 1964; Krantz et al. 1971). Imagine two quantities A and B, lacking in any of the apparent intrinsic structure had by the qualitative system of lengths. Still we might consider how A and B trade-off to produce varying levels in some third quantity, C, and from there extract meaning. For ${\displaystyle i,j=1,2,\dots }$, let ${\displaystyle a_i,a_j}$ be distinct levels of A and ${\displaystyle b_i,b_j}$ distinct levels of B. Furthermore, let ${\displaystyle \succsim }$ now be a partial order over C, and let ${\displaystyle a_i{b}_j}$ be the level of C produced by ${\displaystyle a_i}$ and ${\displaystyle b_j}$. Assume that A and B combine to determine C in an intuitively “additive” way. This assumption can be rigorously characterised in purely qualitative terms, but basically amounts to a sequence of independence conditions on the structure of ${\displaystyle \succsim }$—for example, if ${\displaystyle a_2{b}_i\succsim a_1{b}_i}$ for some ${\displaystyle b_i}$, then ${\displaystyle a_2{b}_i\succsim a_1{b}_i}$ for all ${\displaystyle b_i}$, so the contribution to C made by A is independent of the contribution made by B. Given that ${\displaystyle \succsim }$ has the appropriate structure, we can then extract an ordering over A: say that ${\displaystyle a_2}$ is more than ${\displaystyle a_1}$ just in case ${\displaystyle a_2{b}_i\succ a_1{b}_i}$ for some ${\displaystyle b_i}$. Moreover, we can define ratios of differences in A. Suppose that ${\displaystyle a_1{b}_2\sim a_2{b}_1\succ a_1{b}_1}$. We read this as saying that the change from ${\displaystyle a_1}$ to ${\displaystyle a_2}$ (holding B fixed) produces the same effect in C as the change from ${\displaystyle b_1}$ to ${\displaystyle b_2}$ (holding A fixed); hence, the difference between ${\displaystyle a_2{b}_2}$ and ${\displaystyle a_1{b}_1}$ is twice that between ${\displaystyle a_1{b}_2}$ and ${\displaystyle a_1{b}_1}$, or between ${\displaystyle a_2{b}_1}$ and ${\displaystyle a_1{b}_1}$. Now if ${\displaystyle a_2{b}_2\sim a_3{b}_1}$, then the difference between ${\displaystyle a_1}$ and ${\displaystyle a_3}$, in terms of the contribution to C, is twice the difference between ${\displaystyle a_1}$ and ${\displaystyle a_2}$.

These ratios in relation to A have real meaning, but unlike the case of length that meaning need not correspond to any natural qualitative operations comprehensible in A-terms alone—the meaning is manifest, rather, in the relationships between A, B and C. For a conjoint measurement of a system like this, the goal will be to construct two separate functions, ${\displaystyle A:\mathbf{A}\mapsto \mathbb{R}}$ and ${\displaystyle B:\mathbf{B}\mapsto \mathbb{R}}$ which combine via a specified operation ${\displaystyle F:\mathbb{R}\times \mathbb{R}\mapsto \mathbb{R}}$ to determine a function ${\displaystyle C:\mathbf{C}\mapsto \mathbb{R}}$ such that ${\displaystyle C(a_i{b}_j)=F[A(a_i),B(b_j)]}$ and

Any alternative measures in the same representational system must preserve this relation (as captured by the rule ${\displaystyle F}$) between the three quantities; this constrains what counts as a permissible transformation of each function individually. Think of the numerical representations of A, B and C as a package deal; or, better, as parts of a single representation comprising three functions and a rule linking them together. What’s meaningful in ${\displaystyle A}$, then, will be tied up in how that function relates to the rest of the conjoint representation.¹¹

To help illustrate this in the case of belief, consider next a well-known example from Lyle Zynda (2000). Let actions be represented in the usual way as a functions (${\displaystyle \alpha }$, ${\displaystyle \beta }$, …) from states (${\displaystyle s_i}$, ${\displaystyle i=1,2,\dots }$) to consequences, and let ${\displaystyle {\succsim }_p}$ be Sally’s preference relation. We assume that Sally’s preferences are a function of her beliefs about the state of the world and the desires she has in relation to the consequences of her actions. A decision-theoretic representation of this system will be a conjoint representation consisting in a representation of beliefs, a representation of desires, and a decision rule by which they jointly determine a system of preferences. An expected-utility representation, for example, will comprise a probability function ${\displaystyle P}$ and a utility function ${\displaystyle U}$ such that

Now, if the probability-utility pair ${\displaystyle (P,U)}$ represents ${\displaystyle {\succsim }_p}$ in this manner, then so too does ${\displaystyle (P,U^{\star })}$, where

Since ${\displaystyle U}$ and ${\displaystyle U^{\star }}$ are related by a linear (interval-preserving) transformation, and since utilities are typically understood to be measured on an interval scale (like temperatures in degrees Celsius) the usual response to this fact is that there’s no meaningful difference between the two functions. They represent the same information in different ways—what matters is what’s invariant between them. And since ${\displaystyle U}$ and ${\displaystyle U^{\star }}$ do not have ratios in common, we should say that ratios are not meaningful in the measurement of desire. So far so good. What Zynda notes is that whenever an expected-utility representation ${\displaystyle (P,U)}$ exists, then so too will another decision-theoretic representation ${\displaystyle (P^{\star },U)}$, where

The catch is that this time we need to adjust the decision rule by which ${\displaystyle P^{\star }}$ and ${\displaystyle U}$ jointly determine ${\displaystyle {\succsim }_p}$:

Call this a valuation-maximising representation. If any expected-utility representation ${\displaystyle (P,U)}$ of ${\displaystyle {\succsim }_p}$ exists, then a valuation-maximising representation ${\displaystyle (P^{\star },U)}$ of ${\displaystyle {\succsim }_p}$ also exists, and vice versa. And by analogy with ${\displaystyle U}$ and ${\displaystyle U^{\star }}$, one might be tempted to infer from this something about meaningfulness in ${\displaystyle P}$ and ${\displaystyle P^{\star }}$—namely, that there’s no meaningful difference between the two functions, that what matters is what’s invariant. Consequence: ratios are not meaningful in ${\displaystyle P}$, since they’re not invariant across ${\displaystyle P}$ and ${\displaystyle P^{\star }}$. As Zynda suggests,

One might point out that ${\displaystyle P^{\star }}$ is simply a linear transformation of ${\displaystyle P}$, and argue that in the case of probabilities (like utilities and temperatures) this is a difference that makes no difference. This approach commits … to taking as real properties of degrees of belief at most those properties that are common to both [${\displaystyle P}$ and ${\displaystyle P^{\star }}$] … According to this solution, people really have properties that can properly be called ‘degrees of belief,’ though these are more abstract in nature than subjective probabilities, being purely qualitative … . The concept of degree of belief on this strategy becomes a purely ordinal notion … (2000: 64–65, notation altered for consistency)

But there were some leaps there. While the example does highlight something important about meaningfulness in ${\displaystyle P}$, this is very much not it.

First note that while ${\displaystyle P}$ and ${\displaystyle P^{\star }}$ share their ordinal structure, that’s not all they share. The linear transformation linking ${\displaystyle P}$ and ${\displaystyle P^{\star }}$ preserves lots of properties, not just the ordering. Most importantly, the transformation is bijective, so $P(p)\ne P(q)$ iff $P^{\star }(p)\ne P^{\star }(q)$ and consequently if $P_1\ne P_2$ then $P_1^{\star }\ne P_2^{\star }$. And in just the same way that differences between ordinally equivalent but non-identical probability functions ${\displaystyle P_1}$ and ${\displaystyle P_2}$ can make a difference for our preferences according to the expected utility rule, differences between ordinally equivalent but non-identical ${\displaystyle P_1^{\star }}$ and ${\displaystyle P_2^{\star }}$ will likewise matter according to the valuation-maximisation rule. The same will necessarily be true for any possible “redefinition” of ${\displaystyle P}$. So the example cannot support treating the concept of degree of belief as “a purely ordinal notion” after all—extra-ordinal information still matters.

The reader may note that $P(p)\ne P(q)$ iff $P^{\star }(p)\ne P^{\star }(q)$ precisely because linear transformations preserve ratios of differences. But do not place any weight on this fact, for therein lies the deeper error. Let

The transformation from ${\displaystyle P}$ to ${\displaystyle P^{†}}$, or from ${\displaystyle P^{\star }}$ to ${\displaystyle P^{†}}$, does not preserve difference ratios. Nevertheless, whenever Sally’s preferences ${\displaystyle {\succsim }_p}$ can be given an expected-utility representation ${\displaystyle (P,U)}$, or a valuation-maximising representation ${\displaystyle (P^{\star },U)}$, then they can also be given a schmaluation-maximising representation ${\displaystyle (P^{†},U)}$ such that

In fact, we can even construct “equivalent” decision-theoretic representations where not even orderings are preserved. For any transformation that takes us from ${\displaystyle P}$ to some other ${\displaystyle P^{\ast }}$, then provided the transformation is bijective and therefore invertible, there will be at least one (potentially very complicated) rule by which they can be combined to generate the same preferences relative to ${\displaystyle (P^{\ast },U)}$ as the expected utility rule generates relative to ${\displaystyle (P,U)}$. So there’s approximately nothing that’s preserved across all the belief functions that might figure in one or another decision-theoretic representation—aside from the utterly trivial requirement that different degrees of belief must be assigned different values.

The lesson here is that meaningfulness in the representation of any quantity is only sensibly defined relative to a fixed choice of representational format.¹² Ratios of lengths are meaningful when lengths are represented additively—that is, when the qualitative length system ${\displaystyle \langle \mathbf{O},\succsim ,\circ \rangle }$ is represented in ${\displaystyle \langle {\mathbb{R}}^{\ge 0},\ge ,+\rangle }$—but additive representations are only one among infinitely many ways in which we might choose to measure length. Hölder (1901) showed that the system of lengths can be given a multiplicative representation in the system ${\displaystyle \langle {\mathbb{R}}^{\ge 1},\ge ,\times \rangle }$, and ratios are never invariant across multiplicative representations. Nor will additive and multiplicative representations have ratios in common. In fact, approximately nothing is preserved across all possible numerical representations of length, aside from the trivial requirement that different lengths must be assigned different values. Not even orderings are preserved across all measures of length in that sense, but it would be absurd to conclude that orderings are meaningless.

Likewise for decision-theoretic representations. The fact that ratios vary between ${\displaystyle P}$ and ${\displaystyle P^{\star }}$ implies nothing whatsoever about the meaningfulness of those ratios, because expected utility representations and valuation-maximising representations involve distinct representational formats. Stronger: what Zynda-style examples actually establish is that ratios in ${\displaystyle P}$ really are meaningful relative to expected-utility representations, precisely because transformations of ${\displaystyle P}$ that do not preserve ratios will generally require also making some changes to the decision rule to preserve fit with preferences. But the real trick here is in recognising that the qualitative meaning of those ratios need not be expressible in purely doxastic terms. When we’re modelling beliefs in a decision-theoretic context, the psychological structure we’re trying to represent is not necessarily something internal to system of beliefs itself, considered in isolation from anything else and characterised in purely doxastic terms, but instead at least partly something about the relations that hold between beliefs, desires, and preferences. That is why we cannot alter the probabilistic model of beliefs without making corresponding adjustments to the decision rule: because the meanings of the probabilities in the model are tied up with how they interact with the utilities in the production of preferences. Ratios really are meaningful in the measurement of belief—at least according to expected utility theory—but we should not presume their meaning can be fully captured in purely qualitative doxastic terms and without reference to the role beliefs play as part of a broader psychological system.

This lesson has long been appreciated in the case of utilities. From Ramsey (1931) through von Neumann & Morgenstern (1944) to Savage (1954), the orthodox account of why difference ratios in utility functions are meaningful has appealed to the role that desirabilities play as part of a broader system. Considered wholly in isolation, there’s no immediate reason to suppose that our desires should be measured on anything stronger than an ordinal scale: one desires this more than that. It’s when those desires interact with beliefs in the production of preferences under conditions of uncertainty that the need for a richer measure is manifest. Two utility functions may be ordinally equivalent, but if they vary in their difference ratios then they’ll be differentiated in at least some decision situations—and therein lies the qualitative meaning of those difference ratios. Given the intimate connection between desires and beliefs, it’s a mystery that we should have been inclined to treat the representation of the beliefs any differently.

7. Functional Interpretations

With so much setup, I can in this final section be mercifully brief. The concern with the vagueness interpretations was that a representor ${\displaystyle \mathcal{R}}$ sometimes seems to represent a doxastic state that’s determinately unlike what’s represented by any of the ${\displaystyle P}$ in ${\displaystyle \mathcal{R}}$. Comparativist interpretations agree on this point: where ${\displaystyle \mathcal{R}}$ represents an incomplete confidence ordering, then every ${\displaystyle P}$ in ${\displaystyle \mathcal{R}}$ will determinately misrepresent that ordering. But comparativist interpretations do not play nicely with contemporary theories, which overwhelmingly tend to presuppose the meaningfulness of extra-ordinal information. Pluralists do strictly better on that front, since they allow that ${\displaystyle \mathcal{R}}$ may carry additional representational import beyond the comparative confidence orderings it determines. However, pluralists still presuppose that the psychological structures underwriting our numerical representations of belief—the structures that ultimately explain what is and is not meaningful in those representations—are non-conjoint, purely doxastic qualitative structures. And that’s not obvious either.

So here’s my thought: if it may end up being impossible to appreciate what’s real versus what’s artefact in a formal model of belief without appreciating the role those models play in the psychological theories that make use of them, then why not just take those roles themselves to be what’s real? We do not have to come up with an interpretation of representors that’s independent of the theories in which they figure, since the interpretation of ${\displaystyle \mathcal{R}}$ relative to a psychological theory—of decision making, say, or a theory of belief update, or better still a theory that combines both—can just be the thing that plays the ${\displaystyle \mathcal{R}}$-role in that theory.

In more detail, suppose that ${\displaystyle T}$ is some decision-theory-cum-epistemology in which representors have a role to play. In the usual functionalist manner (à la Lewis 1970), we treat representors as theoretical terms implicitly defined by their role within ${\displaystyle T}$. According to ${\displaystyle T}$ itself, the state that’s designated by a representor ${\displaystyle \mathcal{R}}$ always perfectly occupies the ${\displaystyle \mathcal{R}}$-role that ${\displaystyle T}$ sets out. But ${\displaystyle T}$ might be mistaken, such that nothing perfectly occupies the ${\displaystyle \mathcal{R}}$-role even if something still comes close to doing so. Thus we take the meaning of ${\displaystyle \mathcal{R}}$ relative to ${\displaystyle T}$ to be a function from worlds to whatever it is that does the best job of satisfying the ${\displaystyle \mathcal{R}}$-role at that world, if anything does, and provided it does so well enough. The extension of ${\displaystyle \mathcal{R}}$ relative to ${\displaystyle T}$ will be whatever the meaning designates at our world. Two theories ${\displaystyle T}$ and ${\displaystyle T^{\prime }}$ will typically determine distinct meanings for ${\displaystyle \mathcal{R}}$, and in that sense different interpretations of ${\displaystyle \mathcal{R}}$; though they may also be associated with the same interpretation in the sense of fixing on the same extension for ${\displaystyle \mathcal{R}}$.

Two representors ${\displaystyle \mathcal{R}}$ and ${\displaystyle {\mathcal{R}}^{\prime }}$ are meaningfully distinct according to ${\displaystyle T}$ just in case ${\displaystyle \mathcal{R}}$ and ${\displaystyle {\mathcal{R}}^{\prime }}$ play distinct roles within that theory. On a (hypothetical) comparativist decision theory, ${\displaystyle \mathcal{R}}$ and ${\displaystyle {\mathcal{R}}^{\prime }}$ ought to play the same role just in case they determine the same confidence relations. For the theories we actually have, this won’t be true. Of course, whether we actually ought to treat ${\displaystyle \mathcal{R}}$ and ${\displaystyle {\mathcal{R}}^{\prime }}$ as designating distinct doxastic states depends on what we take the most plausible theories to be—there’s not much point worrying about whether ${\displaystyle \mathcal{R}}$ and ${\displaystyle {\mathcal{R}}^{\prime }}$ are meaningfully distinct according to this or that theory, if we don’t have much reason to think those theories are plausible. Thus, I claim, we have reason to treat ${\displaystyle \mathcal{R}}$ and ${\displaystyle {\mathcal{R}}^{\prime }}$ as meaningfully distinct simpliciter inasmuch as our best theories of rational belief and decision-making posit distinctive roles for ${\displaystyle \mathcal{R}}$ and ${\displaystyle {\mathcal{R}}^{\prime }}$.

Eriksson & Hájek (2007: 204–211) once proposed something much like what I have in mind here. What they propose is that degrees of belief are those things that play the kinds of roles numerical probabilities are supposed to play in the best systematisations of our ideas and intuitions about rational belief and decision-making. They called their view primitivism, but they also note that their proposal is very much in the spirit of functionalism—the main difference being that the functionalist will want to say that our theories implicitly define what “degrees of belief” are via their distinctive roles, whereas they question whether this should really be counted as a “definition” (See Eriksson & Hájek 2007: 210). They prefer instead to say that the concept of “degrees of belief” is a theoretical primitive, and we get a handle on the concept by understanding the roles it plays in the theories that make use of them. It is a difference that makes little difference. The essence of Eriksson & Hájek’s proposal is functionalism, broadly construed, and in that respect is closely related to mine.

But not quite the same. Eriksson & Hájek’s proposed primitives are absolute degrees of belief. That makes sense inasmuch as we’re modelling beliefs in the traditional manner, since everything a probability function says about a total belief state can be derived from what it says about the particular degree of belief it associates with each proposition. But when dealing with representors, we’d be wise not to take absolute degrees of belief as our “theoretical primitives.” What a representor represents cannot always be captured merely by specifying what it says about the (imprecise) degree to which the agent believes each proposition. That is what the summary function ${\displaystyle {\mathcal{R}}^s}$ does, but a summary function can omit information relevant to the role played by the representor it summarises. ${\displaystyle {\mathcal{R}}_{coin}}$ assigns the very same maximally imprecise interval to ${\displaystyle p}$ and to ${\displaystyle q}$, but it would be a mistake to say that Sally’s attitudes towards ${\displaystyle p}$ and ${\displaystyle q}$ are the same. Better instead to let the entire system of beliefs be our primitive, represented by ${\displaystyle \mathcal{R}}$, and characterise that total system of beliefs by the functional role played by ${\displaystyle \mathcal{R}}$ in the best theories we have that make use of such models.

The important point is that the functional interpretation carries no presupposition that meaningful differences between the systems of belief represented by ${\displaystyle \mathcal{R}}$ and ${\displaystyle {\mathcal{R}}^{\prime }}$ must be explicable by reference to purely doxastic qualitative structures—in terms of comparative confidences, say, give or take some other doxastic relations, and without reference to the relations between belief and the rest of our psychology. We can still say, of course, that if ${\displaystyle \mathrm{\forall }P\in \mathcal{R}:P(p)>P(q)}$ and , then ${\displaystyle \mathcal{R}}$ represents greater confidence in ${\displaystyle p}$ than ${\displaystyle q}$ while ${\displaystyle {\mathcal{R}}^{\prime }}$ represents the reverse. The functional interpretation is not committed to the meaninglessness of such relations—quite the opposite. But it’s not committed to saying that everything meaningful in a representor can be expressed in a similar fashion. Sometimes, the most we might be able to say in purely qualitative terms is that ${\displaystyle \mathcal{R}}$ and ${\displaystyle {\mathcal{R}}^{\prime }}$ just represent different systems of belief—as evidenced by their distinctive roles in the production of preferences for instance, or how they give rise to divergent choice behaviour conditional on evidence. That is what sets the functional interpretation apart, and it’s a thing worth having.

Acknowledgements

I started on this paper in 2016, though it was quite different back then. Thanks are due to Miriam Bowen, Seamus Bradley, Jakob Donskov, Jessica Isserow, and Robbie Williams for discussion on various bits and pieces of one version or another. Thanks also to audiences at the University of Leeds, the University of Notre Dame, and the CUNY Cognitive Science Speaker Series. Finally, thanks to two referees for Ergo for their constructive and useful feedback.