1. Introduction

Beliefs come in degrees, or so it seems. Assuming they do, one important question concerns the basis of their numerical representation. It is typical to represent the varying strengths with which propositions might be believed using percentages, or real values between 0 and 1, or with intervals thereof. Moreover, it is typical to assume that these numbers encode more than merely ordinal information. For instance, it seems that we can meaningfully talk about intervals of strengths of belief: an agent—let’s call her α—might believe one proposition much more than she believes another, or she might believe it just a little more. Likewise for ratios: if α is 50% confident that the coin she flips will land heads, then most of us would be happy to say that she has half as much confidence in that event than she has in the coin landing either heads or tails. And, if she is even a little bit rational, then she will probably be at least twice as confident that it will land heads on the next toss than that it will land heads consistently on the next several tosses.

There is, in other words, a widespread prima facie commitment in our understanding of degrees of belief that they can be measured on a ratio scale or something much like it. Given this, we will assume for the remainder of this article that the numbers we use to represent the strengths of our beliefs can, at least in principle, carry cardinal (read: at least ratio and therefore also interval) information. Supposing that is correct, it is just the sort of thing that ought to be explained by any adequate account of what degrees of belief are. We do not get to posit cardinality for free—α’s doxastic states do not come with little numbers attached to them, and they do not literally stand in numerical relationships with one another. Rather, they must have some nonnumerical structure that is in some way similar to and hence representable by the real values in the unit interval, in particular such that both the ordinal and relevant cardinal properties of and relations between those numbers represent something doxastically meaningful. That much is clear enough—the hard part consists in saying exactly what that structure is.

So how is it that we manage to get from the purely nonnumerical stuff in our heads through to numerical representations of our doxastic states that encode interesting cardinal information? A few answers to this question have been suggested. One long-standing tradition seeks to explain where the numbers come from and how they get their meaning, by considering how beliefs interact with preferences (e.g., Ramsey Reference Ramsey and Braithwaite1931b). Others have tried to extract numerical representations out of comparative expectations, a special kind of nonpropositional comparative attitude (e.g., Suppes and Zanotti Reference Suppes and Zanotti1976). Still other potential approaches have yet to be explored. For instance, if you like the idea that degrees of belief are really just outright beliefs about objective probabilities, then you might think that whatever cardinality they possess is derivative on the cardinal information possessed by those probabilities—wherever that comes from.

I am inclined to think that each of these possibilities is worth considering seriously; at least, none of them seem to me either obviously correct or irretrievably hopeless. I have argued elsewhere that the connection with preferences is one promising avenue to explore (Elliott Reference Elliott2019a). But in this article I want to focus on an entirely different kind of approach: comparativism.

For the sake of concreteness, I will take comparativism to be the view that the facts about an agent’s degrees of belief supervene on, and indeed hold in virtue of, the facts about what we will call her confidence comparisons. These are purely ordinal comparative doxastic states such as being more confident that P than that Q, being equally confident that P as that Q, or being at least as confident that P as that Q.Footnote ¹ With that as their starting point, comparativists tend to see degrees of belief and the numerical representations thereof as a kind of theoretical tool, a way to represent and reason about sufficiently coherent systems of comparative confidence. Or to put that another way: the numbers we use to represent our beliefs ultimately describe a purely ordinal structure imposed over a set of propositions by our confidence comparisons, when those comparisons satisfy some minimum threshold of coherence.

On the face of it, comparativism might seem to struggle with providing any plausible explanation of the possibility of cardinal information. After all, individual confidence comparisons contain no more than purely ordinal information, so how could a system composed of nothing more than such comparisons possess anything more than that?Footnote ² Nevertheless, comparativists have what is by now a standard explanation of how cardinality can be generated out of nothing more than ordinal confidence comparisons. By drawing on a well-worn analogy with the measurement of mass, length, and other extensive quantities, comparativists have managed to set down conditions (or axioms) under which meaningful cardinal information might be extracted out of a system of confidence comparisons.

That is the current state of play. However, the axioms to which comparativists typically appeal when addressing this kind of challenge are quite strong indeed. Essentially, they impose a comparative variety of probabilistic (and hence logical) coherence on the agents’ confidence comparisons. And this is a key limitation with the view in its most typical contemporary form: it lacks an adequate account of how ordinary agents—who do not live up to the very strict standards of probabilistic coherence—might nevertheless have beliefs that carry genuine cardinal information. Consequently, in this article I want to explore whether, and how, the standard ‘probabilistic’ axioms might be weakened, while maintaining the same basic strategy for extracting cardinality out of a system of comparative confidences.

Let me say that again, for emphasis: the goal here is to explore whether, and to what extent, the usual probabilistic axioms can be weakened. This is a question of interest to proponents and opponents of comparativism alike and for those who might be on the fence. I stress however that my results are formal, not evaluative. The current article is not intended to be a defense of comparativism. (It would be woefully inadequate if so.) An evaluation of the overall merits and demerits of the comparativists’ view is well beyond the scope of this discussion, and I will not try to address the tricky empirical question of whether and to what extent the weakened axioms are satisfied or even approximated by ordinary agents. Still less is this an article on what our comparative confidences should be like, so I will not have anything much to say about how Ramseyan comparativism relates to arguments for probabilism.

I begin my discussion by reviewing the standard account of how mass can be measured on a ratio scale and how probabilistic comparativism posits an essentially similar process for the measurement of belief (secs. 2 and 3). Following that, I discuss in a little more detail the motivations for seeking more general axioms under which cardinality can be extracted out of a system of confidence comparisons (sec. 4). Finally, I show that the axioms of what I will call probabilistic comparativism can be weakened to a significant extent, although not without limits. I do this by developing what I call Ramseyan comparativism (sec. 5). Moreover, I show that the Ramseyan axioms on confidence comparisons are in one important respect maximally weak: inasmuch as comparativists want to retain the analogy with the measurement of mass as it is usually understood, the Ramseyan axioms are as weak as they come.

2. The Measurement of Mass

Let a and b be any two concrete objects you like, and compare:

Cardinal obviously contains more information than Ordinal, and that information has to come from somewhere. Yet masses do not come with little numbers attached to them. Whatever it is that explains the extra information in Cardinal must ultimately be nonnumerical in nature. So how can we get from the nonnumerical facts on the ground through to numerical masses that encode interesting cardinal information?

The representational theory of measurement gives us a plausible answer.Footnote ³ First, note that Cardinal is true (roughly) if and only if (iff), if you were to take two disjoint objects each as massive as b (call them b ₁ and b ₂, b’s duplicates) and join them together, then the resulting object would be just as massive as a. Call the operation of joining objects together concatenation; we assume that no mass is gained or lost in the act of concatenating. Given this, it is plausible that there is nothing more to the truth of a claim like Cardinal than what we have just said—that is, ‘a is twice as massive as b’ just means something roughly to the effect of ‘a is as massive as the concatenation of two duplicates of b’. By reference, then, to purely ordinal comparisons between duplicates and the concatenations thereof, we have been able to give straightforward nonnumerical meaning to Cardinal.

And we can easily generalize this idea to explain other rational ratio comparisons. For positive integers n, m, say that a is n/m times as massive as b whenever there is some object c such that

1. 1. a is as massive as the concatenation of n duplicates of c, and

2. 2. b is as massive as the concatenation of m duplicates of c.

Now let x designate c’s mass in whatever units you like—let’s say slugs (∼14.6 kg). Intuitively, a must then have a mass of $n\cdot x$ slugs, and b must have a mass of $m\cdot x$ slugs. Hence, a is n/m times as massive as b. Indeed, with a little bit more work, we can generalize the idea even further to explain arbitrary real ratio comparisons. However, for the sake of simplicity we will stick with rational ratios throughout this discussion.

Hiding in the background is a crucial empirical assumption: that the operation of concatenation behaves as a kind of nonnumerical analogue of addition. We rely on exactly this assumption to move from, for example, ‘a is as massive as the concatenation of n duplicates of an object with a mass of x slugs’ to ‘a has a mass of $n\cdot x$ slugs’—that is, we assume that the mass of a concatenation is just the sum of the masses of the concatenands. (Imagine if, instead, concatenation behaved like quaddition: whenever you concatenate up to 57 duplicates together, things are as usual, but concatenate more and the result is always as massive as five duplicates. We could have then used concatenations to define our way up to one object’s being 57 times more massive than another but no further.)

Fortunately, the analogy between concatenation and addition is quite close. Where

$\begin{array}{c}a{\succsim }^{m}b\text{ iff }a\text{ is at least as massive as }b\text{,}\\ a{\sim }^{m}b\text{ iff }a\text{ is exactly as massive as }b\text{,}\\ a\oplus b=\text{ the concatenation of }a\text{ and }b\text{,}\end{array}$

then it is plausible that ≿^m is transitive and complete, and ∼^m is its symmetric part. Furthermore, ⊕ behaves with respect to ≿^m a lot like + behaves with respect to ≥: for all disjoint objects a, b, c,

1. 1. $a\oplus b{\succsim }^{m}b$.

2. 2. $a\oplus b{\sim }^{m}b\oplus a$.

3. 3. $a\oplus (b\oplus c){\sim }^m(a\oplus b)\oplus c$.

4. 4. $a{\succsim }^{m}b\text{ iff }a\oplus c{\succsim }^{m}b\oplus c$.

Now compare these with the following properties of + in relation to ≥, where n and m are nonnegative real numbers:

1. 1. $n+m\ge m$.

2. 2. $n+m=m+n$.

3. 3. $n+(m+k)=(n+m)+k$.

4. 4. $n\ge m$ iff, for any k, $n+k\ge m+k$.

Indeed, if we posit a rich enough space of concrete objects and make one further ‘Archimedean’ assumption (roughly, that no object is infinitely more massive than any other), then we can say something stronger still: if 𝒪 is the set of ∼^m-equivalence classes of concrete objects and ℝ⁺ the positive reals, then the relational system 〈𝒪, ≿^m, ⊕〉 has essentially the same structure as 〈ℝ⁺, ≥, +〉. Thus, we can assign a number to each object in such a way that ≿^m is represented by ≥, and ⊕ is represented by +. And with that in hand, we can start to define up ratios of masses, numerical differences in mass, ratios of differences in mass, and so on. In other words, we have all the basic resources needed to explain how numerical representations of mass manage to carry all sorts of interesting cardinal information.

The upshot: numerical masses represent a fully nonnumerical system of ordinal mass comparisons that have an ‘additive’ structure over concatenations. We are justified in treating ratios of masses as meaningful because there exists an operation on objects that is intuitively and formally like ‘adding’ masses together. And we can apply the same basic idea outlined here to account for the measurement of other (extensive) quantities: a is twice as long as b iff a is as long as two length duplicates of b laid end to end, a has twice the volume of b iff a has the same volume as two volume duplicates of b joined together, and an event e ₁ has twice the duration of e ₂ iff e ₁ can be split into two disjoint events with the same duration as e ₁.

To apply the same idea to the measurement of beliefs, comparativists have therefore historically sought an operation on the relata of confidence comparisons (i.e., propositions) that behaves, with respect to those comparisons, similarly enough to addition to justify treating it as a nonnumerical analogue thereof. As Krantz et al. put it, the strategy is “to treat the assignment of [subjective] probabilities as a measurement problem of the same fundamental character as the measurement of, e.g., mass or duration” (Reference Krantz, Luce, Suppes and Tversky1971, 200). So let’s see how that plays out in practice.

3. Probabilistic Comparativism

In this section I provide an overview of probabilistic comparativism. I begin by laying out some basic notation and assumptions (sec. 3.1), followed by the mathematical underpinnings of the view (sec. 3.2). Finally, I define two specific varieties of probabilistic comparativism—one ‘precise’ (sec. 3.3) and the other ‘imprecise’ (sec. 3.4).

3.1. Notation and Assumptions

Let α be an arbitrary thinking subject whose beliefs we are trying to represent. I will assume that the propositions regarding which α has beliefs can be modeled as subsets of some space of logically possible worlds, Ω. By ‘logically possible’, I mean no more than that the worlds are closed under a consequence relation at least as strong as that of classical propositional logic. So, you can assume that Ω includes metaphysically or even epistemically impossible worlds, if that is what floats your boat—as long as the worlds are classically logically consistent. (I talk more about this assumption in sec. 4.)

Next, let $ℬ\subseteq \wp \left(\Omega \right)$ denote that set of propositions regarding which α has beliefs. Without loss of generality, I assume throughout that ℬ is a Boolean algebra of sets on Ω. So, ℬ contains at least Ω and ∅, and it is closed under relative complements and binary intersections/unions. I also assume throughout that ℬ is finite. Doing this will simplify much of the ensuing discussion and formalities.Footnote ⁴

I assume that α’s full system of confidence comparisons can be modeled with a single binary relation ≿ defined over ℬ, where

$P\succsim Q\text{ iff }\alpha \text{ believes }P\text{ at least as much as she believes }Q\mathrm{.}$

I refer to ≿ as α’s confidence ranking. Consequently, where ≻ and ∼ stand for the comparatives more probable and equally probable, respectively, I am in effect assuming that

$\begin{array}{c}P\sim Q\text{ iff }(P\succsim Q)\text{ \& }(Q\succsim P),\\ P\succ Q\text{ iff }(P\succsim Q)\text{ \& }\neg (Q\succsim P)\mathrm{.}\end{array}$

Nothing about this last assumption should be treated as obvious or trivial. For example, α might be at least as confident in P as in Q without being more confident in P than in Q or without being equally confident in P as in Q. Nevertheless, it will simplify the discussion, and nothing of great importance hangs on it.

Finally, where a function Cr assigns real numbers to the propositions in ℬ, I will say that Cr almost agrees with ≿ iff, for all P, $Q\in ℬ$,

$P\succsim Q\text{ only if }𝒞r\left(P\right)\ge 𝒞r\left(Q\right)\text{,}$

and we will say that Cr agrees with ≿ just in case

$P\succsim Q\text{ iff }𝒞r\left(P\right)\ge 𝒞r\left(Q\right)\text{.}$

For ease of expression, I treat agreement (but not almost agreement) as symmetric: ≿ agrees with Cr just in case Cr agrees with ≿.

3.3. Precise Probabilistic Comparativism

Assuming that Cr agrees with α’s confidence ranking, say henceforth that Cr constitutes a fully adequate model of α’s beliefs whenever

$\alpha \text{ believes }\mathrm{P }\frac{n}{m}\text{ times as much as she believes }Q\text{ iff }𝒞r\left(P\right)=\frac{n}{m}\cdot 𝒞r\left(Q\right)\mathrm{.}$

I assume that full adequacy is worth striving for—after all, most theorists will be happy to make both of the following kinds of inferences:

1. 1. α believes P to degree x and Q to degree y.

2. 2. $x=n\cdot y$.

3. ∴ α believes P n times as much as she believes Q.

And in the other direction,

1. 1. α believes P n times as much as Q.

2. 2. α believes P to degree y.

3. ∴ α believes Q to degree $x=n\cdot y$.

Only full adequacy licenses inferences in both of these directions, and so I take it that full adequacy stands as an important desideratum for any comparativist theory. With that said, we can also say that Cr is L-to-R adequate iff the left-to-right direction of the above biconditional holds, and R-to-L adequate iff the right-to-left direction holds. Comparativists may well want to reject full adequacy in favor of mere L-to-R or R-to-L adequacy, provided that the rejection is well motivated and they are able to explain away any intuitions in support of full adequacy. (I say a little more about this in sec. 5.3.)

Next, let precise probabilistic comparativism denote any comparativist theory that is committed to the following conditional:

Precise Probabilistic Comparativism. If Cr is the unique probability function that agrees with α’s confidence ranking, then Cr is a fully adequate model of α’s beliefs.

Note the stated requirement that the probability function be unique. This is needed to avoid contradiction: for any nontrivial algebra ℬ, there will always be some collection of probability functions on ℬ that agree with one and the same confidence ranking—and since any two probability functions on the same domain will disagree on at least some ratios, any inference from ‘ $𝒞r\left(Q\right)=n/m\cdot 𝒞r\left(Q\right)$’ to ‘α believes P n/m times as much as Q’ will be valid only when the Cr is unique in the relevant sense. In short, R-to-L adequacy presupposes uniqueness, which in turn requires further constraints on ≿.

There are multiple ways to ensure uniqueness. Of particular note is what Stefánsson (Reference Stefánsson2017, Reference Stefánsson2018; cf. also Savage Reference Savage1954; Suppes Reference Suppes1969, 6–7) uses to ensure uniqueness in his recent defenses of probabilistic comparativism:

Continuity. For all nonminimal P, Q, there are P′, Q′ such that $P\sim P^{\prime }$, $Q\sim Q^{\prime }$, and P′ and Q′ are each the union of some subset of a finite set of disjoint propositions R ₁, …, R_m such that $R_i\sim R_j$ for i, $j=1$, …, n.

The interested reader can see Krantz et al. (Reference Krantz, Luce, Suppes and Tversky1971, sec. 5.2) and Fishburn (Reference Fishburn1986) for other conditions sufficient to ensure uniqueness.

Now, probabilistic comparativism clearly has resources to put forward an account of how a system of confidence comparisons might end up carrying cardinal information, in the event that ≿ satisfies the requisite axioms. In particular, consider the following principle, which in essence is just the comparative probability version of how we defined rational ratio comparisons for mass earlier in section 2:Footnote ⁷

General Ratio Principle. α believes P n/m times as much as Q if

1. 1. for $0<n\le m$, there are m nonminimal, equiprobable pairwise pseudodisjoint propositions R ₁, …, R_m such that $Q\sim (R_1\cup \dots \cup R_m)$ and $P\sim (R_1\cup \dots \cup R_n)$, or

2. 2. α believes P n′/m′ times as much as R and believes R n′′/m′′ times as much as Q, where $n/m=n^{\prime }\cdot n^{″}/m^{\prime }\cdot m^{″}$.

So, for instance, suppose that $Q\cap Q^{\prime }$ is minimal. Then, α will take P to be twice as probable as Q inasmuch as $Q\sim Q^{\prime }$ and $(Q\cup Q^{\prime })\sim P$. In this case, Q and Q′ are acting as ‘duplicates’ of one another, and $Q\cup Q^{\prime }$ is their ‘concatenation’.

3.4. Imprecise Probabilistic Comparativism

Say that Cr confirms the General Ratio Principle (GRP) just in case, whenever that principle implies that P is believed n/m times as much as Q, then $𝒞r\left(P\right)=\frac{n}{m}\cdot 𝒞r\left(Q\right)$; otherwise, it disconfirms the GRP. It is easy to check that if any probability function almost agrees with ≿, and ∅ is minimal, then that function will confirm the GRP. This means that it is possible to extend the account of ratio comparisons just given to incomplete confidence rankings.

For ordinary agents, the Completeness axiom is widely considered highly implausible. Consider the following statements, adapted from Fishburn (Reference Fishburn1986):

P = The global population in 2100 will be greater than 13 billion.

Q = The next card drawn from this old and incomplete deck will be a heart.

Are you more confident that P than that Q, or less, or just as confident in either? It is not clear that there must be a fact of the matter. Similar examples abound.Footnote ⁸

There is a natural way of dealing with incompleteness to which comparativists can (and do) appeal. Where ℱ is any set of real-valued functions on ℬ, say this time that the set ℱ agrees with ≿ just in case for all relevant P, Q,

$P\succsim Q\text{iff}\forall 𝒞r\in ℱ:𝒞r\left(P\right)\ge 𝒞r\left(Q\right)\mathrm{.}$

The idea behind a set-of-functions model is to recapture the structure of the confidence ranking by doing something like supervaluating over the functions in ℱ—only what is common to every such function is treated as having representative import. If P and Q are incomparable in terms of relative confidence, then ℱ will contain at least one pair of probability functions that disagree on the relative ordering of P and Q—hence, we still manage to ‘numerically’ represent incomplete ≿ rankings.

Alon and Lehrer (Reference Alon and Lehrer2014) have shown that a set of probability functions agrees with ≿ just in case the latter satisfies the Complete Package minus the Completeness axiom (i.e., the Noncomplete Package). Furthermore, while there will often be more than one set of probability functions F that agrees with ≿, the union of all such sets will always agree with ≿. In sum: whenever ≿ satisfies the Noncomplete Package, there is guaranteed to be a unique set of probability functions that agrees with ≿ and that is maximal with respect to inclusion.

Consequently, if we extend the definitions of full/L-to-R/R-to-L adequacy in the natural way (i.e., by inserting ‘ $\forall 𝒞r\in ℱ$’ in the appropriate locations), we can characterize imprecise probabilistic comparativism by its commitment to:

Imprecise Probabilistic Comparativism. If a nonempty set of probability functions ℱ agrees with α’s confidence ranking and ℱ is maximal with respect to inclusion, then ℱ is a fully adequate model of α’s beliefs.

Imprecise probabilistic comparativism implies the precise version. More precisely, if we assume that ℱ and Cr are essentially the same representation whenever $ℱ=\left\{𝒞r\right\}$, then the two varieties of comparativism amount to one and the same thing whenever exactly one probability function agrees with ≿.

Furthermore, every Cr in a set ℱ that agrees with ≿ will itself almost agree with ≿. So, if we also extend the definition of ‘confirms the GRP’ in the obvious way to sets of functions, it follows that if a set of probability functions ℱ agrees with ≿, then ℱ confirms the GRP. The upshot is that both the precise and imprecise versions of probabilistic comparativism can extract cardinality from comparative confidences in basically the same way; the latter is a natural generalization of the former.

4. Why Generalize?

We have seen now that conformity to the Noncomplete Package is sufficient for the union of pseudodisjoint sets to behave like addition. But it is by no means necessary. It is possible to weaken those axioms still further while maintaining the analogy, and I think it is of some importance for comparativism that this can be done. In this section I say why.

The basic reason is that the axioms of the Noncomplete Package are, in conjunction, quite strong—it is not likely that they are jointly satisfied by any ordinary agents. Since I think it is especially troubling, I will focus on one issue in particular: in the context of the (individually rather weak) axioms Nontriviality and Nonnegativity, Scott’s Axiom immediately generates a probabilistic version of the classical problems of logical omniscience. Those three axioms entail that if $P\subseteq Q$ and P, $Q\in ℬ$, then $Q\succsim P$. Consequently,

That is, any confidence ranking that is (i) defined over propositions taken from a space of worlds that is closed under ⇒ and (ii) agrees with a (set of) probability function(s) will ipso facto be ‘coherent’ with respect to ⇒ in the manner just described. In section 3.1 it was assumed that ⇒ is at least as strong as the consequence relation we find in classical propositional logic, and it is implausible that ordinary agents’ confidence rankings are everywhere and always coherent with respect to that logic. I say more about that in a moment. But the point can also be put in a much more general way: we are (probably) not omniscient with respect to any very interesting logics, so unless ⇒ is extremely weak indeed, the confidence rankings of any ordinary agents will (probably) falsify at least one of Nontriviality, Nonnegativity, or Scott’s Axiom.

How might a comparativist respond to this fact? Four obvious (but also obviously nonexhaustive) options are:

1. 1. Argue that ordinary agents’ comparative confidences do conform to the Noncomplete Package after all, because they are probabilistically coherent after all.

2. 2. Argue that ordinary agents’ comparative confidences do conform to the Noncomplete Package after all, once we define propositions over a richer space of worlds.

3. 3. Argue that because ordinary agents’ comparative confidences do not conform to the Noncomplete Package, they therefore do not ground any cardinal information (or not the same kind of information).

4. 4. Accept that ordinary agents’ comparative confidences do not conform to the Noncomplete Package and seek weaker axioms under which cardinality can be extracted from comparative confidences.

The fourth seems to me clearly the best option. After all, nothing about comparativism per se ties it irrevocably to specifically probabilistic representations of degrees of belief, and if more general conditions exist then it only makes sense for comparativists to find and use them. But if you prefer one of the others, or something else not listed, then so be it—there is no harm in developing ideas in many different directions. I will, however, here give some reasons to think that the fourth option should be preferred.

Regarding the first: I take it for granted in the following discussion that we are not (classically) logically omniscient. But maybe we are—sure, and I am not unsympathetic to the idea that we ordinary agents really are probabilistically coherent. But since this is usually met with an incredulous stare let’s just move on already.

The second option seems to be the more common way of arguing that the Noncomplete Package can actually be satisfied by ordinary (and ordinarily irrational) agents. As I have noted, if the entailment relation ⇒ is weak enough, then logical omniscience might not look so bad. So what would happen if we remove the assumption that the worlds in Ω are closed under any interesting logic?

In a little more detail, the idea is this. If we help ourselves to a rich enough space of possible and impossible worlds, then it is well known that we can construct a probability function properly so-called on that enriched space that ‘mimics’ the behavior of a nonprobabilistic function defined over the smaller space of classical possible worlds.Footnote ⁹ So what looks like comparative confidences that are inconsistent with Nontriviality, Nonnegativity, or Scott’s Axiom when they are defined for propositions qua sets of possible worlds can in fact be rerepresented using (sets of) probability functions, if we make use of enough impossible worlds. Hence, to apply the probabilistic comparativists’ explanation of cardinality to ordinary agents, we do not need to weaken the axioms all. We can keep the Noncomplete Package as long as we just make sure to use enough impossible worlds.

That seems easy enough, but I do not think that this is a viable strategy for the comparativist to adopt. I will set out the reasons for this very briefly, since most of the relevant issues are discussed at length in Elliott (Reference Elliott2019b). The problem is that once Ω includes enough impossible worlds for the strategy to work (roughly, for any impossibility, there is an impossible world that verifies it), then most subsets of Ω will be meaningless and consequently not representative of any proper contents of belief. Moreover, for any meaningful subset P of Ω, none of P’s subsets or supersets will be meaningful, nor will any subset of $\Omega \setminus P$ be meaningful. In short, having too many impossible worlds in Ω renders useless for the purposes of comparativism any set-theoretic definition of ‘concatenation’ along the lines described in section 3. Furthermore, any algebra of propositions defined on a space of possible and impossible worlds that is rich enough to represent the contents of belief will contain only meaningful propositions just when the relevant space of worlds is closed under a consequence relation that is, for all intents and purposes, at least as strong as classical propositional logic.

Of course, comparativists do not have to define their concatenations set theoretically as I have done in section 3.2. But the only other place that we will plausibly find the structure required to define an appropriate concatenation operation is in the logical relations among the contents of the propositions. That is, we could define concatenations in terms of disjunctions of inconsistent contents (or disjunctions of contents whose conjunctions are minimal). But defining the concatenation operation in this way brings us straight back to where we started vis-à-vis the problem of logical omniscience, and appealing to impossible worlds will be of absolutely no help here.

So there is no easy way to pursue either the first or the second route: if you want to tie the possibility of cardinality to the Noncomplete Package, then you will be tying it to very strong conditions of logical omniscience. And consequently you will need to face up to the empirical and intuitive evidence that ordinary agents just are not that good at classical logic.

Could we instead take the third route and argue that ordinary agents whose comparative confidences do not satisfy the Noncomplete Package cannot have beliefs that carry ratio and interval information? This does not strike me as very plausible. For example, the literature on the conjunction and disjunction fallacies already strongly suggests that ordinary agents do not have comparative confidences that respect even relatively simple bits of classical logic. So imagine that α has just committed the conjunction fallacy—she thinks it is more plausible that Linda is a bank teller (B) and active in the feminist movement (F) than that she’s a bank teller. Are we going to say now that there is no meaningful way to answer the question of how much more α believes $B\cap F$ over B? Of course not. Similarly, I am not logically omniscient, and (like most people) I have probably fallen foul of various probabilistic fallacies before. My comparative confidences do not satisfy the Noncomplete Package. Maybe they do not even come close to satisfying those axioms. None of this prevents me from believing some things much more than other things, or at least twice as much as other things.

Our capacity to believe one proposition much more than another, or (at least) twice as much as another thing, and so on, is not hostage to any presupposition of logical coherence; still less should it depend on a condition of probabilistic representability. Most philosophers will see no inconsistencies at all in holding both that (a) ordinary agents’ beliefs cannot be faithfully represented by (a set of) probability functions and that, (b) for arbitrary P and Q, an ordinary agent might believe P much more than Q, or (at least) twice as much as Q. These claims should be uncontroversial—only someone caught firmly in the grips of a deeply unrealistic picture of belief would think to deny it. Or at least I will say this: if you want to argue otherwise, then you will be facing a difficult uphill battle. Better, I think, to seek more general axioms under which cardinal information can be extracted from a system of comparative confidences.

5. The Ramseyan Alternatives

What I am calling Ramseyan comparativism is inspired by a brief remark from Frank Ramsey in “Probability and Partial Belief”: “‘Well, I believe it to an extent 2/3’, i.e. (this at least is the most natural interpretation) ‘I have the same degree of belief in it as in $P\vee Q$ when I think P, Q, R equally likely and know that exactly one of them is true’” (Reference Ramsey and Braithwaite1931a, 95). In a recent paper, Weatherson (Reference Weatherson2016, 223–24) has also suggested that Ramsey’s remark points toward a version of comparativism that is weaker than probabilistic comparativism. However, neither Ramsey nor Weatherson take their discussion beyond this initial suggestion, and (as we will soon see) there is a bit of work that needs to be done in order to flesh the idea out in full.

In the remainder of this article, I develop precise Ramseyan comparativism (secs. 5.1 and 5.2) and then an imprecise version (sec. 5.3). Following that, I prove an important result about the axioms under which Ramseyan comparativism supports the analogy with the measurement of mass (sec. 5.4).

5.1. The Main Ideas

First, it will be useful to introduce another definition (the term ‘n-scale’ comes from Koopman [Reference Koopman1940]):

Definition 4. A set 𝒫 of n pseudodisjoint propositions is an n-scale of P iff (i) $P\notin 𝒫$, (ii) $\cup 𝒫\sim P$, and (iii) for all Q, Q′ in 𝒫, $Q\sim Q^{\prime }$.

We can take this as a comparativist characterization of what it is for an agent to think that Q is as likely as a disjunction of equiprobable propositions at most one of which is true. So, for example, if α thinks Q is as likely as $P\cup P^{\prime }$, where P and P′ are equiprobable and pseudodisjoint, then {P, P′} is a 2-scale of Q. We will also assume that α is certain of P’s truth just in case P is maximal, and we will represent certainty in P with $𝒞r\left(P\right)=1$. This is something the Ramseyan view shares with probabilistic comparativism, where in order to fix the scales the values of the minimal and maximal propositions need to be stipulated.

In light of definition 4, Ramsey’s idea can be recast as follows: α believes P to degree n/m when $P\sim (Q_1\cup \dots \cup Q_n)$, where the Q ₁, …, Q_n belong to an m-scale {Q ₁, …, Q_n, …, Q_m} of some maximal proposition R. A good start—but there is a natural extension that will be helpful to incorporate into what follows.

Consider, to begin with, the following situation. Let ℬ designate the power set of $\Omega =\{{\omega }_1,{\omega }_2,{\omega }_3\}$, and let P _〈n〉 and P _〈nm〉 designate the possible worlds propositions {ω_n} and {ω_n, ω_m}, respectively (e.g., $P_{〈12〉}=\{{\omega }_1,{\omega }_2\}$). Suppose now that ≿ is transitive and reflexive, and (where the square brackets indicate equiprobability)

$\Omega \succ \left[\begin{array}{c}{P}_{〈13〉}\\ P_{〈23〉}\end{array}\right]\succ \left[\begin{array}{c}{P}_{〈12〉}\\ P_{〈3〉}\end{array}\right]\succ \left[\begin{array}{c}{P}_{〈1〉}\\ P_{〈2〉}\end{array}\right]\succ \varnothing \mathrm{.}$

We can represent ≿ with figure 1, where the relative sizes of the boxes containing the ω_i correspond to the order of propositions in the confidence ranking.

Figure 1.

Indirect R-Scalability.

Now Ω is maximal, and {P _〈12〉, P _〈3〉} is a 2-scale of Ω, so Ramsey would say that

$𝒞r\left(P_{〈3〉}\right)=𝒞r\left(P_{〈12〉}\right)=\frac{1}{2}\mathrm{.}$

However, P _〈1〉 and P _〈2〉 do not belong to any n-scale of Ω, so Ramsey’s idea does not yet give us any strength with which they are believed. But since {P _〈1〉, P _〈2〉} is a 2-scale of P _〈12〉, it is only reasonable to say that

$𝒞r\left(P_{〈1〉}\right)=𝒞r\left(P_{〈2〉}\right)=\frac{1}{4}\mathrm{.}$

We can capture the foregoing by means of the following definition:

Definition 5. For integers n, m such that $m\ge n\ge 0$, $m>0$, P is

1. 1. 0/m valued if P is minimal and m/m valued if P is maximal, and

2. 2. n/m valued if $P\sim (Q_1\cup \dots \cup Q_{n^{\prime }})$, where the Q ₁, …, Q _n′ belong to an m′-scale of an $n^{″}/m^{″}\text{-valued}$ proposition, and $n^{\prime }\cdot n^{″}/m^{\prime }\cdot m^{″}=n/m$.

The new, generalized version of Ramsey’s idea now amounts to the claim that α believes P to degree n/m if P is n/m valued. As such, define a Ramsey function as follows:

Definition 6. $𝒞r:ℬ\mapsto [0,1]$ is a Ramsey function (relative to ≿) iff, for all $P\in ℬ$, if P is n/m valued, then $𝒞r\left(P\right)=n/m$.

The close connection between Ramsey functions and the GRP should at this point be apparent, and it should likewise already be clear that the way Ramsey proposes to measure degrees of belief is not too different from the strategy the probabilistic comparativists want to adopt. In fact, in the present terminology, the first (noninductive) clause of the GRP essentially states that for $m\ge n$, P is believed n/m times as much as Q whenever 𝒫 is an m-scale of Q, and ${𝒫}^{\prime }\subseteq 𝒫$ is an n-scale of P. In this case, for any Ramsey function Cr, $𝒞r\left(P\right)=n/m\cdot 𝒞r\left(Q\right)$. With respect to n/m-valued propositions, Ramsey functions always confirm the GRP.

Essentially, a Ramsey function either directly or indirectly scales every middling n/m-valued proposition relative to some maximal proposition, which has a stipulated value. With respect to pairs of propositions that cannot be so scaled, however, a Ramsey function may disconfirm the GRP. An especially clear example where this would occur can be seen in figure 2. Where

$\Omega \succ P_{〈23〉}\succ \left[\begin{array}{c}{P}_{〈12〉}\\ P_{〈13〉}\end{array}\right]\succ \left[\begin{array}{c}{P}_{〈2〉}\\ P_{〈3〉}\end{array}\right]\succ P_{〈1〉}\succ \varnothing \mathrm{.}$

In this case, the only nontrivial n-scale is the 2-scale {P _〈2〉, P _〈3〉} of P _〈23〉. According to the GRP, then, we should be able to say

$𝒞r\left(P_{〈2〉}\right)=𝒞r\left(P_{〈3〉}\right)=\frac{1}{2}\cdot 𝒞r\left(P_{〈23〉}\right)\mathrm{.}$

However, since P _〈23〉 cannot be scaled relative to Ω, Ramsey’s suggestion gives us no means of fixing values for P _〈2〉, P _〈3〉 and P _〈23〉.

Figure 2.

Failure of R-Scalability.

Call any proposition that is n/m valued R-scalable. All of the propositions other than Ω and ∅ represented in figure 2 are not R-scalable. Ramsey says nothing about how to measure propositions that are not R-scalable, although perhaps this is not a very troubling gap in his proposal. One might simply assume that such cases do not exist. Let N designate the set of R-scalable propositions, then:

R-Scalability. $𝒩=ℬ$.

R-Scalability is not implied by the Complete Package. However, given that package, it is equivalent to Continuity. (See the appendix for a proof.)Footnote ¹⁰ In other words, precise probabilistic comparativists do not seem to have anything to fear from an axiom like R-Scalability. (Nevertheless, I discuss below how the Ramseyan comparativist can do without it.)

R-Scalability merely guarantees that every proposition in ℬ is R-scalable. Importantly, this is not yet enough to ground a minimally plausible comparativist theory. There are still two additional problems that can arise in the absence of further assumptions about the structure of ≿:

1. 1. We need to ensure that definition 6 is consistent. Without further assumptions, it is possible that, for example, $P\sim Q$, where for some R, P belongs to a 2-scale of R and Q belongs to a 3-scale of R. This is clearly unacceptable: α cannot believe P to the degrees 1/2 and 1/3 simultaneously. If Ramsey functions are to be well defined, we will need to ensure that if P is both n/m valued and n′/m′ valued, then $n/m=n^{\prime }/m^{\prime }$.

2. 2. We need to ensure that any Ramsey function relative to ≿ will agree with ≿. Without further assumptions, there is no guarantee that $𝒞r\left(P\right)\ge 𝒞r\left(Q\right)$ if or only if $P\succsim Q$. For instance, P could be 1/2 valued and Q 1/4 valued, yet $Q\succsim P$. This is also undesirable: if the order of the values we assign propositions does not match up to the confidence ranking, then there can be no plausible sense in which those values are a measure of the strengths with which those propositions are believed.

In the presence of R-Scalability, we can kill these two birds with a single stone by adding the following rather strong axiom:

R-Coherence. If P is n/m valued and Q is n′/m′ valued, $P\succsim Q$ iff $n/m\ge n^{\prime }/m^{\prime }$.

R-Coherence is sufficient to avoid both worries, as established by the following representation theorem:

Theorem 1. (i) ≿ satisfies R-Coherence iff there exists a Ramsey function Cr with respect to ≿, and (ii) ≿ also satisfies R-Scalability iff Cr is the unique Ramsey function relative to ≿ that agrees with ≿.

The proofs for this theorem and the two that follow below can be found in the appendix.

5.2. Precise Ramseyan Comparativism

We will say from now on that one accepts precise Ramseyan comparativism just in case one accepts the following conditional:

Precise Ramseyan Comparativism. If Cr is the only Ramsey function relative to α’s confidence ranking, then Cr is a fully adequate model of α’s beliefs.

We can now characterize precisely the respects in which precise Ramseyan comparativism is more lenient than probabilistic comparativism. To start with, it is easy to see that R-Coherence is implied already by the Complete Package. Indeed, if any probability function Cr agrees with ≿, then Cr is also a Ramsey function relative to ≿. Moreover, where the Complete Package plus R-Scalability holds, then the unique probability function that agrees with ≿ just is the unique Ramsey function that agrees with ≿. This is important, since (in light of what we said earlier) it means that precise Ramseyan comparativism is a generalization of any version of precise probabilistic comparativism that makes use of Continuity.

In the other direction, R-Scalability and R-Coherence together obviously imply Completeness and Preorder. However, they do not imply any of Nontriviality, Nonnegativity, or Scott’s Axiom. For a simple (albeit extreme) example in which all three of those axioms fail, assume that $\Omega =\{w_1,w_2,w_3,w_4\}$, ≿ is transitive and reflexive, and

$\left[\begin{array}{c}{P}_{〈4〉}\\ P_{〈24〉}\\ P_{〈124〉}\\ P_{〈234〉}\end{array}\right]\succ \left[\begin{array}{c}\varnothing \\ \Omega \\ P_{〈23〉}\\ P_{〈34〉}\end{array}\right]\succ \left[\begin{array}{c}{P}_{〈2〉}\\ P_{〈14〉}\\ P_{〈123〉}\end{array}\right]\succ \left[\begin{array}{c}{P}_{〈1〉}\\ P_{〈3〉}\\ P_{〈12〉}\\ P_{〈13〉}\\ P_{〈134〉}\end{array}\right]\mathrm{.}$

It is straightforward (albeit a little tedious) to check that R-Scalability and R-Coherence are satisfied in this case. The only nontrivial n-scales (i.e., $n>1$) that can be defined using this ranking are:

1. 1. The 2-scale {P _〈23〉, P _〈34〉} of the maximal propositions;

2. 2. The 2-scale {P _〈123〉, P _〈14〉} of ∅, Ω, P _〈23〉, and P _〈34〉;

3. 3. The several n-scales composed out of minimal propositions, each of some other minimal proposition.

Consequently, $𝒞r\left(\Omega \right)=𝒞r(\varnothing )=1/2$ because {Ω} and {∅} are 1-scales of P _〈23〉 and P _〈34〉, where the latter are 1/2 valued, and $𝒞r\left(P_{〈2〉}\right)=1/4$, because {P _〈2〉} is a 1-scale of P _〈14〉 and P _〈123〉, where the latter are 1/4 valued. Every other proposition is either maximal or minimal and assigned either 1 or 0 accordingly. That the example violates Nontriviality and Nonnegativity is obvious; to see that it violates Scott’s Axiom it suffices to consider the two short sequences P _〈13〉, P _〈24〉 and P _〈12〉, P _〈34〉.

The interesting ‘work’ here is of course being done entirely by R-Coherence. This axiom imposes a limited kind of additive structure on ≿, specifically with respect to confidence rankings between propositions constructed out of members of the same n-scale of any n′/m′-valued proposition. Roughly, within an n-scale, ≿ behaves “pseudoprobabilistically”—but not every proposition is constructible out of the members of an appropriate n-scale, and across n-scales ≿ can behave quite irrationally indeed.

6. Conclusion

Let’s take stock. The standard comparativist strategy for explaining cardinality is based on a purported analogy with the measurement of certain extensive quantities like length or mass. So, for instance, to say that P is n times more likely than Q, we just need to be able to say that P is as likely as the union of n ‘duplicates’ of Q, where the ‘duplicates’ are propositions that are equiprobable and pairwise (pseudo)disjoint. The two Ramseyan varieties of comparativism I have outlined offer an account of when this kind of ‘adding’ is meaningful that generalizes the axioms assumed by the more common probabilistic comparativism, thus applying to a wide range of confidence rankings that are not probabilistically representable.

In particular, we have shown that comparativists can in principle do without any appeal to Nontriviality, Nonnegativity, and Scott’s Axiom and can avoid the problems that those axioms bring in their wake. This is an interesting result by itself, since it establishes that comparativists can preserve their favorite explanation of cardinality without necessarily committing to the stronger conditions required for probabilistic representability. Moreover, we have been able to show that the union of (pseudo)disjoint sets behaves like addition only if the comparative confidence ranking satisfies Preorder and R-Coherence. Inasmuch as comparativists want to retain the analogy with the measurement and mass as it is usually understood (i.e., in terms of the union of either disjoint or pseudodisjoint propositions), then Ramseyan comparativism is as general as it gets.

It remains to be seen whether it is correct to say that an agent α considers P to be n times more likely than Q iff P is as likely for her as the union of n pseudodisjoint duplicates of Q. But we now know the minimal conditions required for the analogy with mass to hold, so we can ask: (a) Are Preorder and R-Coherence plausibly satisfied by actual agents—or at least, by the kinds of agents who we are happy to say have degrees of belief that carry cardinal information? And, (b) if so, does the GRP in those cases accurately predict our considered judgments about the degrees of belief of such agents? These are questions that I have not considered in this article, but they will need careful consideration in future discussions on the viability of the comparativist view.