2. The continuous uniform distribution: its standard numerical representation

The continuous uniform distribution over a given interval of real numbers $[a,b]$ —or, more generally, over a given set of outcomes that can be parameterized as such—is the probability distribution that satisfies the following constraints:

(U1). Each real number in $[a,b]$ is equiprobable. (More precisely, each singleton included in $[a,b]$ is equiprobable.)

(U2). The probability of any subinterval of $[a,b]$ is proportional to the length of that subinterval. Equivalently, given subintervals $[{a_1},{b_1})$ and $[{a_2},{b_2})$ , $[{a_1},{b_1})$ is $({b_1} - {a_1})/({b_2} - {a_2})$ times as probable as $[{a_2},{b_2})$ , provided ${a_2} \ne {b_2}$ .

The foregoing description may be regarded as a pretheoretic characterization of the continuous uniform distribution over $[a,b]$ . To understand it, one need only an intuitive grasp of the notions of equiprobability and probability ratio (as in one proposition being twice as probable as another proposition). Nonetheless, this distribution may be mathematically represented in multiple ways. I now describe its standard representation in terms of real-valued numerical probability. In section 4, I describe how it can be represented in terms of qualitative probability.

The standard numerical representation of the continuous uniform distribution involves a real-valued numerical probability function P whose probability density function f is as follows:

$$f(x) = \left\{ {\matrix{ {{1 \over {b - a}}} \quad {{\rm{if}}\;a \le x \le b,} \cr 0 \quad{{\rm{otherwise}}.} \cr } } \right.$$

For simplicity, I focus on the case in which $a = 0$ and $b = 1$ in what follows, and I call this case “the” continuous uniform distribution. In this case, for any $c,d \in [0,1]$ , $P(\{ c\} ) = P(\{ d\} ) = 0$ . So, P satisfies (U1). Additionally, the probability of a given subinterval $[{a_1},{b_1}) \subseteq [0,1]$ is simply given by $P([{a_1},{b_1})) = {b_1} - {a_1}$ . So, given subinterval $[{a_2},{b_2}) \subseteq [0,1]$ , ${{P([{a_1},{b_1}))} \over {P([{a_2},{b_2}))}} = {{{b_1} - {a_1}} \over {{b_2} - {a_2}}}$ , provided ${a_2} \ne {b_2}$ . As a result, P satisfies (U2) as well. Hence it is appropriate to call P a “representation” of the continuous uniform distribution.

It is customary in probability theory to define P not only for subintervals of $[0,1]$ but for various other subsets as well (namely, the “measurable” subsets). The standard way to do so is to have P assign every Borel set in $[0,1]$ its Lebesgue measure (see Billingsley Reference Billingsley1995, chap. 3). The measure-theoretic details of this procedure do not matter for our purposes, but it yields a number of probability assignments that one would intuitively expect. For example, this procedure yields that $P([0,{1 \over 2}) \cup [{3 \over 4},1)) = {1 \over 2} + {1 \over 4} = {3 \over 4}$ .

It is well known that, when P is extended in the preceding manner, it satisfies countable additivity. In this respect, P is unlike any probability function that might be taken to represent a fair lottery over a countably infinite set of outcomes. Accordingly, there has been little worry among defenders of countable additivity that the probability distribution that P represents—that is, the continuous uniform distribution—is conceptually coherent. Later, once we consider a qualitative representation of the continuous uniform distribution, I will argue that this complacency has been misplaced.

3. Qualitative probability

In what follows, I understand qualitative probability to be the binary relation of one proposition being at least as probable as another proposition. I employ the following notation:

• $A \underline \succ B$ : A is at least as probable as B.

• $A \approx B$ : A and B are equiprobable.

• $A \succ B$ : A is strictly more probable than B.

It is standard to define $ \approx $ and $ \succ $ in terms of $ \underline \succ $ , as follows:

• $A \approx B$ if and only if $A \underline \succ B$ and $B \underline \succ A$ .

• $A \succ B$ if and only if $A \underline \succ B$ and $B \nsucceq A$ .

In what follows, I assume the preceding definition of $ \approx $ but not that of $ \succ $ , as the latter is more controversial.^{Footnote 2}

The central object of formal theories about qualitative probability is a qualitative probability structure $\langle \Omega ,{\cal F}, \underline \succ \rangle $ , which consists of a nonempty set $\Omega $ of possible outcomes, an algebra ${\cal F}$ on $\Omega $ , and a binary relation $ \underline \succ $ on ${\cal F}$ . In what follows, I generally focus on just a given qualitative probability relation, though it is important to bear in mind that every such relation is also associated with some $\Omega $ and some ${\cal F}$ . As is standard, I also treat propositions as being members of ${\cal F}$ .

A number of axiomatizations of qualitative probability have been developed in the literature (for a survey, see Krantz et al. Reference Krantz, Duncan Luce, Suppes and Tversky1971, chap. 5). I do not assume any particular axiomatization in what follows, but I do assume that any qualitative probability relation $ \underline \succ $ satisfies at least the following constraints for any propositions $A,B,C$ :

• Nonnegativity. $A \underline \succ \emptyset $ .

• Additivity. Suppose $(A \cap C) = (B \cap C) = \emptyset $ . Then, $A \underline \succ B$ if and only if $(A \cup C) \underline \succ (B \cup C)$ .

• Transitivity. If $A \underline \succ B$ and $B \underline \succ C$ , then $A \underline \succ C$ .

• Weak regularity. If $A \ne \emptyset $ , then $\emptyset \nsucceq A$ .

Note that, if we assume the preceding definition of $ \succ $ , then weak regularity is a consequence of the following:

• Regularity. If $A \ne \emptyset $ , then $A \succ \emptyset $ .

That said, I do not assume regularity in what follows. I also do not assume that $ \underline \succ $ satisfies the following:

• Totality. $A \underline \succ B$ or $B \underline \succ A$ .

Nonnegativity, additivity, and transitivity are entailed by all major axiomatizations of qualitative probability. Weak regularity, regularity, and totality are entailed by some axiomatizations, while others are silent on them.^{Footnote 3} Because weak regularity is the most controversial constraint I will assume, I will say much more about it later.

Historically, the main interest in studying qualitative probability has been to prove various “representation theorems” (for a survey of such theorems, see Fishburn Reference Fishburn1986). These are theorems that state conditions under which a given qualitative probability relation can be “represented by” some numerical probability function. More precisely, these theorems state conditions under which, for a given qualitative probability relation $ \underline \succ $ , there exists some numerical probability function P such that $A \underline \succ B$ if and only if $P(A) \ge P(B)$ .^{Footnote 4} Although I am not primarily concerned with representation theorems in this article, one particular class of representation theorems is worth noting in the present context. These are representation theorems that state conditions under which a given qualitative probability relation can be represented by a countably additive probability function. Most notably, Villegas (Reference Villegas1964) shows that a necessary condition for a qualitative probability relation $ \underline \succ $ to be representable by a countably additive probability function is that $ \underline \succ $ satisfies the following constraint:

Monotone continuity. Consider propositions ${A_1},{A_2}, \ldots $ , and B. Suppose

1. (i) ${A_1} \subseteq {A_2} \subseteq \ldots ,$

2. (ii) $A = { \cup _i}{A_i}$ ,

3. (iii) $B \underline \succ {A_i}$ for all i.

   Then, $B \underline \succ {A}$ .^{Footnote 5}

Additional representation theorems have revealed further connections between monotone continuity and countable additivity (see Chuaqui and Malitz Reference Chuaqui and Malitz1983; Chateauneuf and Jaffray Reference Chateauneuf and Jaffray1984; Schwarze Reference Schwarze1989). Such theorems generally assume that qualitative probability satisfies nonnegativity, additivity, transitivity, and weak regularity.^{Footnote 6} This fact suggests that monotone continuity is, in some sense, a qualitative analogue of countable additivity when these constraints are assumed.

Indeed, there is more to say about the connection between monotone continuity and countable additivity than merely that they both appear in certain representation theorems. Consider the following constraint on a given numerical probability function P:

Continuity. Consider propositions ${A_1},{A_2}, \ldots $ Suppose

1. (i) ${A_1} \subseteq {A_2} \subseteq \ldots $

2. (ii) $A = { \cup _i}{A_i}$ .

   Then, $P(A) = \mathop {\lim }\nolimits_{i \to \infty } P({A_i})$ .

Continuity is sometimes included in the axiomatization of numerical probability,^{Footnote 7} and indeed, countable additivity entails continuity. Moreover, note that continuity entails a numerical version of monotone continuity. In particular, suppose that P satisfies continuity and that $P(B) \ge P({A_i})$ for all i, for suitable ${A_i}$ and B. Then, passing to the limit, it follows that $P(B) \ge P(A)$ —which is just a numerical version of the consequent of monotone continuity. Thus, if P is a countably additive probability function, then any qualitative probability relation that P represents satisfies monotone continuity.

The foregoing considerations suggest an argument for a conditional conclusion: if we should accept countable additivity as a constraint on numerical probability, then we should accept monotone continuity as a constraint on qualitative probability. This conclusion follows if we assume that any countably additive probability function indeed represents some qualitative probability relation. However, even if we reject this assumption, it is hard to see why we should accept countable additivity yet reject monotone continuity.^{Footnote 8} As we just saw, continuity entails a structurally analogous numerical version of monotone continuity. So, it seems that monotone continuity at least partly captures the intuitive motivation for accepting continuity. Moreover, because countable additivity entails continuity, it seems that monotone continuity at least partly captures the intuitive motivation for accepting countable additivity as well. These considerations are by no means decisive, but prima facie, it seems there is no reason to accept countable additivity yet reject monotone continuity.

In recent work, DiBella (Reference DiBella2018) shows that any qualitative probability relation that satisfies constraints analogous to nonnegativity, additivity, transitivity, and weak regularity and that represents a fair lottery over a countably infinite set of outcomes violates monotone continuity. Given the aforementioned connections between monotone continuity and countable additivity, this result is perhaps unsurprising. After all, it is well known that any numerical probability representation of such a lottery violates countable additivity. What is perhaps more surprising is that any qualitative probability representation of the continuous uniform distribution violates monotone continuity as well. I prove this fact in the next section.

4. The continuous uniform distribution and monotone continuity

Let me begin with a clarification. When I speak of a qualitative probability relation that represents the continuous uniform distribution, I use the term represents not in the sense of the aforementioned representation theorems but rather in the informal sense in which a given mathematical model might be said to represent some collection of pretheoretic ideas. For example, it is plausible that the concept of a set in Zermelo–Fraenkel set theory represents at least some pretheoretic conception of collection (for an argument to this effect, see Boolos Reference Boolos1971). As I explained in section 2, the standard numerical representation of the continuous uniform distribution readily satisfies the pretheoretic constraints (U1) and (U2) that characterize the distribution. I now explain how these constraints can be satisfied by a qualitative probability relation—and why any such relation violates monotone continuity.

Suppose $ \underline \succ $ is some qualitative probability relation that represents the continuous uniform distribution. Clearly $ \underline \succ $ must satisfy the following constraint:

• Fairness. For any $i,j \in [0,1]$ , $\{ i\} \approx \{ j\} $ .

If $ \underline \succ $ satisfies fairness, then (U1) is readily satisfied. Indeed, fairness plausibly encapsulates what it means for a probability distribution to be fair, namely, that each outcome is equiprobable.

We might also stipulate that $ \underline \succ $ satisfies the following constraint:

• Length ordering. For any $[{a_1},{b_1}),[{a_2},{b_2}) \subseteq [0,1]$ , $[{a_1},{b_1}) \underline \succ [{a_2},{b_2})$ if and only if $({b_1} - {a_1}) \ge ({b_2} - {a_2})$ .

Although length ordering does not explicitly make reference to “probability ratios”—so it is not immediately obvious whether (U2) is satisfied—there are methods of making sense of probability ratios purely in terms of qualitative probability such that (U2) is plausibly satisfied if length ordering is satisfied. In particular, note that any two half-open subintervals of equal length are equiprobable according to length ordering. For example, $[0,1/4) \approx [1/4,1/2)$ . Because $[0,1/2)$ is simply the union of these two pairwise disjoint subintervals, it seems intuitively plausible that $[0,1/2)$ is twice as probable as each of $[0,1/4)$ and $[1/4,1/2)$ —which is exactly what (U2) says.^{Footnote 9}

In the appendix, I construct a qualitative probability structure that satisfies nonnegativity, additivity, transitivity, weak regularity, fairness, and length ordering. This construction shows that there indeed exists a qualitative probability representation of the continuous uniform distribution (in the aforementioned sense).

That said, the stipulation that $ \underline \succ $ satisfy length ordering is unnecessary to yield a violation of monotone continuity. It suffices merely that (1) $ \underline \succ $ satisfies nonnegativity, additivity, transitivity, weak regularity, and fairness and that (2) the algebra ${\cal F}$ over which $ \underline \succ $ is defined satisfies the following weak “measurability” constraint:

• Measurability. ${\cal F}$ contains every singleton and half-open interval of the form $[a,b)$ included in $[0,1]$ .

To see how monotone continuity is violated, first note that the following constraint is a simple consequence of nonnegativity and additivity:

• Monotonicity. If $A \subseteq B$ , then $B \underline \succ A$ .^{Footnote 10}

I will sometimes employ monotonicity in what follows. Here is the main result of this section.

Let i be an arbitrary positive integer. Note that $0 \lt {p_i} \lt 1$ . So, $(0,{p_i}] \subseteq (0,1)$ . Thus, by monotonicity, $(0,1) \underline \succ (0,{p_i}]$ . Moreover, by fairness, $\{ {p_i}\} \underline \succ \{ 0\} $ . So, by additivity, $(0,{p_i}] = [\{ {p_i}\} \cup (0,{p_i})] \underline \succ [\{ 0\} \cup (0,{p_i})] = [0,{p_i})$ . Thus, by transitivity, $(0,1) \underline \succ [0,{p_i})$ . That is, $B \underline \succ {A_i}$ .

Next, by weak regularity, $\emptyset\, \nsucceq \,\{ 0\} $ . Thus, by additivity, $\emptyset \cup B\, \nsucceq \,\{ 0\} \cup B = A$ . That is, $B\, \nsucceq \,A$ —in violation of monotone continuity. □

The preceding result suggests that the continuous uniform distribution and fair lotteries over countably infinite sets of outcomes are not as dissimilar as previous writers on fair infinite lotteries have thought. Once we view such lotteries from a qualitative perspective—and, crucially, assume weak regularity—each fares the same with respect to monotone continuity.

Additionally, as I argued in the previous section, if we should accept countable additivity as a constraint on numerical probability, then we should accept monotone continuity as a constraint on qualitative probability. Thus, if one accepts countable additivity and thereby rejects the conceptual coherence of fair lotteries over countably infinite sets of outcomes, then one should accept monotone continuity and thereby reject the conceptual coherence of the continuous uniform distribution. At least, the latter claim follows if one should also accept the assumptions of theorem 1. As I said earlier, the most controversial among these is weak regularity. So, whether one should have differing attitudes toward fair lotteries over countably infinite sets of outcomes and the continuous uniform distribution largely hinges on the status of weak regularity. In section 6, I argue that one should accept weak regularity—at least, in the special case in which qualitative probability is interpreted as rational comparative confidence. However, before I do that, I will prove a generalization of theorem 1.

5. Arbitrary fair infinite lotteries and monotone continuity

Consider a fair lottery over an arbitrary infinite set $\Omega $ of outcomes. Clearly any qualitative probability relation $ \underline \succ $ that represents such a lottery must satisfy the following constraint:

Generalized fairness. For any $\omega ,\omega^\prime \in \Omega $ : $\{ \omega \} \approx \{ \omega^\prime\} $ .^{Footnote 11}

To yield a violation of monotone continuity in this more general case, I will assume that the algebra ${\cal F}$ over which $ \underline \succ $ is defined satisfies the following constraint:

Measurability*. ${\cal F}$ is a $\sigma $ -algebra on $\Omega $ that contains every singleton included in $\Omega $ .

I now prove the following generalization of theorem 1.

First, let i be an arbitrary positive integer. Note that $({A_{i + 1}}\backslash \{ {\omega _1}\} ) = \{ {\omega _2}, \ldots ,{\omega _{i + 1}}\} \subseteq B$ . So, by monotonicity, $B \underline \succ ({A_{i + 1}}\backslash \{ {\omega _1}\} )$ . Next, by generalized fairness, $\{ {\omega _{i + 1}}\} \underline \succ \{ {\omega _1}\} $ . Thus, by additivity, $[\{ {\omega _{i + 1}}\} \cup \{ {\omega _2}, \ldots ,{\omega _i}\} ] \underline \succ [\{ {\omega _1}\} \cup \{ {\omega _2}, \ldots ,{\omega _i}\} ]$ . That is, $({A_{i + 1}}\backslash \{ {\omega _1}\} ) \underline \succ {A_i}$ . Hence, by transitivity, $B \underline \succ {A_i}$ .

Next, by weak regularity, $\emptyset\, \nsucceq \,\{ {\omega _1}\} $ . Thus, by additivity, $\emptyset \cup B\, \nsucceq \,\{ {\omega _1}\} \cup B = A$ . That is $B \underline \succ {A}$ —in violation of monotone continuity.^{Footnote 12} □

Although this result is not a full generalization of theorem 1—because it requires qualitative probability to be defined over a $\sigma $ -algebra—it does show that violations of monotone continuity are quite typical among qualitative probability representations of fair infinite lotteries (if the Axiom of Choice is true). As with theorem 1, the key assumption being made here about qualitative probability is that it satisfies weak regularity. I now defend weak regularity.

6. In defense of weak regularity

Although the qualitative constraints of regularity and weak regularity have not received much attention in recent philosophical literature, the following constraint on any numerical probability function P has:

In what follows, I refer to the qualitative constraints as qualitative regularity and weak qualitative regularity to distinguish them from quantitative regularity.^{Footnote 13}

Different versions of quantitative regularity vary with respect to different types of modality as well as different types of probability. For example, the following is an epistemic version of quantitative regularity:

One can also consider versions of quantitative regularity that are formulated with respect to physical possibility (and objective chance), logical possibility (and logical probability), and so on (for further discussion, see Hájek, forthcoming, sec. 2).

Although quantitative regularity may seem to be an intuitively plausible principle, several objections have been leveled against it in cases in which a given numerical probability function is defined over an infinite set of outcomes (for a review of such objections, see Benci, Horsten, and Wenmackers Reference Benci, Horsten and Wenmackers2018, sec. 4). Though fewer objections have been leveled specifically against qualitative regularity or weak qualitative regularity, perhaps the most prominent such objection is Williamson’s (Reference Williamson2007) “infinite coin toss” argument.^{Footnote 14} That said, I will not defend weak qualitative regularity by offering new criticisms of Williamson’s argument here (for criticisms of Williamson’s argument, see Weintraub Reference Weintraub2008; Howson Reference Howson2017). Instead, I will defend weak qualitative regularity by offering a new argument for it. Then, I will discuss the import of my argument to Wiliamson’s.

By providing an argument for weak qualitative regularity, I also aim to fill a notable gap in the literature, namely, that virtually all extant pro-regularity arguments have concerned quantitative regularity rather than either of its qualitative counterparts.^{Footnote 15} Nonetheless, as I later explain, it is perfectly consistent to accept weak qualitative regularity (or even qualitative regularity) yet to reject quantitative regularity.