1. Introduction

Regularity is usually taken to be the thesis that all contingent propositions should be given probabilities strictly between zero and one (advocates of regularity conditions include Kemeny Reference Kemeny1955; Shimony Reference Shimony1955; Jeffreys Reference Jeffreys1961; Stalnaker Reference Stalnaker1970; Lewis Reference Lewis1980; Appiah Reference Appiah1985). There are some well-known problems for regularity. For instance, if you throw a dart with a perfectly defined tip at a dartboard, on the standard probability calculus, the probability that it hits a particular point is going to be zero, even though it is possible for the dart to hit that point.Footnote ¹ Another standard example is that a fair coin that will be flipped an infinite number of times has a zero probability of landing heads each time but nonetheless can do so (after all, the all-heads sequence is just as possible as any other infinite sequence of throw outcomes).

However, such cases may simply be taken to show that we need to expand the range of values that probability assignments take beyond real numbers, to include, for instance, hyperreal values such as infinitesimals in a coherent nonstandard analysis setting. Thus, one might assign an infinitesimal probability to any particular outcome of the dart throw experiment (cf. Bernstein and Wattenberg Reference Bernstein, Wattenberg and Luxemberg1969; Skyrms Reference Skyrms1980, 187) or to any particular sequence of coin throws. Recently, Williamson (Reference Williamson2007) has argued that in the coin-toss case this cannot be done, but his argument relies on a number of assumptions from classical probability theory, such as how independence is to be handled. The regularist can still hope that some extension of the classical probability calculus will save regularity or can deny one of Williamson’s intuitions about the probabilities to be assigned in the cases he constructs (e.g., Weintraub Reference Weintraub2008).

Alan Hájek (Reference Hájek2011) suggested, however, that once one extends the set of probability values, say to the hyperreals, one should be able to extend the probability domains—we may think, say, of darts thrown in a space-time with a hyperreal-based continuum—to make the problem of zero-probability events recur. He conjectured that the probability-value-extension “arms race” is a contest the defender of regularity cannot win: no matter how far she extends the set of probability values, there will be a domain where regularity fails even given that set of probability values. I shall prove Hájek’s conjecture under very mild assumptions about probability values, via a counting argument. In the case in which the probability values are partially ordered with an appropriate addition operator and we have a generalization of the Kolmogorov axioms with finite additivity, a no-go result holds that if the sample space has sufficiently large cardinality as compared to the set of probability values, there is no regular probability assignment. The formal results will be presented in section 2, and then in section 3 I will argue on philosophical grounds that the kinds of high-cardinality sample spaces that the formal results discuss are in fact possible. An appendix with some proofs follows. While the results become somewhat neater given the axiom of choice (AC), our main results require only the standard Zermelo-Fraenkel axioms.Footnote ²

2. The Formal Results

Formal Kolmogorovian probability theory involves a sample space Ω, a σ-algebra F on Ω, and a probability measure P on F. A σ-algebra on Ω is a nonempty set of subsets of Ω closed under complementation and countable unions. The members of F are known as the measurable subsets of Ω. The probability measure, then, is a function from F to the reals satisfying the following conditions:

• (K1) for all A in F.

• (K2) .

• (K3) for any pairwise disjoint sets in F (where a collection of sets is pairwise disjoint, provided that no two have any elements in common).

We then say that P is regular, provided that for every nonempty member A of F (it then follows from the axioms that for every member A of F other than Ω).

For our formal no-go results, we shall weaken these assumptions. First, we shall loosen the assumption that probabilities are real numbers. Instead, we shall be working with a set V of values with a partial ordering ≤, that is, a transitive binary relation such that for all x and y in V if and then , and for all x in V we have . I shall write “” for “ and .” We shall also assume that V has an addition operation “+” and a distinguished element 0, satisfying the following two axioms:

• (V1) for all x in V.

• (V2) For all x and y in V, if then .

Moreover, we shall weaken the countable additivity axiom K3 to finite additivity. If we do that, it will also be natural to weaken the assumption that F is a σ-algebra to the assumption that it is an algebra of subsets of Ω, that is, a nonempty set of subsets of Ω closed under complementation and finite unions. Thus, suppose that we have a V as above. A V-probability assignment will then be a function P from the algebra F to V satisfying:

• (K1∗) for all A in F.

• (K3∗) for any disjoint sets A and B in F.

The total probability axiom K2 has been dropped entirely because our formal results do not require it. If we really wanted to, we could replace K2 with the nontriviality assumption:

• (K2∗) .

But we will not need to. We will, as before, say that P is regular, provided that for every nonempty A in F. The range R of the V-probability assignment shall be the set of values that P can take: .

We will write that and say that the cardinality of A is less than or equal to that of B, provided that there is a one-to-one function from A to B, and write that and say that A and B have the same cardinality, provided that there is a one-to-one function from A onto all of B. Then means that and . For convenience, I will use as an abbreviation for , where a ₀ is not in A; thus, is the same cardinality as , if A is infinite, and one plus the cardinality of A, if A is finite. I will also write for the denial of ( is equivalent to if AC holds). At this point, no interpretation is needed for the symbol outside of a context like that above.

Our main formal result follows and is proved in the appendix. A total ordering < on a set A is a transitive binary relation on A such that for all x and y in A exactly one of the following relations holds: , , or .

To see how the theorem works, consider as a special case classical Kolmogorovian probabilities on a finite space with n elements. Let and let for . Suppose P is a regular probability assignment on the power set of Ω. Then by K3, we have , and so . Thus, the range of P must contain at least values. Contrapositively, if is not less than or equal to the cardinality of the range of P, the assignment P will not be regular, and hence for some , which is what the theorem says.

Theorem 1 will follow from theorem 2, below. But first we give two corollaries.

For by AC, Ω has a well ordering (a well ordering on A is a total ordering on A such that every nonempty subset of A has a minimal element) and hence a total ordering.Footnote ³ If , then a fortiori , where R is the range of P, and hence , where the move from to uses the Cantor-Schröder-Bernstein theorem (Takeuti and Zaring Reference Takeuti and Zaring1971, 71). But then theorem 1 shows that there is no regular V-probability assignment on all the subsets of Ω.

The above corollary made use of AC. Here is one that does not use AC but merely the standard Zermelo-Fraenkel axioms. Given a set R, let be the Hartogs number of R, that is, the least ordinal α such that there is no one-to-one function from α to R. An ordinal is a set α with the properties that (a) if , then every member of x is also a member of α, and (b) α is well ordered by the membership relation ∊. (The finite ordinals are the natural numbers constructed inductively by setting and .) It is then a theorem of Zermelo-Fraenkel set theory, without any need for AC, that a Hartogs number exists for every set (Hrbacek and Jech Reference Hrbacek and Jech1999, 130).

This corollary could be proved from theorem 1, but we will instead get it as a direct consequence of corollary 4, below.

One may want to generalize the notion of probability and regular probability even further, allowing for a set V of values that has neither a distinguished zero nor an addition operation. Suppose now that V is any partially ordered set. If Ω is a sample space and F an algebra of subsets of Ω, then we say that P is a generalized V-probability, provided that:

• (K3∗∗) If for sets A and B in F, then .

Moreover, we say that P is a regular generalized V-probability, provided that whenever A and B are measurable sets such that A is a proper subset of B, then . We then have the following analogues of our formal results above.

The following lemma implies that, given V1 and V2, V-probabilities are generalized V-probabilities and regular V-probabilities are regular generalized V-probabilities.

Theorem 1 then follows from theorem 2. For under the conditions of theorem 1, by theorem 2, the function P is not a regular generalized V-probability assignment, and so by lemma 1, P is not a regular V-probability assignment, and hence for some nonempty A. But then let x be any member of A and observe that by K1∗ and lemma 1, from which it follows by partial ordering that , which is the conclusion of theorem 1. Corollary 3 follows from theorem 2 just as corollary 1 followed from theorem 1. Finally, corollary 4, given lemma 1, immediately implies corollary 2. In the appendix, we will prove lemma 1, theorem 2, and corollary 4.

3. The Philosophical Results

Corollary 2 tells us that no matter how rich and nonstandard we make our probability values, no matter how far in a hierarchy of hyperreals we go, as long as our sample space is large enough, we will run into zero-probability possibilities. There is, however, a philosophical question: Are there any such high-cardinality sample spaces of possible situations? I will offer one family of constructions.

Plausibly, there are kinds K of concrete objects such that for any nonzero cardinality k, it is possible that there are exactly k objects that are Ks. For example, we might simply take K to be the kind concrete object. While it is controversial whether there could be no concrete objects at all,Footnote ⁴ it is very plausible that for any nonzero cardinality k there could be exactly k concrete objects. Or we might follow Pruss (Reference Pruss2001) and take K to be a kind of elementary particle like photon, which allows for more than one instance of the kind to be in the same quantum state (you cannot run this argument on the basis of electrons). Moreover, while physicists appear to think of space-time as having at most the cardinality of the continuum, there does not seem to be a principled reason to limit space-time to be Archimedean in nature.Footnote ⁵ One can easily mathematically construct large cardinality non-Archimedean space-times (just use the upward Löwenheim-Skolem theorem), and it is plausible that such a space-time could be populated with a large cardinality of entities. At least we should not rule out such possibilities a priori. Or we might take K to be a kind like universe, for example, defining a universe as a maximal spatiotemporally connected aggregate of concrete objects. Pace David Lewis (Reference Lewis1986), it is very plausible indeed that for every nonzero cardinality k (again, the case will be controversial) there could have been exactly k universes. Many versions of string theory predict a multiverse containing a large finite cardinality (and others predict an infinite cardinality) of universes, so hypotheses about the cardinality of the set of universes are within the realm of science.

Given such a kind K, we can construct a counterexample to regularity as follows. We imagine the regularist to have assigned to all propositions probabilities that take values in some large set V satisfying V1 and V2 and satisfying propositional analogues of K1∗ and K3∗, namely, that for every proposition p and whenever p and q are incompatible. Let be the Hartogs number of V. Let be the αth element in the aleph sequence (Takeuti and Zaring Reference Takeuti and Zaring1971, 81). Then the number of initial ordinals β such that equals . Now let Ω be the space of all such initial ordinals β (this will be well ordered under the ordinal ordering) and let F be any algebra of measurable sets that contains all singletons and all ordinal intervals of the form (we could just let F be the power set of Ω if we think every subset of Ω should be measurable). Then . Since the range R of P is a subset of V, it follows that , and, by theorem 1, it follows that P is not regular.Footnote ⁶

The singletons in F correspond to propositions of the form “there are exactly β₀ Ks,” while the ordinal intervals correspond to propositions of the form “the cardinality of Ks is infinite but less than β₀,” and it is very plausible that such propositions should have probabilities on a Bayesian program. After all, for certain kinds K, we might well have to engage in scientific reasoning about such propositions.

We can also extend such examples to generalized V-probability assignments. While for V-probability on propositions, regularity says that only impossible probabilities p have V-probability 0, for generalized V-probability on propositions, regularity would say that if p entails q and , then q also entails p (i.e., p and q are logically equivalent). Extended examples would show that generalized V-probabilities do not save regularity. The basic lesson of this article is that no matter how large our range of probability values is, we can choose a sample space of possible situations that is large enough that we will have to assign probability zero to some possibility or at least (this is the case of generalized V-probabilities) assign the same probability to a set as to a proper subset of it, even if we have significantly relaxed the axioms of probability.