2.1. The Best Systems Account

The regularity theory of law holds that laws are true generalizations; the frequentist account of chance holds that chances are actual frequencies. A naive regularity account holds that all true generalizations are laws; a naive frequentist account holds that all chances are exactly equal to actual frequencies.

Both naive views are clearly false. Not all true generalizations are laws: some, like the fact that all of the eggs in my refrigerator are brown, are merely accidental. Laws support counterfactuals, but this generalization about my refrigerator does not: were I to buy white eggs, they would not become brown when I put them in the fridge. Similarly, not all chances precisely match their frequencies. For suppose the actual number of coin flips was odd; then it would be impossible for exactly half of the flips to be heads. But it is ridiculous to suppose that a fair coin could not be flipped an odd number of times. Similarly, the quantum mechanical probability of some events is an irrational number, but as actual relative frequencies are ratios of integers they cannot be irrational.

Refined frequency and regularity accounts circumvent these counterexamples by restricting the generalizations that are laws and loosening the link between chances and frequencies. The most developed such account is Lewis’s (Reference Lewis and Jeffrey1980, Reference Jeffrey1994) Best System Account (BSA) of laws and chances. Lewis’s account holds that the laws and chances are those generalizations and chance statements that form a deductive system that maximizes three virtues: strength, simplicity, and fit.

We will call a set of generalizations and chance statements a ‘lawbook’. Each lawbook receives a score for each virtue. Lawbooks get higher marks in strength for implying many true propositions (or, equivalently, ruling out many worlds). Lawbooks get higher marks in simplicity for being syntactically shorter: by having fewer and shorter sentences.Footnote ³ Lawbooks get high marks in fit for having chances that are close to the frequencies of the world. The points in each category are tallied; the lawbook with the most overall points comprises the scientific laws of the world.

The BSA escapes the counterexamples to the naive regularity and frequentist accounts by evaluating lawbooks holistically and invoking a simplicity consideration. Because lawbooks get higher marks for simplicity when they have fewer sentences, not every true generalization will be a law—only those whose contribution to the inferential power of the lawbook outweighs their cost in syntactic length. Similarly, Lewis suggests that a lawbook may contain probabilities that diverge from the actual frequencies for simplicity considerations. For example, a lawbook may assign 0.5 to a coin flip coming up heads despite the actual frequency of heads being 0.4999999.

Now for some brief criticism: Lewis’s account of fit, like his account of strength, applies to worlds as a whole, rather than local events. Just as a system may be strong but allow inferences only from global states of the world to other global states, a system may fit well but provide little information about the likelihood of local events. Such systems are almost completely useless to embedded agents, who can neither discover nor apply them. This should worry us; our laws should be ours, both available and useful to agents like us.

Furthermore, it is not clear how simplicity should lead us away from the actual frequencies of the world. In what sense is .4999999 more complex than .5? Both are rational numbers. Perhaps we should count the number of decimal places. But if this is correct we have a problem: quantum mechanics often assigns events irrational probabilities. These have infinite decimal expansions and so are by this measure maximally unsimple. But since irrational numbers are not ratios of integers, they cannot be the actual value of frequencies. So simplicity measured in this way cannot explain why quantum chances diverge from frequencies.

I will explore an alternative constraint in section 3.2. The alternative constraint I offer is more closely tied to our epistemic access to the laws. Strength is counterbalanced in laws, not by a virtue measuring the ease by which they can be expressed but instead by a virtue measured by how easily they can be discovered.Footnote ⁴ First, however, I will provide an overview of the orthodox view of the second face of chance: its relationship to our partial beliefs.

2.2. The Chance-Credence Link

The connection between chance and credence is codified by chance-credence norms. Three are currently active in the literature: the Principal Principle (Lewis Reference Lewis and Jeffrey1980), the New Principle (Hall Reference Hall1994; Lewis Reference Jeffrey1994; Thau Reference Thau1994), and the General Principal Principle (Ismael Reference Ismael2008). I will discuss each of these in turn and then give reason to prefer the New Principle.

where T_i is a chance theory and ch_i(A) is the chance that theory assigns to A. As a constraint on initial credences, the Principal Principle tells us to match our credences (where $b\left(A|E\right)$ indicates our credence in A given some evidence E) to the chances (where ch(A) is the chance of A). That is, if we think that the correct chance theory assigns a chance of x to A, we should have a credence of x in A, whatever other (admissible) evidence we have.

Unfortunately there is a problem combining the Principal Principle with the reductionist account of chances outlined in section 2.1. For recall that fit requires the chances to match the frequencies as much as simplicity will allow. Hence, arbitrary mismatches between the chances and the frequencies are metaphysically impossible. As is customary I will call such mismatches undermining futures, because they provide evidence against the chance function, and call chance functions that allow them modest chance functions. A modest chance function is so called because it does not assign itself a chance of 1—it is not certain that it is the right chance function.

As an illustration, suppose T is a chance theory that assigns .5 to coin flips landing heads. If I know that the coin will be flipped exactly 400 times, what credence should I assign to every flip coming up heads? Because T will only be the chance theory if the frequency of heads is near 0.5, if T is the chance theory this implies that all heads is false. We should give no credence to the metaphysical impossibilities,Footnote ⁵ so we should set $b\left(\mathrm{allheads}|T\right)=0$. But if $\mathrm{ch}\left(\mathrm{heads}\right)=.5$, then $\mathrm{ch}\left(\mathrm{allheads}|400\mathrm{flips}\right)=3.87\times {10}^{\mathrm{-121}}>0$. That is, the chance of every flip coming up heads, given that the coin was flipped 400 times, is extremely low but above zero. So by PP, $b\left(\mathrm{allheads}|T\right)=\mathrm{ch}\left(\mathrm{allheads}\right)>0$: the Principal Principle tells us to set our credence in all heads conditional on the chance theory to its chance, which is nonzero. But this sequence of flips is metaphysically impossible given Lewis’s account of the metaphysics of chance. Hence, the Principal Principle leads to inconsistent constraints on our credences: either we should have nonzero credence in the metaphysically impossible, or we should diverge our credences from the chances.

In response to this worry, reductionists about chance have embraced the New Principal (Hall Reference Hall1994; Lewis Reference Jeffrey1994; Thau Reference Thau1994):

NP tells us to set our credences to the chances conditional on all of our evidence, including the fact that they are the chances. In the special case of non-self-undermining chance functions, $\mathrm{ch}\left(T\right)=1$, $\mathrm{ch}\left(A|T\right)=\mathrm{ch}\left(A\right)$, and NP agrees with PP. But if a chance function assigns positive chance to propositions with which it is incompatible, $\mathrm{ch}\left(T\right)<1$, and so for undermining propositions U, $\mathrm{ch}\left(U|T\right)=0\ne x=\mathrm{ch}\left(U\right)$. Thus, $b\left(U|T\right)=\mathrm{ch}\left(U|T\right)=0$ without contradiction.

Finally, there is Ismael’s (Reference Ismael2008) General Principal Principle:

Because GPP places no direct constraints on an agent’s conditional credences, it is compatible with GPP that an agent has conditional credence $b\left(A|T_i\right)=0$ when ${\mathrm{ch}}_i\left(A\right)\ne 0$. However GPP has other unpalatable consequences: Pettigrew (Reference Pettigrew2013a) shows that GPP is inconsistent with Bayesian conditionalization and that GPP requires us to assign credence 0 to chance functions that are themselves modest but that assign a nonzero probability to immodest chance functions.

Because NP is the least internally problematic principle, and because it fits best with other expert principles, I focus on it for the remainder of the article.

3.1. Measuring Accuracy

There is a growing literature on accuracy measures for credence functions (Joyce Reference Joyce1998; Leitgeb and Pettigrew Reference Leitgeb and Pettigrew2010). The idea is simple: we take one credal function to be maximally accurate at a world. Call this the vindicated function. If our aim is truth, we will take the vindicated function at a world w to be the truth function at w, v_w, where the truth function assigns 1 to truths and 0 to falsehoods.

We then devise a measure of distance between credal functions. The accuracy of a credal function then is its distance from the vindicated function. The distance measure is a function D(d_i, d_j) that measures the distance between two credal functions. If both functions are the credences of agents, the distance measure will tell us by how much they disagree. The standard distance measure takes distance to be the sum of the squared differences between the credences of the agents in each proposition:

where we assume for simplicity that the two agents have credences in the same (finite) set of propositions, F.

Now that we have a maximally accurate function v_w and a distance measure, we have all we need to compare the accuracy of partial beliefs at a world w. For we can define the inaccuracy of a credal function b(*—*) as distance from maximal accuracy:

Now that we have in hand a notion that will allow us to compare the accuracy of credal functions, we can state the thesis that the fit of a chance theory varies with respect to its accuracy; chance theory T has a higher fit score than T* at a world w if and only if ch_T is less inaccurate than ${\mathrm{ch}}_{T^{*}}$.

I will not argue directly for Fit as Accuracy. Rather, I will show how Fit as Accuracy provides a better explanation of the normative force of the NP, more directly explains the fact that chances are probabilities, and overcomes putative objections. Fit as Accuracy does better than traditional measures of fit in some respects and worse in none.Footnote ⁶

3.2. Fit, Accuracy, and the BSA

Minimizing inaccuracy is a good first step for our account of the chance function. But it should be clear that the chance function cannot merely be that credence function that minimizes inaccuracy. For we know what this function is: it is the truth function. And we know that the truth function is not the chance function.

The BSA, of course, has a response to this: fit trades off against simplicity. But Lewis is unclear about how one chance function can be simpler than another, as I discussed in section 2. Here I will make two new proposals.

4.1. An Epistemic Utility Argument for the Principal Principle

Pettigrew’s (Reference Pettigrew2012) proof is based on the notion that chance is to credence as truth is to full belief. Pettigrew (Reference Pettigrew2013b) gives a distinct proof that I will not consider here. Our credences, holds Pettigrew, should aim at the chances just as our beliefs should aim at the truth. To represent this goal, Pettigrew introduces the chance-based Brier score as a measure of epistemic utility:

where E is the total evidence admissible to an agent at a time. We will say that CBBS measures epistemic utility in the presence of E. Taking CBBS as a measure of epistemic utility follows from the claims that (1) the ideal credence function at w is ${\mathrm{ch}}_w\left(*|*\right)$ and (2) the disutility of a credal function is proportional to its distance from the ideal function, as measured by the sum of squared differences. Note that the CBBS is just Inaccuracy from section 3 with the ‘vindicated’ function taken to be the chance function rather than the truth function.

The remaining premises in Pettigrew’s argument are imported from Joyce (Reference Joyce1998). First, we assume (3) dominance: if credence function b has a higher epistemic utility than b* at all worlds, and there is no other credence function that has a higher epistemic utility than b at all worlds, then it is irrational for an agent to employ b*.

Pettigrew shows that 1, 2, and 3 together imply that agents are irrational to adopt credal functions that do not obey the axioms of probability and the Principal Principle—and that by slightly tweaking CBBS we can show that agents must obey the New Principle. Pettigrew’s proof relies on taking the chance function to be a probability function.

Pettigrew provides little support for claim 1, that the ideal credence function at w in the presence of E is ${\mathrm{ch}}_w\left(*|E\right)$. His argument rests on the claim that chance is to credence as truth is to full belief, which Hájek (Reference Hájek2011) has defended. But it is difficult to see how this claim could be defended without a metaphysical account of chance. We take truth to be the aim of belief not as a basic posit but because it comports well with our theories of truth. On the most naive correspondence theory, the true propositions represent the actual world. This world dependence is what makes truth an appropriate aim for belief in the actual world; in order to accept the claim that our credences should aim at the chances, we need a similar account of the world dependence of the chances.

Pettigrew also fails to support the claim that the chances must be probabilities. To my knowledge, there are two defenses of the claim that the chances must be probabilities. The first relies on frequentist or hypothetical frequentist accounts of chance: frequencies, as ratios of outcomes, obey the probability axioms, so the chances must be probabilities.Footnote ⁹ But this justification relies on a false theory of chance, as discussed in section 2.1. The second justification goes via the Principal Principle—the chances are those things to which we are rationally required to match our credences. It is irrational to have nonprobabilistic credences; thus, the chances must be probabilities. But in the context of Pettigrew’s proof, this is circular. A new vindication is given below.

4.2. Vindicating Chance

If the chances obey our two constraints, we have an argument for the claim that the chance function is the ideal credence function and a reason to believe that the chance function is a probability function. I will start by arguing that it is irrational to fail to respect the chance function arrived at by taking the lower bound of simplicity, that is, the least simple chance function compatible with Evidential Equivalence.

Recall that the chances, according to Fit as Accuracy and Evidential Equivalence, are the credence function that maximizes accuracy while also maximizing discoverability (secs. 3.2.1, 3.2.2) and treating evidentially indistinguishable setups as equivalent. The chance function, then, is the most accurate credence function that obeys the same evidential constraints that we do. In this sense, then, chance is to credence not as truth is to belief but as knowledge is to belief.Footnote ¹⁰

Now suppose that someone knowingly fails to match her credences to the chances. Then she is either employing a credence function that fails to respect Evidential Equivalence or employing a credence function that respects Evidential Equivalence but is accuracy dominated by another such function. If she fails to respect Evidential Equivalence, then her credences in the outcomes of chance setups depend on more than her evidence; this is irrational. And if she respects Evidential Equivalence but is accuracy dominated, she is also irrational—because she is knowingly employing a credence function further from the truth than she could. Hence, if she fails to respect the chances she is irrational.

This gives us reason to believe that our credences should aim not directly at the truth but instead at the objective probabilities. For just as our goal for full belief is not merely to have true beliefs but to have knowledge, our aim for partial belief should not be merely to have accurate beliefs but to have accurate and well-supported beliefs. Since our credences cannot be more accurate (while retaining evidential support) than the chance function, we should take the objective chances at our world to be the target of our credences. Finally, this account of accuracy puts the claim that chances are probability functions on more solid ground.

By making accuracy a constitutive feature of chance, we are able to show that the same constraints that require our credences to be probabilities require the chances to be probabilities: namely, any nonprobabilistic chance function will be accuracy dominated by a probabilistic one (see Joyce Reference Joyce1998). But since our chances must be the most accurate credal function at some grain of simplicity, they will not be accuracy dominated. So they will be probabilities.