1. Introduction

How should Humeans understand chance? If Humeanism demands Humean Supervenience (HS), there are significant problems. HS demands that chance supervene on history, but intuitively, different chances are consistent with identical histories. Further, combination of HS with David Lewis's Principal Principle (PP), which specifies the intuitive relation between chance and rational credence, plausibly yields a contradiction—the paradox of undermining futures. Humeans have responded to the paradox with a veritable cottage industry of solutions. Lewis, Ned Hall, Michael Strevens, Jonathan Schaffer, and others have offered proposals for finessing the ‘Big Bad Bug’. To properly discuss every proposal would require a substantial review article, but it will suffice to say that each—typically by its author's own admission—is less than ideal. Some intuitively compelling claim about chance, one reflected in our practice and not merely our philosophical preconceptions, must go by the board. The proposals show great subtlety, but they are not what we would have wanted when first we began to theorize about chance. The overall impression is of a square peg tapped with exquisite delicacy into a round hole.

However, Humeanism does not demand supervenience. If chances are projections of our confidences, and our standards for projection are not so stringent as to associate a unique chance with each total history, there need be no paradox. This paper fleshes out such a proposal, yielding a view that does not conflict with our practice, and throwing a justification of PP into the bargain. Of course, being situated at the projectivist end of the Humean spectrum, this is a disreputable kind of theory. However, as I shall argue, it satisfies the standard contemporary motivations for Humeanism; it delivers the features of our chance practice that its Humean competitors have had to abandon; and it is defensible: popular misconceptions and bluff no-nonsense realist posturing notwithstanding, there are no knock-down refutations of projectivism with regard to chances (or scientific laws, or ethics, or rationality for that matter). Hence, at the very least, it merits serious consideration.

2. The Problem

In “A Subjectivist's Guide to Objective Chance,” David Lewis ([Reference Lewis and Jeffrey1980] 1986) argued persuasively that PP captures the intuitive connection between rational credence and chance:

Lewis (Reference Lewis1986, 92) characterized admissible information as “the sort of information whose impact on credence about outcomes comes entirely by way of credence about the chance of those outcomes.” Plausibly, historical information and chance theories will be admissible. Thus, if H is the actual history to the present, T the actual theory of chance, and P the chance function determined by H and T, $C(A\vert HT) =P(A) $ . The problem derives from the conjunction of PP with Lewis's HS: For worlds like ours “… all else supervenes on the spatiotemporal arrangement of local qualities throughout all of history, past and present and future” (Lewis Reference Lewis1994, 474). The exact nature of the supervenience base is unimportant here; supervenience of the chances on the nonmodal facts suffices to generate the paradox as follows.

On any remotely plausible analysis of chance that respects HS, the present chance of an outcome must supervene on all of history, not merely the past and present. Let F be an undermining future, one that conjoined with H would determine different present chances than the actual chances and for which $P(F) \neq 0$ . PP yields $C(F\vert HT) =P(F) \neq 0$ . However, since F is undermining, the conjunction of F and H is inconsistent with T, and $C(F\vert HT) =0$ . We have a contradiction in our theory of rational credence, born of the marriage of HS and PP. For example, consider a world that contains one type of chance process, say, coin flips. Suppose that at some time, the laws determine that only a finite number of flips will occur, and that the chances of outcomes are independent and identically distributed. A world where half the flips come out heads will have a chance theory that, at some time t, assigns $P(F) \neq 0$ , where F is undermining, specifying for instance, that 90% of such flips come up heads.

In response, Lewis (Reference Lewis1994) and Hall (Reference Hall1994) each proposed the New Principle (NP) that purportedly captures the correct connection between rational credence and chance. Lewis formulated the most important case of NP as follows:

If F is undermining, then $P(F\vert T) =0$ . The contradiction is evaded. Lewis and Hall argue that in typical applications, the credences furnished by NP will closely approximate those furnished by PP. Both Hall (Reference Hall1994) and Strevens (Reference Strevens1995) offer intuitive justifications for NP. However, even if these justifications are legitimate, the counterintuitive inapplicability of PP to standard situations remains. Further, NP is incompatible with the intuitive requirement that if the chance of A is nonzero at some time, then A is possible at that time (a requirement more rigorously formulated by Bigelow, Collins, and Pargetter (Reference Bigelow, Collins and Pargetter1993) as the Basic Chance Principle (BCP)). Jonathan Schaffer (Reference Schaffer2003) evades these undesirable features by the simple expedient of assigning chance zero to all undermining futures: His L*-chances are defined by conditionalizing the L-chance distribution, the distribution delivered by Lewis's (Reference Lewis1994) best system analysis of laws and chances, on the chance theory T. Thus, L*-chances are compatible with BCP and can be plugged into PP without fear of undermining. Of course, they do not remotely resemble chances. L*-chance distributions—as Schaffer recognizes, and as is clear from a related discussion by Frank Arntzenius and Ned Hall (Reference Arntzenius and Hall2003)—support single-case chances of outcomes that vary from trial to trial. For cases where we have a finite number of coin flips, the L*-chance will generally depend on the outcomes of previous flips. Schaffer additionally proposes that our notion of chance is vague, with both L*-chances and L-chances as acceptable precisifications. By shuttling between the two, depending on what intuitive requirement is to be saved, one may claim that one has a vague notion of Humean chance that meets all our desiderata. Delicate hammer work, but a square peg nevertheless.

3. A Solution: Projecting Chances

Considerations of space demand that I am selective in my presentation. I shall restrict my discussion to the case of coin flips. How it might be generalized is, I think, reasonably clear. Before proceeding, a note about the scope of the analysis: It is intended to apply even if coin flips are, at bottom, deterministic processes. In that case, the chance laws are to be understood as laws that furnish predictions from a type of initial data that is insufficiently precise to allow perfectly reliable prediction of the outcomes. Similarly, on this analysis, classical statistical mechanics furnishes chance laws, even though classical physical systems are deterministic.

Science has two chief goals: explanation and prediction. We can plausibly characterize our nonchancy predictive goals as follows. We demand accuracy and adequacy: Our predictive apparatus must only make true predictions, and it must, at least in principle, provide predictions of all actual events, excepting only boundary conditions. However, our chancy predictive goals are more modest. Whether we are reduced to pursuing them by a fundamental indeterminism in the world, the limited precision of our measurements, or the difficulty of exactly solving systems of the order of 10²³ coupled partial differential equations, they are for situations where we have no rational basis for selecting a predictive apparatus that specifies the outcomes of experiments. Thus, rather than specifying outcomes, our predictive apparatus should specify credences regarding outcomes. The question is then: What, plausibly, is the goal with respect to which we should evaluate these credences?

I shall equate our chancy, predictive goal with gambling success, and I shall take maximization of expected utility as the basis for rational decision making in gambling situations. Suppose you and I are to play a betting game on the outcomes of coin flips. I propose odds and a stake. If I propose odds of 2:1 on heads and a stake of $15, then whoever bets on heads puts up $5, the other bettor puts up $10, and the winner takes the entire $15. More generally, if the odds ratio on heads is $a\thinspace{:}\thinspace b$ , you have credence p in the outcome heads, and we measure the stake in units of utility. You will have the following expected utilities for bets on head and tails:

Maximization of expected utility dictates that you pick the side of the bet with the greater utility. If U(heads) = U(tails) you should be indifferent between the sides of the bet. In that case, we have that $a/ b=p/ 1-p$ . This odds ratio is the fair-betting quotient determined by your credences. Thus, it is only rational to pick a particular side of a bet, if the odds ratio proposed is less than or equal to the fair-betting quotient determined by your credences, $p/ 1-p$ . Note that I am not claiming we should interpret the fair-betting quotient as specifying the odds at which we are disposed to bet. The naive equation of degrees of belief with betting dispositions seems psychologically unrealistic. Further, we must recognize that scientific hypotheses may correspond to bets that cannot be effectively decided between human agents. Nevertheless, I shall assume that we can make judgments of the fairness of sets of odds, even though the gamble may not be effectively decidable or we may have no desire to gamble. I shall take these subjectively fair odds as a reliable indicator of our credences. Our predictive goal is then, in a sense to be further clarified, attuning our odds for gambling success.

All we need to pursue our chancy predictive goal is a set of credences regarding the outcomes of chance trials. Why, then, should an agent not merely have such credences that she updates in accord with the data, say, by Bayesian conditionalization, coupled with some reasonable inductive prejudices? In other words, why do we need talk of chances or chance laws at all? Our explanatory goals provide the reason. It's inductively rational to expect unobserved cases to resemble the observed. Thus, in normal circumstances, one's credences in future outcomes being heads should increase if one determines that the outcome of a flip is heads. Correspondingly, one's credence regarding the outcome of some future coin flip conditional on other future outcomes should not, in general, equal one's unconditional credence. My credence for heads on the third flip I will perform, conditional on the first two outcomes being heads, should exceed my credence conditional on the first two outcomes being tails, and also exceed my unconditional credence. This is where the explanatory goal of science motivates us to think in terms of chances: An agent who did not form chance hypotheses, but as above, formed her credences directly, would lack the resources to distinguish between systems whose behavior should be explained in distinct ways.

Consider the radioactive decay of uranium atoms. In certain finely tuned situations, when such atoms are bunched together, they decay more quickly than isolated ones. This is readily explained. Decaying uranium atoms produce neutrons, which if of appropriate energy, can trigger neighboring atoms to decay. Hence, one can reasonably expect an unisolated atom to decay more quickly than an isolated one. This is an example of a Markov process, a process where the outcome of a chance trial is dependent on the outcomes of other trials. The explanation is founded on a set of credences for which the probability of a particular decay conditional on the decay of neighbouring atoms exceeds that of a decay conditional on there being no such neighboring decay. Even if we didn't have the story about neutrons from one decay triggering another, we could legitimately explain the phenomenon in terms of those conditional probabilities, and hence in terms of the mutual proximity of the uranium atoms. The comparison of the conditional credences is an essential part of the explanation, and our having such conditional credences is mandated by data for which the relative frequency of decays conditional on adjacent decays is significantly higher than the frequency conditional on no adjacent decay.

Returning to our inductively rational sets of credences for coin flips, we see that they also have this Markovian structure. Our credence for the outcome of a flip conditional on previous outcomes depends on the nature of the previous outcomes. If we use these credences to explain the outcomes of coin flips, we should explain an outcome of heads that is preceded by a run of heads in Markovian terms. Figuratively, we must think of coins as having a ‘memory’ of their previous outcomes. However, without wishing to belabor the obvious, there are distinct explanatory possibilities we might reasonably countenance for coin flips. It might, indeed, be that coins have ‘memories’, but of course, previous outcomes might be explanatorily irrelevant. So, we need chance hypotheses. At least we need something other than directly formed credences, if we are to have sets of credences corresponding to the distinct explanatory possibilities regarding the behavior of such systems.

I shall now characterize these sets of credences. A full account of how we form our opinion regarding coin flips should include a specification of the sets of credences corresponding to Markovian explanations. These explanatory possibilities should not be dismissed out of hand. However, such a treatment is beyond the scope of this paper, and I restrict my discussion to the non-Markovian options. Given this constraint, what form should candidate predictive-explanatory hypotheses for coin flips take? Several considerations suggest themselves. First, however closely we scrutinize the setups of coin flips, we cannot identify any feature whose instantiation guarantees that the coin invariably produces the outcome heads or the outcome tails. Therefore, we should be prepared to countenance predictive-explanatory options for which there is no fact regarding the setup that is a perfectly reliable indicator of the flip's outcome.Footnote ¹ Second, as per the above discussion, candidate hypotheses on which we take the outcomes of previous flips as explanatorily irrelevant to the succeeding outcomes should determine credences conditional on arbitrary specifications of previous outcomes that equal the unconditional credences. Third, if we scrutinize the setups of different coin flips, we find that there is nothing about the setup of any flip that distinguishes it from any of its fellows with regard to indication of outcome produced. It doesn't matter if you flip, or I flip, or we face the coin heads up prior to flipping; apparently, none of these variations is explanatorily relevant to the relative frequencies of outcomes. So, we should be prepared to countenance predictive-explanatory options for which such factors have no predictive or explanatory relevance, ones for which the unconditional credences determined by a hypothesis do not differ from flip to flip. So, as per the first point, our hypothesis should only specify credences regarding outcomes. Further, as per points two and three, the credences (conditional and unconditional) for all flips should be identical. Therefore, the credences determined by each candidate predictive-explanatory hypothesis are completely specified by a value for p, where $0\leq p\leq 1$ , with p the credence assigned to heads and $(1-p) $ the credence assigned to tails, for any trial. Hence, for a sequence of n trials, commitment to some package of credences commits us to a family of outcome sequences of differing relative frequencies. For $p=.5$ , as the relative frequencies deviate from .5, we find that the associated credences decrease. Thus, for a sequence of 10 flips, the package specifies degree of belief $1/ 2^{10}$ in sequences with 10 heads, and degree of belief $10/ 2^{10}$ for sequences with 9 heads, etc. In the infinite limit the package specifies credence 1 for sequences with relative frequency of heads .5 and credence 0 for the possibilities with relative frequencies that deviate from the chance value, albeit the latter sequences are clearly possible according to this construction.

More formally, each of the chance hypotheses defines a measure as follows. A probability space is a triple $\langle S,\,F,\,P\rangle $ , where S is a nonempty set called the outcome space, F is a sigma-field on S, i.e., a set of subsets of S that includes S and is closed under complementation and countable union, and P is a function from F to the real numbers satisfying the probability axioms:

1. 1. $P\left(A\right)\geq 0$ , for any $A\in F$

2. 2. $P\left(A\right)=1$

3. 3. If {A _i} is a countably infinite collection of pairwise disjoint sets, each of which is an element of F, then

   

Take S as the set of all infinite sequences of possible outcomes of coin flip; we can define a cylinder set as the set of all sequences $s\in S$ that agree on outcomes in a finite number of places, e.g., the set of sequences that have heads on the fifth, third, and thirty-seventh tosses and tails on the sixth, constitute a cylinder set. Assigning probabilities to the cylinder sets in the natural way (e.g., for the above set, if p is the probability of heads and $(1-p) $ is the probability of tails, then the assigned probability is $(1-p) \cdot p^{3}$ ) there is a unique countably additive extension of the probability function P to the sigma-field F generated by Ø and the cylinder sets.Footnote ²

Such measures are the ones intuitively supported by chance laws for coin flips in the absence of inadmissible information, and they were constructed from an elementary consideration of the role of credences in prediction and explanation. I shall later establish the Humean credentials of my view, but for the moment let us note that no non-Humean ontology was invoked. Further, the considerations adduced in the construction of these sets of credences are not philosophical in character. We have engaged only in the kind of reasoning that, plausibly, a scientist might follow to determine what credences she should consider as options for predicting and explaining coin flips. I don't think there's much more to chance than sets of credences like these. There's more to the analysis that comes in the form of a normative construal of chance law claims. Roughly, chance law claims mean ‘these are credences that ought to be used for prediction and explanation’. In addition, we must construct a formal confirmation theory. However, we should first clarify the nature of our predictive goals.

It might look as if a close relative of the problem of undermining futures rears its head for this view. If I commit to the $p=.5$ package of credences as the one that ought to be used in prediction and explanation, then in the absence of inadmissible information, I accept those credences as specifying the gambling odds I ought to adopt. However, those credences allow the possibility that every coin flip from here to eternity comes up heads, and hence, that my odds will let me down horribly. Indeed, committing to that package of credences is entirely compatible with knowing that another set of odds will serve me better. For instance, in a three-flip game with constant stakes, I know the best credences for bets on heads are those that match the actual frequency of outcomes: either 1 (3 heads), $\frac{2}{3}$ (2 heads, 1 tail), $\frac{1}{3}$ (1 head, 2 tails), or 0 (3 tails).Footnote ³ How can I rationally commit to a package as specifying the odds I ought to adopt, while simultaneously recognizing that another set of odds could serve me better? A little care in characterizing my commitment that the $p=.5$ package satisfies our predictive goals disposes of the worry. I am not committing to the claim that my choice of betting odds will be vindicated in any game. Nor do I commit to the specified odds being better than every other set of odds I could adopt for the game in question. However, while I do not expect the $p=.5$ odds to serve me better than every other set of odds, I do expect them to serve me better than any other particular set of odds. For instance, suppose you have degree of belief $\frac{1}{3}$ in heads. You will be willing to post odds that provide, from my point of view, a disproportionately large payoff for heads. Consequently, I will evaluate you as having a bad set of credences, in that they direct you to accept bets that according to my credences have a negative expected utility. I think you're taking a bad risk, although I cannot guarantee it, and I expect to make a killing if we continue to play rationally with our respective odds. Correspondingly, given my commitment to the $p=.5$ package, adopting fair-betting odds that deviate from those it specifies would dispose me to place bets which, according to my $p=.5$ credences, have negative expected utility. In a three-flip game, adoption of any one of the fair-betting quotients that might turn out to be the best, 1, $\frac{2}{3}$ , $\frac{1}{3}$ or 0, would be irrational for just that reason. So, there's no irrationality implicit in committing to the expectations determined by such a package of credences and holding it as specifying the fair-betting odds one should use in gambling; in fact, the former rationally requires the latter.

4. Semantics and Confirmation Theory

We shall now construct a semantics for chance law claims. I think of laws in terms of rules that generate predictions and explanations. Suppose we want to predict the evolution of two billiard balls using Newton's laws of motion. We take the specification of the initial positions and velocities of the particles and use the differential equation in Newton's second law to compute their trajectories as a function of time. Depending on the initial velocities and positions, we may predict that the two balls collide and bounce off each other, or we may predict that they just travel along in straight lines and never collide. Each such specification of the physical state of a two-ball system over time is a model of Newton's laws, and the differential equation in Newton's second law I will call a Model Generating Rule (MGR), by which I mean only that it's a rule that yields descriptions of particular cases by generating a model for each specification of boundary conditions. We can characterize Chance MGRs (CMGRs) as follows. In a chancy world, differing evolutions may proceed from distinct tokens of one type of boundary or initial conditions: Sometimes a coin flip yields the outcome heads; sometimes tails. Thus, CMGRs associate multiple models with a single specification of initial or boundary conditions, and the chance function specifies the credence to be associated with each model. A CMGR that specifies a .5 chance for heads and a .5 chance for tails specifies two possible evolutions from the initial state, where the initial state is an unexceptional coin flip, and directs us to associate credence .5 with each model:

Since CMGRs are rules that directly associate credences with the outcomes of single trials, the generated packages of expectations will, as noted in the previous section, match those intuitively supported by chance laws.Footnote ⁴ In particular, each CMGR favors a certain relative frequency, but does not determine that only that frequency can obtain, even in the limit. By contrast, actual and modal relative frequency analyses of chances impute to chance claims such a mistaken commitment.Footnote ⁵

Each MGR is to be understood as a candidate law. The analysis of law claims is as follows: Assertion of ‘It is a law that P’ expresses endorsement of P for the purposes of prediction and explanation. Thus, we may paraphrase ‘It is a law that P’ as ‘P is a rule that should be used for prediction and explanation’. Note that this analysis does not require there to be some fact of the matter that determines P as the rule to use; the function of the sentence is merely to single P out. Nor are these claims autobiographical. As our semantics shall make explicit, chance law claims are about rules for prediction and explanation, not mental states.

We shall now develop our normative logic, exploiting Gibbard's (Reference Gibbard1990) factual-normative worlds approach. The consistency relations are rationality constraints on the use of MGRs in the construction of explanations and predictions:

1. 1. Law-Law Consistency. A set of atomic law claims is consistent iff no application of the MGRs endorsed to a set of boundary conditions (a) produces inconsistent physical state specifications, or, (b) assigns non-single-valued or probabilistically incoherent degrees of belief to any physical state propositions.

2. 2. Law-Physical State Consistency. A law claim (or set of law claims) is consistent with a physical state specification iff the application of the specified MGR (or set of MGRs) to the boundary conditions of that physical state yields a model consistent with the physical state.

An example of 1(a): the lawhood of Newton's second law, $F=ma$ (force equals the product of mass and acceleration), is inconsistent with the lawhood of say, $F=ma^{2}$ , since the application of both equations to a physical state where F and m are nonzero will yield conflicting values for the acceleration: $F/ m$ and $(F/ m) ^{1/ 2}$ . Such a model is useless for prediction and explanation. Examples of 1(b): The lawhood of ‘the chance of heads on a normal flip is .5' and ‘the chance of heads on a normal flip is .6' is inconsistent since they direct us to associate two degrees of belief with the one proposition. The lawhood of ‘the chance of heads = .5', ‘the chance of tails = .5', ‘the chance of (heads and tails) = 0', and the chance of (heads or tails) = .75' are inconsistent because they endorse rules that jointly assign incoherent degrees of belief to the set of propositions {outcome = heads, outcome = tails, outcome = heads or outcome = tails, outcome = heads and outcome = tails}. Again, such sets of law-claims approve MGRs that, working in tandem, fail our goals of prediction and explanation. A failure to meet constraint 2 entails that the MGR (or set of MGRs) makes a false prediction regarding that physical state, hence failing our predictive goals. Suppose I assert that it is a law that the chance of heads is .5 and the chance of tails is .5, and that there are no other possible outcomes of a coin flip than heads or tails. I flip a coin and find that it lands on its edge. My MGR wrongly told me that any coin flip would produce a head or a tail. Thus, the rule is predictively inadequate and endorsement of the rule for the purpose of prediction is practically inconsistent with that physical state specification. On the other hand, if the specified physical state data matches some model generated by the MGR (i.e., if that physical evolution is possible according to that MGR) then the lawhood of that MGR is consistent with that physical state. Why? First, it is predictively adequate in the weak sense that its predictions have not conclusively failed for that physical state. Second, we know that even if a physical state of that type is rather unexpected according to that MGR, it does not preclude the possibility that if the physical state were to be extended (say, by performing more coin flips) it would turn out to be a very good predictor. Thus, our standard of consistency between law claims and physical state claims should demand no more than mere predictive adequacy.

An atomic law claim A is a normative logical consequence of a set of law claims Ψ iff every (degree of belief, model) pair generated by the MGRs referred to in Ψ is a (degree of belief, model) pair generated by the MGRs referred to in $\Psi \cup \{ A\} $ . For example, the law that the chance of two heads in a row is .25 is a normative logical consequence of the law that the chance of heads, on any trial, is .5. A physical state claim is a normative logical consequence of a set of law claims iff it is true of every model generated by the MGRs specified in that set. Hence, notably, P follows from ‘It is a law that P’.

Let w be a complete specification of the physical state of a world. Let n be a set of atomic law claims: consistent with each other, closed with respect to normative logical consequence, and consistent with w in the sense specified above. Call each such pair, 〈w, n〉, a factual-normative world. These represent the rationally cotenable options among which we discriminate, or over which we spread our degrees of commitment (to be characterized below) as we acquire knowledge of the physical state. Note that this set of possibilia is the intuitive one for laws and physical states. Projectivist and non-Humean sets of possibilia differ only in that the norms in the former are replaced in the latter by factual non-Humean features. Further, this set of factual-normative worlds has been constructed from a fairly elementary consideration of the nature of prediction and explanation. This seems to me a notable result.

We can use this set of possible worlds to specify the content of claims of arbitrary complexity, solving the Frege-Geach problem (Geach Reference Geach1965). The content of a sentence S is specified as follows. Replace each occurrence in S of the normative predicate ‘is a law’ by the corresponding descriptive predicate ‘is an MGR in system n’, and relativize any claim about the physical state to w, to obtain the sentence S _wn. For example:

S _wn is a purely descriptive sentence and we say that it holds in 〈w, n〉 exactly if ‘the chance of heads is 1’ is an MGR in n or the 54 coin flips specified did not come up heads in w. More generally, S holds in 〈w, n〉 iff S _wn holds in 〈w, n〉. Then, the content of S is represented by the set of worlds O _s, for which S holds. Thus, we can straightforwardly define consistency and other logical relations over sentences of arbitrary complexity, e.g., any pair of sentences will be inconsistent, if their contents hold no worlds in common.

To complete the analysis we need a confirmation theory. Our factual-normative semantics furnish a set of consistency relations for such norms, and hence, prima facie, they can enter into the same sorts of inferential relations as factual beliefs. Indeed, the case can be made that whether we choose to account such commitments as beliefs may ultimately be little more than a matter of taste.Footnote ⁶ However, what similarities exist between full commitment to a norm and belief in a factual claim should not seduce us into casually speaking of degrees of belief in norms. We must develop an account of degrees of commitment. A successful theory of this kind will explain the appearance that we have degrees of belief in chance norms, and at that point, we may consider it harmless to speak that way. However, the right must be earned, not assumed.

What are degrees of commitment? When we fully commit to a law claim, we are committing to a strategy for prediction and/or explanation. Sometimes we do not know which predictions and explanations to give: We are uncertain which commitments to make. As with factual propositions, this uncertainty may come in degrees, characterized as follows. If we are to recover a Bayesian story regarding the confirmation of chance claims, we shall need interpretations for each of the quantities in Bayes’ Theorem. Letting K specify setup conditions for a chance experiment whose outcome is reported by e, our degree of commitment to the chance norm T is to be updated from $C(T) $ to $C(T\vert e) $ on learning e, where $C(T\vert e) $ is given by:

So, the question is how should we interpret $C(T\vert e) $ , $C(T) $ , and $C(e\vert T\,\& \,K) $ , where T is a norm?

We begin with $C(e\vert T\,\& \,K) $ , initially restricting the discussion to standard cases of law confirmation, i.e., cases for which K is admissible and specifies the setup of an experiment. The inadmissible cases are discussed below with reference to the justification of PP. T expresses endorsement of a rule for mapping from setup conditions to credences. Since K does not include inadmissible information, a rational agent will form the conditional credence by making an induction regarding the outcome of the experiment. Granted commitment to the endorsed rule as the one that should be used for making inductions, it is rational to adopt the credences generated by the application of that MGR to K as one's own, i.e., $C(e\vert T\,\& \,K) $ involves conditionalization, not on a proposition, but on a specification of credences. Thus, if the application to K of the MGR endorsed in T yields the probability distribution $P(e) $ , it is rational to treat $P(e) $ as an expert function, where an expert function is defined as follows: If C is my personal probability function, then P is an expert function for me concerning some set of propositions S iff $C(e\vert P(e) =p) =p$ , for all e in S.Footnote ⁷ Hence, $C(e\vert T\,\& \,K) =C(e\vert P(e)) $ .

Can we interpret $C(T) $ as a fair-betting quotient for T, even though T is a norm and not a factual claim? We can identify, in an intuitive way, our degrees of commitment to some norms with fair-betting quotients. Suppose we interpret subjunctive conditional claims normatively (Ward Reference Ward2002), taking a conditional such as ‘if this coin were to be flipped an infinite number of times, the relative frequency of outcomes of heads would be $\frac{1}{2}$ ' to be expressing an endorsement (for purposes of prediction and explanation) of an inference, from the claim that the situation specified in the antecedent obtains, to the consequent claim. The statement recommends a certain policy for prediction. If we were to flip the coin an infinite number of times and the limit relative frequency of heads were not $\frac{1}{2}$ , that predictive policy would have definitively failed. If the relative frequency were $\frac{1}{2}$ the predictive policy would have been vindicated. So, we can reasonably identify our degree of commitment to such a norm with our fair-betting quotient for its ‘correctness condition’ being met, were the experiment to be performed. Normative claims about relative frequencies or particular outcomes correspond to some subset A of our outcome space, the subset for which the prediction is correct, and their probability is the measure of that subset. However, a chance norm, T _n, does not apparently correspond to any such subset. Since it is normatively consistent with arbitrary sequences of outcomes, it lacks a correctness condition.

However, each chance norm does correspond to a probability measure, $C(\cdot \vert T_{n}) $ , i.e., the assignment of credences recommended by the norm. An agent who countenances some chance norm is one who is disposed to adopt the corresponding measure as her own, a disposition that would be manifested by a rational agent only under the impact of appropriate data. I will assume that a rational agent updates her probabilities by Bayesian conditionalization. Hence, she countenances the measures corresponding to the members of some subset of {T _m}, and only those members, iff her credence function, $C(\cdot) $ , has the form

where summation (here and below) is over m, and the coefficients can be modified under conditionalization so that, for any T _n countenanced, we can obtain $a_{n}=1$ and $a_{m}=0$ for all $m\neq n$ .Footnote ⁸

The coefficients are determined as follows. Writing ‘lim r.f.’ for ‘limit relative frequency’ and substituting, we have $C(\mathrm{lim}\,\mathrm{r.f.}\,=n) =\sum a_{m}C(\mathrm{lim}\,\mathrm{r.f.}\,=n\vert T_{m}) $ . Thus, since $C(\mathrm{lim}\,\mathrm{r.f.}\,=n\vert T_{m}) =\delta _{nm}$ (i.e., 1 if $n=m$ , and 0 otherwise), we obtain $a_{n}=C(\mathrm{lim}\,\mathrm{r.f.}\,=n) $ . The coefficient, a _n, corresponding to each norm, T _n, is just our confidence that the limit relative frequency would equal that chance value n.

This yields the following reasonable characterization of degrees of commitment to chance norms. A rational agent who countenances such chance norms has as credence function, $C(\cdot) =\sum C(\mathrm{lim}\,\mathrm{r.f.}\,=m) C(\cdot \vert T_{m}) $ . In general, $C(\cdot) =\sum C(\mathrm{lim}\,\mathrm{r.f.}\,=m) C(\cdot \vert \mathrm{lim}\,\mathrm{r.f.}\,=m) $ . Thus, for each n, $C(\cdot \vert \mathrm{lim}\,\mathrm{r.f.}\,=n) =C(\cdot \vert T_{n}) $ , i.e., for an agent who countenances T _n, the measure obtained by conditionalizing on the limit relative frequency equaling the chance value, n, is just the measure specified by the chance norm T _n. That the limit relative frequency would equal the chance value is not a correctness condition for T _n, but for such an agent, $C(\mathrm{lim}\,\mathrm{r.f.}\,=n) $ is her confidence that the credences specified by T _n are the ones that she should use. Further, that the limit relative frequency would equal n, is the unique condition of this kind. It is the only proposition which, conditionalized on, in general yields the credence function $C(\cdot \vert T_{n}) $ , whatever norms the agent chooses to countenance, or not to countenance, in addition to T _n. Hence, in general, $C(\mathrm{lim}\,\mathrm{r.f.}\,=n) $ is her degree of commitment to T _n.

Such expressions as $C(T_{n}\vee T_{m}) $ , her degree of commitment to $T_{n}\vee T_{m}$ , can be interpreted as $C(\mathrm{lim}\,\mathrm{r.f.}\,=n\vee \mathrm{lim}\,\mathrm{r.f.}\,=m) $ , since the latter is, in general, her confidence that the coin would have a limit relative frequency which, conditionalized on, gives the T _n measure or the T _m measure, i.e., we can interpret it as her confidence that either the T _n credences or the T _m credences are the ones she should use. Clearly, we can similarly interpret degrees of commitment to arbitrary semantic compounds of chance norms. Further, since $C(\mathrm{lim}\,\mathrm{r.f.}\,=n) $ , $C(\mathrm{lim}\,\mathrm{r.f.}\,=m) $ , $C(\mathrm{lim}\,\mathrm{r.f.}\,=n\vee \mathrm{lim}\,\mathrm{r.f.}\,=m) $ , etc., are probabilities, degrees of commitment to chance norms and their compounds manifestly conform to the axioms of probability. Finally, $C(T_{n}\vert e) $ is just $C(\mathrm{lim}\,\mathrm{r.f.}\,=n\vert e) $ .

I have two final comments on confirmation. First, on this story, our goal in chance theorizing is not finding the facts about the chances; there is no fact that determines any chance claim as true. How then can a projectivist be justified in asserting a chance law, i.e., in holding that a set of CMGRs is the one that ought to be used? All of the constructed packages of expectations tell us that the longer the sequence of trials (i.e., the more data we have) the more confident we can be that the observed relative frequency is close to the limit relative frequency that would be obtained if the flips were infinitely extended. Hence, ceteris paribus a set of confidences regarding limit relative frequencies is better than another if it is based on more experimental trials, in the sense that it is inductively rational to be more confident that it is tightly focused around the limit relative frequency that would be obtained. Thus, our pursuit of chancy predictive success is regulated, as any sensible epistemology will demand, by the goal of maximization of experimental data, and an agent is justified in asserting a chance law insofar as a significant number of trials confer on it a high degree of commitment. Second, of course, any Bayesian theory must ultimately provide some story about prior probabilities. I have nothing original to say on the topic here. However, note that on our semantics, ‘the chance equals p’ is entirely distinct from ‘I/you/we are confident that the chance = p’. Thus, the absence of objective constraints on priors would yield subjectivity in rational opinion regarding the chances; it would not make the chances depend on our opinion. There's no problem peculiar to projectivism here.

5. Vindication and Justification

The paradox of undermining futures depends on the existence of futures, F, for which the actual chance $P(F) \neq 0$ , but which conjoined with H, determine different present chances than the actual chances. However, as shown, for the projectivist, distinct chance law claims for coin flips are mutually inconsistent, but individually consistent, with any given outcome sequence composed entirely of heads and/or tails. Hence, there are no undermining futures, and the paradox does not arise. Indeed, since the set of possibilities, as represented by factual-normative worlds, is isomorphic to the intuitive set of possibilities, the view squares with our intuitions in being inconsistent with HS, but vindicating BCP.

PP is justified as follows. The chance packages for coin flips were determined by appeal to our goals of prediction and explanation, and it is rational to employ them in forming our credences when we are making an induction regarding the outcome of a chance trial. If e is a statement regarding the outcome of a trial and K merely specifies admissible information, say the setup of the experiment, then induction is the rational mode of credence formation regarding e. Hence, as discussed in the previous section, with $P(\cdot) $ the chance probability distribution obtained by application to K of the MGR endorsed in T:

Cases where K specifies inadmissible information break into two types. If K entails the truth or falsity of e, rational belief formation regarding e no longer involves induction. Hence, the chance law is irrelevant to rational credence formation. In cases where K provides inadmissible information that does not entail the truth or falsity of e, plausibly, PP should be generalizable. Rational credence formation still involves induction, but the expert function on which we conditionalize will differ from cases where K is admissible. For example, suppose T endorses the .5 MGR, K states that if the outcomes of the next three coin flips do not include the coin landing on its edge, there will be two heads and one tail, and e states that the third flip will be heads. Application of the MGR only to the information that three coin flips will be made yields an expert function that specifies credence .5 for e, and hence the conditional credence is .5. But, the inadmissible information makes it rational to restrict the expert function to sequences of three flips that yield two heads and one tail. Thus, the distribution specified by the expert function should be renormalized over such sequences. Each possible outcome sequence gets credence $\frac{1}{3}$ . Thus, the renormalized expert function gives $P_{R}(e) =\frac{1}{3}$ . In this case, rational credence is given by

Plausibly, a more general version of PP should be justifiable, one that does not fall mute regarding rational credence for all cases of inadmissible information.

6. A Projectivist Defense of Humeanism

Why call this view projectivist? When we speak of chances we speak as if we were describing a feature of the world. However, on this analysis, chance talk is evaluative discourse regarding rules for making predictions and explanations; it merely expresses attitudes to such rules. Hence, the surface appearance of our discourse embodies a projection of the expressed attitude onto the world.

How does it relate to other antirealist proposals regarding chance? Simon Blackburn ([Reference Blackburn and Mellor1980] Reference Blackburn1993) has presented a projectivist construal that considerably influenced the theory offered here. However, the work antedates discussion of the paradox of undermining, and is, I think, susceptible to a related problem. James Logue (Reference Logue1995) also offers a projectivist account. In addition, there are subjectivist construals of chance along the lines developed by de Finetti (Reference de Finetti1937), Jeffrey (Reference Jeffrey1983), and Skyrms (Reference Skyrms1984). All the above have some similarities with the current proposal. However, they do not give explanation the prominent motivating role it has here, a role that permits us to dispense with the exchangeability assumptions that are fundamental to subjectivist accounts and Logue's projectivism. I shall not argue the point here, but I think it plausible that an account founded on consideration of possible explanations, rather than exchangeability assumptions, is closer to our practical chance reasoning, and better positioned to explain it. Further, none of the above proposals provides a semantics for chance claims. Hence, on this count, the current proposal is better equipped to explain our practice.

Is the view Humean? I have no desire to fight over a label, but it seems appropriate. What motivates contemporary Humeans is the denial that non-Humean laws and chances feature in our best scientific explanations. As Lewis (Reference Lewis1994, 474) put it:

The projectivist view does not inflate our ontology. Thus, it answers this motivation. What it denies is that the motivation mandates acceptance of HS, which is the root of the problem of undermining futures.

In motivating Humeanism about chance in particular, Lewis (Reference Lewis1994, 484) said (and Humeans typically concur):

Ironically, in this regard, Lewis and most other Humeans offer only a promissory note, or in Strevens's case, an outright denial that PP or any chance-credence principle is justifiable.Footnote ⁹ Projectivism provides an explicit justification of PP. So, the view has the advantage of demonstrably satisfying this motivation for Humeanism about chance.

Of course, since projectively construed chance claims are not factual claims, projectivists cannot concur with Lewis's supposition that ‘chance’ names some feature of reality. At this point, realists will wind up the familiar arguments. For example, if there is no fact of the matter regarding the chances, it follows that there are no chances, and all chance ascriptions are false and rationally unassertible. Not so. On a projectivist construal, to deny there are chances is to deny there are any chance MGRs that ought to be used for prediction and explanation. That denial would be justified only if we were justified in holding that all candidate sets of chance laws were normatively inconsistent with the physical state, i.e., only if we believed a condition were met, which would preclude the existence of chance laws on a realist construal. If an agent observes a typical sequence of coin flips, there will be many chance law claims that, projectively construed, are normatively consistent with the data, and for which she will have confirming evidence. She will typically have nonzero degrees of commitment for such chance law claims, and practically speaking, will be in the same position as on a realist construal that assigns her corresponding degrees of belief about chance facts. Projectivist antirealism about chances, unlike, for example, van Fraassen's (Reference van Fraassen1989) antirealism about laws, is not an error theory; it does not entail the rejection of chance claims. Realists mistakenly attribute it this consequence precisely because they fail to properly construe the discourse in projectivist terms, i.e., as having a function other than correctly describing the facts. Alternatively, it will be claimed that projectivism is a thinly-disguised subjectivism, or makes the chances mind-dependent. Considerations of space demand that I give these objections short shrift, but this is all rather uncharitable. We have a Bayesian confirmation theory that gives an account of how a rational agent will modify her opinion in response to the evidence. We have a semantics that makes it explicit that projectively construed chance claims are about rules for prediction and explanation not about mental states. Further, these objections are not specific to the projectivist construal of chance laws and they are adequately answered elsewhere.Footnote ¹⁰

Finally, how exactly does the above semantics and confirmation theory provide an explanation of the scientist's ‘naïve’ chance reasoning, and hence, a viable alternative to other proposals? In particular, since the projectivist holds that chance claims are not reducible to claims about the physical state, she must surely expect typical agents to be non-Humeans about chance. If so, whither the defense of Humeanism? I happily concede that a reflective agent may well interpret her chance theorizing as being about non-Humean chances. Indeed, since chance claims are manifestly not reducible to claims about the physical state, that is what we should expect, at least in the absence of philosophically imposed Humean scruples. However, this is just philosophical theorizing on the agent's part, and I would say to her what I would say to a philosopher: While your philosophical theorizing provides the most accessible explanation of your scientific activity, it does not obviously provide the best. An alternative explanation is that our scientific goals include prediction and explanation. In our pursuit of these goals we countenance various candidate predictive-explanatory rules, including those constructed in this paper. The rational use of these candidates in prediction and explanation demands conformity with the projectivist semantics and confirmation theory. However, it does not demand that agents can formulate the semantics and provide an analysis of degrees of commitment. That we are not self-conscious projectivists, although our inferential practices properly involve no appeal to non-Humean ontology, is explained by two things: (i) typical agents do not possess a philosophical or psychological theory of why we are disposed to pursue explanations and (ii) the projectivist reading of prima facie factual discourse with its attendant factual-normative worlds semantics is not at all obvious. Hence, it is unsurprising that naïve theorizing about such claims should lead to a non-Humean view. However, by the same token, that does not undermine an explanation of our intuitions that views us as agents attempting to predict and explain a merely Humean world. We have a Humeanism that is superior to nonprojective Humeanism in three significant respects: (i) it vindicates those intuitions that are reflected in our practice, i.e., it permits that outcomes may be independent and identically distributed, and agrees with the intuitions captured by PP and BCP, (ii) it justifies PP, and (iii) where it differs with our intuitions (in being antirealist), it explains the divergence. This is no square peg in a round hole.