the metaphysics of chance

The Metaphysics of Chance

Rachael Briggs*University of Sydney

Abstract

This article surveys several interrelated issues in the metaphysics of chance. First, what is therelationship between the probabilities associated with types of trials (for instance, the chance that atwenty-eight-year old develops diabetes before age thirty) and the probabilities associated withindividual token trials (for instance, the chance that I develop diabetes before age thirty)? Second,which features of the the world fix the chances: are there objective chances at all, and if so, arethere non-chancy facts on which they supervene? Third, can chance be reconciled withdeterminism, and if so, how?

1. Single-Case vs. Repeatable Chances

What are chances, and to what do they attach? There are two general types of answers.First, chances might be repeatable—they might attach to the outcomes of types of proba-bilistic trials. A roll of two fair dice is a trial type—distinct trials in the same world cancount as rolls of two fair dice.

Second, chances might be single-case—they might attach to possible outcomes ofparticular token trials. For example, the first crapshoot at the party hosted by NathanDetroit on April 20, 1955 is a token trial. As each possible outcome of a token trial isassociated with a proposition—the proposition that the token trial in question has thatoutcome—one can treat single-case chances as attaching to propositions.

Single-case chances are useful for decision-making. A doctor deciding whether toprescribe a medication to a patient (say, 40-year-old John Smith) needs to know theprobability that the drug will induce dangerous side effects in Smith, on this trial. Informa-tion about the chances of dangerous side effects among patients in general, or malepatients, or 40-year olds, are relevant only insofar as they bear on the chance that thisSmith himself will suffer dangerous side effects.

Repeatable chances are useful for gathering information about the chances. To find theprobability of an outcome on a particular trial type (say, the probability of a drug’s induc-ing dangerous side effects in 40-year-old men), one can repeat many trials of that typeand observe the frequencies of outcomes, where the frequency of an outcome O is givenby the following ratio.

Number of trials resulting in O

Total number of trials

Frequencies are not an infallible guide to chances—an outcome’s frequency can (withnonzero probability) diverge from its chance even over a large number of trials.Nonetheless, they are a good guide to chances—in a sense that will be cashed outmathematically in the next section.

Philosophy Compass 5/11 (2010): 938–952, 10.1111/j.1747-9991.2010.00345.x

ª 2010 The AuthorPhilosophy Compass ª 2010 Blackwell Publishing Ltd

It would be useful to link information about single-case chances to information aboutrepeatable chances—that way, we could bring evidence about the chances to bear onour decision making. There seems to be an obvious way of translating between the twotype of chance. Suppose that the first crapshoot at Nathan Detroit’s party belongs tosome type T. Then the probability of an outcome (say, rolling two even numbers) at thistoken crapshoot ought to be the probability of rolling two even numbers on a trial oftype T.

But would-be translators must proceed with caution: a token trial can belong tomany different types at once. One and the same event may be a roll of two dicedrawn from the briefcase of Sky Masterson (who carries both fair dice and biasedones), a roll of two fair dice drawn from Sky Masterson’s briefcase, and a roll of twofair dice from a particular height, at a particular angle, with a particular velocity. Ifthese different trial types are associated with different probability assignments, then itis not clear which probability assignment (if any) applies to the token crapshoot inquestion.

This puzzle of connecting the single-case probabilities associated with token trials tothe repeatable probabilities associated with type trials is usually known as the reference classproblem or the problem of single-case probabilities (see (Hajek 2007) and (Colyvan et al. 2001)for discussions of how the problem plays out for various accounts of chance).

2. Mathematical Interlude

A little mathematical background will be helpful for subsequent philosophical discussion.Most philosophers agree that chance is governed by the axioms of the probability

calculus set forth by Kolmogorov (1950). In informal terms, Kolmogorov’s axioms treatpossibilities like points in logical space, events or propositions like regions of logical space,and probabilities like the areas of regions in logical space.

In formal terms, Kolmogorov’s axioms introduce a mathematical object called a proba-bility space, which is a triple ÆX, F, Pæ. X is a set of possibilities, F is a Borel field over X(that is, a set of subsets of X closed under countable union, countable intersection, andcomplementation), and P is a probability function assigning a real number to each memberof F, and satisfying the following three conditions.

Non-negativity: "(E 2 F)P(E) ‡ 0(In other words, no event has a negative probability.)

Tautology: P(X) ¼ 1(In other words, the probability that some possibility or other will be instantiated is 1.)

Countable additivity: If {A1,A2,…} is a set of events such that for every Ai,Aj 2 {A1,A2,…}, Ai\Aj ¼ ;, then P(A1\A2\…) ¼ P(A1) + P(A2) + …(In other words, if there is some set of events no more numerous than the natural num-bers, and any two of those events are incompatible with each other, then the probabilitythat at least one will occur is the sum of the probabilities of the individual events.)1

In addition to the concept of probability, Kolmogorov introduces the concept of condi-tional probability, where the conditional probability of B given A, written P(B|A), is theprobability of B on the assumption that A. If probability is like area, then conditionalprobability is like proportional area: P(B|A) is the percentage of A possibilities that arealso B possibilities. (See Fig. 1.) More formally, the ‘ratio formulation’ of conditionalprobability states that PðBjAÞ ¼ PðA\BÞ

PðAÞ whenever the ratio on the right-hand side iswell-defined.

The Metaphysics of Chance 939

ª 2010 The Author Philosophy Compass 5/11 (2010): 938–952, 10.1111/j.1747-9991.2010.00345.xPhilosophy Compass ª 2010 Blackwell Publishing Ltd

Kolmogorov treats the ratio formulation as a definition of conditional probability.Hajek (2003) argues that conditional probability should be taken as primitive, and theratio formulation treated as an extra axiom. This alternative strategy allows for the possi-bility that P(B|A) is well-defined even when the corresponding ratio is not. (This looksright: the probability that a fair coin lands heads given that you flip it is 1/2, even if thereis no well-defined probability that you flip it.)

Conditional probabilities can be used to generate new probability functions. WhereP is a probability function and E is a proposition in the domain of that function, PE

(‘P conditionalized on E’) is a function that takes each proposition A in the domain ofP and assigns it the probability P(A|E). Where P is a probability function, PE is too.In a sense, PE is the probability function that results from changing P to accommo-date the new information that E is certain, while keeping the change as minimal aspossible.

Conditional probabilities have another useful property, related to averaging. One canimpose a partition E1…En on the probability space X, where the Ei’s are mutually incom-patible (i.e., it is impossible for two of them to happen at once) and jointly exhaustive(i.e., one of them must happen). Then for any proposition A, the probability of A isequal to a weighted average of its conditional probabilities given each Ei, where theweightings are determined by the probabilities of the Eis. In other words, P(A) ¼P

iP(Ei)PEi(A). (See Fig. 2.)

Events A and B are independent just in case the occurrence of A is irrelevant to theprobability of B, and vice-versa. In formal terms, A and B are independent just in caseP(A|B) ¼ P(A). More generally, events A1…An are independent just in case, for anyway of dividing these events into two non-overlapping sets, the conjunction of all theevents in one set is independent of the conjunction of all the events in the other. Twotoken trials are independent just in case each outcome of one is independent of eachoutcome of the other.

We can now state precisely the relationship between observed chances and frequencies.Suppose one performs a long sequence of independent trials of the same type, called IIDfor ‘independent and identically distributed’. The following two results hold:

Weak law of large numbers: As the number of trials approaches infinity, the frequencyof an outcome will converge on its chance with probability 1.

For any positive real numbers: � and d, no matter how small, there is some numberof trials large enough that the probability that the frequency of an outcome among thosetrials lies within � of its chance is greater than or equal to 1)d.

A

B

P

B

PA

Fig 1. P(B|A) is the proportion of A’s probability, or ‘area’, lying in B. In this image, A and B are independent, sinceP(B) = PA(B) – that is, B has the same area in the left- and right-hand side of the picture.

940 The Metaphysics of Chance


Because chances are a type of probabilities, the weak and strong laws of large numberslink chances to frequencies. But note that neither law constitutes a non-circular analysisof chance in terms of frequency. The laws tell us not that the frequencies inevitablyapproach the chances over long sequences of trials, but only that the chance of thefrequencies’ approaching the chances is very high.

3. Exchangeability

That covers the mathematics of chance. But what are chances? An extreme viewpoint,most famously championed by de Finetti (1964), holds that all probabilities are individualdegrees of belief, or credences. Credences are grounded in facts about the believer—inparticular, in facts about which bets the believer is disposed to accept. (Some authorschoose to construe ‘bets’ rather broadly. Ramsey (1931), for instance, suggests that run-ning for the train is a way of betting that the train will be on time.)

De Finetti claims that there are no objective chances, and that all probabilities arecredences. This immediately raises two questions. If probabilities are purely subjective,then why is there so much agreement about the probabilities associated with some tri-als—for instance, casino games or quantum measurements? And why are past frequenciesso useful in probabilistic predictions about the future?

De Finetti’s answers to these questions rest on the concept of exchangeability. Imaginea long sequence of trials (e.g., coin tosses). We can pick out finite initial segments ofthis sequence (the first two coin tosses, the first three coin tosses, and so on). For eachfinite initial segment, there will be a sequence of possible outcomes (for instance, thefirst three coin tosses might come up HHH, HHT, HTH, …). The entire sequence oftrials is exchangeable (for an individual) just in case for each of its finite initial seg-

E1 E2

E3E4

PE1PE2

PE4PE3

Fig 2. Where E1, . . . , En is a partition on P, P(A) =Pn

i¼1 PðEiÞPEiðAÞ: In other words, P(A) is obtained by multiplyingeach PEi

ðAÞ by the probability of P(Ei) (‘shrinking’ the area of A within Ei proportional to Ei’s ‘scale’ in the centralpicture) and then adding these probabilities together.



ments, sequences of outcomes that agree with respect to outcome numbers are assignedequal credence. (So if our sequence of coin tosses is exchangeable for someone, thatperson must assign equal credences to the outcome sequences HHT and HTH for thefirst three trials, because those two sequences contain the same number of heads andthe same number of tails.)

Exchangeability is weaker than independence: every sequence of IID trials is exchange-able, but not vice-versa. But every exchangeable probability function over an infinitesequence of exchangeable trials can be expressed as an average of IID distributions overthe same trials. (The result breaks down when the number of trials is finite; see Diaconis(1977) for discussion.)

To see how this works, imagine a coin of unknown bias, which you are to fliprepeatedly. For you, the coin tosses are exchangeable: heads followed by tails is exactlyas likely as tails followed by heads. But they are not independent: if the coin landsheads on the first toss, it’s more likely to land heads on the second toss (because in thatcase it’s more likely to be biased toward heads). Your exchangeable credence functioncan be expressed as a mixture of the credence functions you would have, if you knewthe coin’s bias. Each of these credence functions treats the coin tosses as IID. (De Fi-netti denies that there really is such a thing as the coin’s bias, but grants that bias is auseful heuristic device.)

Exchangeable credence functions satisfy a number of useful convergence results. Therough idea is this: two observers who consider the same sequence of trials exchangeable,and who start out suitably open-minded about the outcomes of those trials, are certain toconverge toward the same credences about the outcomes of future trials in the long run.(Like the law of large numbers, this result has both a weak version, which says that theobservers’ credences will converge with probability 1, and a strong version, which saysthat for any positive real numbers � and d, there is some sufficiently large number n suchthat both observers should place credence at least 1)d in their credences being within� of each other after n trials have been observed.)

The fact that exchangeable credence functions can be expressed as the average of IIDprobability functions explains why there might seem to be objective chances associatedwith coin tosses. And the convergence results explain both widespread agreement and theusefulness of past frequencies. Critics, however, claim that exchangeability alone cannotexplain all the important features of chance. Levi (1977) points out that exchangeabilitycannot explain why there seem to be objective chances associated with types of probabi-listic trials of which there are only a few tokens—even for a one-off coin toss, thereseems to be an objective chance of heads. Gillies (2000) argues that premises about theobjective chances associated with trials are needed to justify treating those trials asexchangeable in the first place.

4. The Principal Principle

Instead of denying the existence of objective chances altogether, Lewis (1986b: 87)defines objective single-case chances in terms of their relationship to rational credence.The link takes the form of the

Principal principle: Let C be any reasonable initial credence function. Let t be anytime. Let x be any real number in the unit interval. Let X be the proposition that thechance, at time t, of A’s holding equals x. Let E be any proposition compatible with Xthat is admissible at time t. Then



CðAjXEÞ ¼ x

A few terms need clarifying. Initial credence functions are meant to exist prior to anycontingent evidence. The credence function of a reasonable person, Lewis thinks, comesfrom conditionalizing a reasonable initial credence function on the person’s total evi-dence—that is, the strongest proposition the person knows. And an admissible propositionis one ‘whose impact on credence about outcomes comes entirely by way of credenceabout the chances of those outcomes’.

The upshot of the Principal Principle is that one’s credence in a proposition A, condi-tional on the proposition that A’s chance is x, should be x—provided one has no admissi-ble information. So someone who is certain that A’s chance is x (and whose totalevidence is admissible) should believe A to degree x. Someone who is not sure what A’schance is (and whose total evidence is admissible) should set her credence in A equal to aweighted average of A’s possible chances—where the weight of each possible chance isequal to her credence that it is A’s actual chance.

Lewis goes on to make substantive claims about which propositions are admissible.Information about the past is admissible, as is information about how the chances dependon the past. Furthermore, the two types of information are admissible when takentogether.

Thus, Lewis (1986b: 97) can rewrite the Principal Principle as follows, where Htw is‘the complete history of world w up to time t: the conjunction of all propositions thathold at w about matters of particular fact no later than t’, and Tw is ‘a complete theory ofchance for world w… a full specification, for world w, of the way chances at any timedepend on history up to that time’.

CðAjXHtwTwÞ ¼ x

Lewis argues that the Principal Principle, together with the information that conjunc-tions of the form HtwTw are always admissible, provides much useful information aboutthe nature of chance. Using the Principal Principle, he argues that the chance function ischaracterized by Kolmogorov’s axioms; that the chances at later times evolve from thechances at earlier times by conditionalization on intervening history; and that if thechance of a proposition A1 is not counterfactually dependent on the truth of a proposi-tion A2 (that is, if A1’s chance would be the same if A2 were to occur as if A2 were notto occur), then the chance function treats A1 and A2 as probabilistically independent.

Although the Principal Principle provides some important information about objectivechance, it says little about what in the world answers to the name ‘chance’. That is atopic for the next two sections.

5. Humean Supervenience

What, if anything, fixes the facts about chance? Are they sui generis, or are they reducibleto some other sorts of facts? Many authors opt for the second answer: chance facts super-vene on non-chancy facts—that is, if you fix all the non-chancy facts about the world,you have thereby fixed the chances.

Which non-chancy facts fix the chances, exactly? One popular answer, defended byLewis (1994), is that the chances supervene on the totality of occurrent facts—those thatcan be defined without appeal to possibility, or related concepts. Excluded from the



realm of occurrent facts are laws, dispositions, and counterfactuals, and causation. Lewis(1986a) calls his view ‘Humean Supervenience’ after David Hume, ‘the great denier ofnecessary connections’ between distinct existences.2

One version of Humean Supervenience is actualist frequentism, according to whichthe chance of an outcome on a token trial is simply the frequency that outcome on trialsof the same type in the actual world. Actualist frequentism has some implausible conse-quences, however: it forces trials that are the sole representatives of their types to haveoutcome probabilities of 0 or 1, and it requires chances to be expressible as ratios of natu-ral numbers in worlds with only finitely many token trials.

A more sophisticated Humean view, defended by Lewis (1994), is the best-system analy-sis, according to which the laws of nature are the theorems of whichever logical system bestsummarizes the occurrent facts. Lewis’s version of the theory incorporates chances by lettingthe best system include history-to-chance conditionals, which ‘specify exactly what chanceswould follow any given initial segment of history Lewis (1994: 484)’. This is one way ofsolving the problem of the single case—the history-to-chance conditionals assign each tokentrial to a unique type, based on the history of the world up to the time of the token trial.

What makes a theory ‘best’? At minimum, the best theory cannot entail any falsehoods.Beyond that, the best theory must make the optimal tradeoff between simplicity (havinga small number of axioms stateable in an appropriate language), strength (making a largenumber of predictions) and, in the case of probabilistic theories, fit (roughly, assigning ahigh probability to the actual world; though see Elga (2004) for criticisms and refine-ments of Lewis’s definition).

Lewis (1994: 484) claims that the best-system analysis possesses a key advantage overother reductive theories of chance: unlike its rivals, it can explain why chances satisfy thePrincipal Principle. In one sense, the Principal Principle needs no justification: it’s a plati-tude that must be satisfied by anything deserving the name ‘chance’. But in anothersense—the sense Lewis has in mind—philosophers purporting to analyze chances in otherterms should explain why the entities they call ‘chances’ satisfy the Principal Principle.

Lewis claims that there is likely to be a good explanation of why best-system chancessatisfy the Principal Principle. (After all, who could deny that one should arrange one’scredences according to the best system for organizing information?) Furthermore, heclaims, philosophers proposing non-Humean accounts cannot explain why their so-called‘chances’ satisfy the Principal Principle.

Be my guest—posit all the primitive unHumean whatnots you like. But play fair in namingyour whatnots. Don’t call any alleged feature of reality ‘‘chance’’ unless you’ve already shownthat you have something, knowledge of which could constrain rational credence. I think I see,dimly but well enough, how knowledge of frequencies and symmetries and best systems couldconstrain rational credence.I don’t begin to see, for instance,how knowledge that two universalsstand in a certain special relation N* could constrain rational credence about the future coin-stantiation of those universals.

Sturgeon (1998) and Hall (2004) argue that Lewis is overly optimistic about the best-sys-tem analysis. Hall also argues that Lewis is overly pessimistic about non-Humean theories.

Sturgeon disputes Lewis’s claim that the best system is a guide to rational belief. Whyshould human thinkers care about simplicity and strength? It might be rational for anindividual to use a system that is gerrymandered and weak, but makes up for these weak-nesses in psychological convenience and plausibility. Hall grants that the best system is agood guide to credence in general, but argues that it is no guide to credence about trialsthat are known to be rare outliers. The best system would be the same regardless of the



outcomes of these trials, so knowledge of the best system cannot constrain credence aboutthese trials.

Hall also shows that one can formulate a non-Humean theory, ‘primitivist hypotheticalfrequentism’, which lets one derive the Principal Principle from other assumptions.According to primitivist hypothetical frequentism, the chance of heads on a coin toss is xiff, were there an infinite sequence of token tosses of the same type, the frequency ofheads on those trials would converge to x. Using an exchangeability assumption, and theclaim that a rational person’s credence in the coin’s landing heads must be independentof whether the coin will be tossed in the future, Hall derives an instance of the PrincipalPrinciple.

One might also question whether non-Humean accounts of chance need to justify thePrincipal Principle. Maudlin (2007) and Gillies (2000) claim that chances are implicitlydefined by our best scientific theories, and that attempts at reductive analysis of chanceare liekly to end in failure. (Maudlin is concerned solely with fundamental physics; Gilliesis also concerned with the special sciences.) As these authors do not aim to reduce chanceto anything else, we cannot ask why their proposed reduction of chance picks outsomething fit to satisfy the chance role.

Justifying the Principal Principle is not the worst of the Humean’s worries. HumeanSupervenience is that (at least in its usual guises, frequentism and the best-system analysis)conflicts with the Principal Principle (Hall 1994; Lewis 1994; Thau 1994). According toboth actualist frequentism and the best-system analysis, the non-chancy facts that fix thechances are the outcomes of probabilistic trials. This raises the possibility of underminingfutures: future sequences of trial outcomes that have nonzero chance at a time t, but that,if they were to occur, would fix different chances at t.

For instance, if the world consists of a single fair coin that is tossed 10,000 times, theninitially, there is a small but nonzero chance that the coin will land heads 70% of thetime. But if actualist frequentism is true, then a coin that lands heads 70% of the time isnot fair, but is biased 70% toward heads.

Briggs (2009) gives the following summary of the problem (along with a criticaloverview of responses to it):

Right now, there is some undermining future F with nonzero chance. By the PrincipalPrinciple, it is rational to place nonzero credence in F, conditional on the current facts aboutchance. But F is incompatible with the current facts about chance; in a world where F obtains,those facts do not. Therefore, it is rational to place zero credence in F, conditional on the cur-rent facts about chance. Contradiction!

6. The Principle of Indifference

Instead of identifying the chance of an outcome with its actual frequency, one mightidentify the chance of an outcome with its frequency among possible trials. For instance,one can compute the chance of rolling a five and a six on a pair of standard dice as 2/36,by observing that there are 36 possible outcomes of such a toss, two of which show a fiveand a six (because the five might be shown on either the first die or the second).

The idea that chances are frequencies among possible trials is closely related to whatKeynes (1921): 42) calls the

Principle of indifference: If there is no known reason for predicating of our subjectone rather than another of several alternatives, then relatively to such knowledge theassertions of each of these alternatives have an equal probability.



The Principle of Indifference says that to compute the probability of an event, youshould first divide the space of epistemically possible alternatives into a set of symmetricpossibilities, then distribute probability equally among the possibilities. Thus, the proba-bility of an outcome O is given by the following ratio.

Number of possibilities in which O occurs

Total number of possibilities

The ‘probabilities’ here might be interpreted as chances, or as rational credences basedon chances. The second interpretation is more natural on Keynes’ formulation, as he rela-tivizes probabilities to an epistemic state. The lack of ‘known reason for predicating ofour subject’ one alternative rather than another might then be construed as a lack ofadmissible information.

A serious obstacle to any possible-frequency-based account of chance is a set of exam-ples known as Bertrand’s paradoxes, after the work of Bertrand (1907). These examplesseem to establish that the Principle of Indifference is inconsistent. van Fraassen (1989):303) illustrates the basic problem using the example of a perfect cube factory.

A precision tool factory produces iron cubes with edge length £ 2 cm. What is the probabilitythat a cube has length £ 1 cm, given that it was produced by that factory?

The Principle of Indifference suggests a uniform probability distribution over lengths,so that the correct answer appears to be 1/2. But suppose we redescribe the situation asfollows:

A precision tool factory produces iron cubes with volume £ 8 cm3. What is the probability thata cube has volume £ 1 cm, given that it was produced by that factory?

The Principle of Indifference suggests a uniform probability distribution over volumes, sothat the correct answer appears to be 1/8. But to for a cube to have length 2 cm just isfor it to have volume 8 cm3—both answers can’t be right!

Castell (1998) and de Cristofaro (2008) propose restricted versions of the Principle ofIndifference which they claim avoid Bertrand’s paradox. Shackel (2007) classifiesresponses to Bertrand’s paradoxes into two types, and argues that neither type of responseis promising.

7. Determinism and Chance

Let us turn to another question. Can there be chances (other than 0 and 1) in a worldwith deterministic fundamental laws? Probabilities crop up in casino gambling, classicalstatistical mechanics (Albert 2000; Sklar 1993), and ecology (Sober 2010)—situations withdeterministic dynamics. But do the probabilities in question really deserve the name‘chance’?

Schaffer (2007) coins the term ‘compatiblist chance’ for any type of chance that iscompatible with determinism. He argues that compatibilist chance must violate someappealing platitudes, while incompatibilist chance need not. One of these platitudes is thePrincipal Principle, taken together with Lewis’s assumption that information about historyand the laws of nature is always admissible.

Chances that satisfy the Principal Principle, Schaffer argues, must be incompatibilist. Bythe Principal Principle, someone who knew that A’s chance was x (for 0 < x < 1), and



who also had complete information about history and the laws of nature, could not assigncredence 0 or 1 to A. So history and the laws of nature could not jointly entail either Aor �A, and determinism would have to be false.

Proponents of compatibilist chance typically accept some version of the Principal Prin-ciple, but restrict which information about history and fundamental laws counts as admis-sible. Loewer (2001) suggests that in addition to the Lewisian chances fixed by historyand the fundamental laws, there are macroscopic chances suitable for statistical mechanics andBohmian mechanics. (Loewer goes on to argue that macroscopic chances are bestexplained by a Humean best-system analysis, but for the moment, I’ll focus on his weakerclaim that macroscopic chances are conceptually coherent.)

To determine the macroscopic chance function at t, one starts with a chance distribu-tion over initial conditions for the universe, then conditionalizes that distribution on apartial description of the world’s history up to t in terms of macroscopic variables. (Mac-roscopic variables represent coarse-grained features of large systems, such as temperatureand pressure in statistical mechanics, or the wave function in Bohmian mechanics. Thecontrast class is microscopic variables, which represent fine-grained features of large sys-tems, such as the precise positions of and momenta of all the molecules in statisticalmechanics, or individual particle positions in Bohmian mechanics.)

The macroscopic chances figure in a macroscopic version of the Principal Principle,which uses the macroscopic chance function in place of the Lewisian chance function,and a macroscopically admissible proposition in place of the admissible proposition. Infor-mation about the laws remains admissible, but only macroscopic information about theworld’s history up to t, is admissible at t.

Ismael (2009) also defends a type of macroscopic chance. She argues that because wenever know the state of a physical system with exact certainty, we need informationabout chances as well as information about fundamental laws in order to predict thebehavior of the world around us. Ismael, unlike Loewer, holds that macroscopic chancesattach to states of (types of) relatively isolated physical systems rather than states of theentire world, and takes macroscopic chances as basic rather than deriving them from aninitial chance distribution.

Schaffer (2007) claims that in addition to violating the Principal Principle, compatibilistchance violates the

Lawful magnitude principle: If A’s chance at t in w is x, then the laws of w entail ahistory-to-chance conditional of the form: if the occurrent history of w through t is H,then A’s chance at t in w is x.

For further discussion of the Lawful Magnitude Principle and its applications outsidethe realm of deterministic chance, see (Schaffer 2003; Fisher 2006; Lange 2006). A keyfeature of deterministic laws, says Schaffer (2007: 130), is that they do not mentionchances: ‘There is no chance parameter hidden in F ¼ ma’. Therefore, if the laws of nat-ure are deterministic, then they do not entail any history-to-chance conditionals, and nochances can satisfy the Lawful Magnitude Principle.

Hoefer (2007) argues for compatibilist chance and against the Lawful Magnitude Prin-ciple. He adopts the best-system analysis of chance, but argues that instead of a single uni-fied best system, there should be separate systems for the chances and the laws. Glynn(2010) argues that advocates of compatibilist chance can save the Lawful Magnitude Prin-ciple by positing probabilistic laws for the special sciences in addition to the fundamentalphysical laws. The appropriate history-to-chance conditionals can then be found in thenon-fundamental laws.



For both Hoefer and Glynn, the same token trial may be associated with several differ-ent outcome chances at different levels of description. This is consistent with the Princi-pal Principle, as the information that a token trial falls under a particular narrowdescription (e.g., as the roll of two fair dice from a particular height, at a particular angle,with a particular velocity) may be inadmissible with respect to the chances associated withthe trial under a broader description (e.g., as the roll of two fair dice).

8. The Method of Arbitrary Functions

Defenders of compatibilist chance face at least two pressing questions, both touched on inthe previous section. First, is there any reason to expect such chances satisfy the PrincipalPrinciple? And second, what is the relationship between compatibilist chances and thelaws of nature?

A technique that may help to answer both questions is the method of arbitrary func-tions, originally proposed by Poincare (1912), developed by Hopf (1934), and recentlydefended in the philosophy literature by von Plato (1983) and Strevens (2003). The basicthought behind the method of arbitrary functions is this: one begins with loose, qualita-tive constraints on classes on probability functions, then uses information about dynamiclaws to derive more precise, quantitative information about the probability functions inquestion. A little more discussion of the formalism will help to clarify the relationshipbetween compatibilist chances and the laws of nature. (The Principal Principle will haveto wait until the end of the section.)

Suppose we have a setup whose underlying dynamics are deterministic, so that the ini-tial conditions, together with the laws, fix the outcome. We can then classify the possibleinitial conditions of the setup according to the outcomes they will produce. Consider, asan example, a roulette wheel painted with alternating red and black wedges of equal size,and equipped with a stationary pointer. The wheel is spun and allowed to come to a stopwith the pointer aimed at either a red or a black wedge. The outcome of any given trialis fixed by a single parameter: the initial speed with which the wheel is spun. We canclassify values of this parameter—possible initial speeds—as ‘red’ or ‘black’ depending onthe color of the wedge where the pointer will eventually land; see Fig. 3.

Thus, the space of initial conditions can be divided into red regions (that is, regionswhich produce red outcomes) and black regions (that is, regions which produce blackoutcomes). Furthermore, our concept of velocity provides a natural measure over thesizes of these regions: the region where the wheel’s initial speed is between 3 and 4 m/sis equal in size to the region where the wheel’s initial speed is between 4 and 5 m/s.

As Fig. 3 shows, the space of initial conditions is divided into small, rapidly alternatingred and black regions. If you compare one of the red regions with an adjoining blackregion, then the ratio between the sizes of those regions—what Strevens calls the strikeratio—is roughly constant. (In the roulette wheel example, the ratio is roughly 1:1, but itmay differ in other examples.) This means that in any contiguous region containing manysmall red and black regions, the ratio between the size of the red portion and the size ofthe black portion is roughly constant.

Next, consider how probability might be distributed over possible values of the initialparameters. (In the case of the roulette wheel, there is only the one initial parameter,speed). We can represent this probability distribution by a density function. The probabilitythat the values of the initial parameters fall within some region is given by integrating thedensity function over that region (intuitively, taking ‘the area under the curve’, as shownin Fig. 4).



Poincare and Hopf prove limiting results for a variety of chance setups. They imaginevarying either the dynamics of a chance setup (for instance, by dividing the roulettewheel into ever more, ever thinner slices, as suggested by Poincare) or the range of likelyinitial conditions (for instance, by allowing the roulette wheel in the above example tobe spun with greater and greater force, as suggested by Hopf). As they vary these parame-ters, they hold the strike ratio constant, and they require that the density function overinitial conditions remain continuous. They show that in the limit, as the outcome of atrial grows more and more sensitive to fine-grained variations in initial conditions, or asthe range of likely initial conditions expands toward infinity, the probability of eachoutcome approaches its strike ratio.

Strevens appeals to approximations rather than to limiting conditions. He shows thatfor a variety chance setups with microconstant dynamics, the chance of each outcome isapproximately equal to its strike ratio provided the that density function over initialconditions is sufficiently smooth.

The method of arbitrary functions suggests that the dynamic laws of nature are crucialin fixing the chances—but not in accordance with the Lawful Magnitude Principle.Instead of history-to-chance conditionals, the laws of nature provide a way of transform-ing relatively smooth chance distributions over initial conditions into relatively stable andcircumscribed chance distributions over outcomes.

What about the Principal Principle? It can be linked to the method of arbitrary func-tions in one of two ways, depending on how we interpret the density functions. Strevens(2003: 72) suggests that these density functions be interpreted as Humean chance func-tions. The Principal Principle could then be justified by a version of the best-systemapproach that Loewer (2001) uses to defend compatibilist chance.

Another option, suggested by Savage (1973), is to interpret the density functions ascredence functions—either the credence functions of actual individuals, as Savage does,or as the credence functions of suitably rational and well-informed individuals. This inter-pretation suggests another way of linking the method of arbitrary functions with the Prin-cipal Principle. The proposition X (that A’s chance is x) can be identified with theproposition that the dynamics of the experiment are microconstant and the strike ratio ofA is x. Then where C is the credence function of a suitably rational agent who has nomacroscopically inadmissible information, the method of arbitrary functions shows thatC(A|X)�x.

Fig 3. Initial speeds for the roulette wheel graphed on the xaxis labeled with red and white bars according to theoutcomes they produce. (White coloring is used to represent black outcomes).



Strevens (1998) argues that the method of arbitrary functions captures the symmetry-based reasoning that the Principle of Indifference is meant to capture, but in a more con-sistent way. Symmetry considerations justify using smooth initial density functions, andthe method of arbitrary functions lets us transform these initial density functions into finalchance functions over outcomes.

9. Conclusion

Our opinions about chance serve as a guide to belief and action—a point succinctlyexpressed in the Principal Principle. Our opinions about chance are in turn guided byobservation, in the form of data about frequencies. Combining these two roles for chancerequires assigning each token trial to a unique type. Two very different solutions aregiven by de Finetti—who holds that token trials are alike just in case one treats them asexchangeable—and Lewis—-who holds that token trials are alike just in case the laws ofnature group them together. Even once these foundational questions are settled, there is afurther practical task: given a token trial, how are we to determine its type?

The nature of the chances that guide our beliefs is a hotly contested matter. HumeanSupervenience strikes many authors as an appealing way of linking the metaphysics ofchance to the epistemic role of chance, but it generates a paradox when combined withthe Principal Principle. The Principle of Indifference provides a way of groundingchances in frequencies without tying those chances too tightly to the actual world, but itfaces a challenge in the form of Bertrand’s paradox.

The compatibility of chance and determinism is another contested area. The worldappears to contain compatibilist chances in a multitude of places, including casino gam-bling, statistical mechanics, and ecology. But advocates of compatibilist chances need toexplain how these chances relate to the laws of nature, and why they ought to satisfy thePrincipal Principle. The method of arbitrary functions is a promising place to look foranswers to both questions.

Acknowledgements

The author thanks John Cusbert, Elise Hedemann, and Stephen Robertson for theirhelpful comments.

Fig 4. A relatively smooth probability distribution over initial conditions. The probability of a red outcome is equalto the sum of the areas of the red bars.



Short Biography

Rachael Briggs is a postdoctoral fellow at the University of Sydney, where she studiesdecision theory, formal epistemology, and metaphysics (particularly the metaphysics ofchance). Her articles on these topics appear in The Philosophical Review, Nous, Syn-these, Oxford Studies in Metaphysics, and Oxford Studies in Epistemology. Her currentresearch focuses on formal social epistemology and its relationship to issues in socialchoice theory. She holds a PhD in philosophy from MIT and a BA in philosophy fromSyracuse University.

Notes

* Correspondence: University of Sydney, Sydney, NSW, Australia. Email: [email protected].

1 Some authors, motivated by concerns about infinity, prefer a weaker finite additivity principle.2 According to Lewis’s usage, Humean Supervenience also entails a commitment to the claim that all occurrentproperties can be localized to point-sized regions of spacetime. Quantum mechanics provides good reasons to rejectthis additional commitment of Humean Supervenience (Forrest 1988), as does classical electromagnetic theory(Robinson 1989). I will ignore Lewis’s claims about localizability to focus on his claims about modality.

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