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The Third Way on Objective Probability:

A Sceptic’s Guide to Objective Chance

CARL HOEFER

The goal of this paper is to sketch and defend a new interpretation or ‘theory’ of ob-

jective chance , one that lets us be sure such chances exist and shows how they can

play the roles we traditionally grant them. The account is ‘Humean’ in claiming that

objective chances supervene on the totality of actual events, but does not imply or

presuppose a Humean approach to other metaphysical issues such as laws or

causation. Like Lewis ( 1994 ) I take the Principal Principle (PP) to be the key to

understanding objective chance. After describing the main features of Humean ob-

jective chance (HOC), I deduce the validity of PP for Humean chances, and end by

exploring the limitations of Humean chance.

1. Introduction

The goal of this paper is to sketch and defend a new interpretation or

‘theory’ of objective chance , one that lets us be sure such chances exist

and shows how they can play the roles we traditionally grant them.

My subtitle obviously emulates the title of Lewis’s seminal

1980

erence in perspective. The view developed below shares

two major tenets with Lewis’s last (

ﬀ

1994

) account of objective chance:

) The Principal Principle tells us most of what we know about ob-

jective chance.

) Objective chances are not primitive modal facts, propensities, or

powers, but rather facts entailed by the overall pattern of events and

processes in the actual world.

But it di

ers from Lewis’s account in most other respects.

Another subtitle I considered was ‘A Humean Guide … ’ But while

the account of chance below is compatible with any stripe of Humean-

ism (Lewis’s, Hume’s, and others’), it presupposes no general Humean

philosophy. Only a sceptical attitude about probability itself is presup-

posed (as in point (

ﬀ

) above); what we should say about causality, laws,

Mind, Vol. 116 . 463 . July 2007

doi:10.1093/mind/fzm549

paper

‘A Subjectivist’s Guide to Objective Chance’ — while indicating an

important di

(

550 Carl Hoefer

modality and so on is left a separate question. Still, I will label the

account to be developed ‘Humean objective chance’.

ned, or at most making

brief allusions to long-run frequencies. The case is reminiscent of quan-

tum mechanics, which the physics community uses all the time, appar-

ently successfully, without having to worry about the measurement

problem, or what — in the world — quantum states actually represent .

Perhaps a theory is not needed; perhaps we can think of objective prob-

ability as a theoretical concept whose only possible de

ﬁ

nition is merely

implicit. Sober (

2004

) advocates this no-theory theory of objective prob-

abilities.

nd this position unsatisfactory. To the extent that we are serious in

thinking that certain probabilities are objectively correct , or ‘out there

in the world’, to the extent that we intend to use objective probabilities

in explanations or predictions, we owe ourselves an account of what it

is about the world that makes the imputation and use of certain proba-

bilities correct. Philosophers are entitled to want a clear account of

what objective probabilities are, just as they are entitled to look for

solutions to the quantum measurement problem. 1

The two dominant types of interpretation of objective probability in

recent years are propensity interpretations and hypothetical or long-run

frequency interpretations. Propensity interpretations come in a wide

range of

) shows), and not all of them involve

deep modal/causal/metaphysical implications. For example, some phi-

losophers who advocate the theoretical term/implicit de

ﬂ

avours (as Gillies (

2000

ﬁ

nition

nd in sci-

ence, in some cases at least, as propensities. But for the purposes of this

paper, I will restrict the term ‘propensity’ to the metaphysically robust,

causally e

ﬁ

cacious, dispositional sort of property postulated by some

philosophers’ accounts of objective chance.

) advocates his no-theory theory on grounds of the severe shortcomings of the

traditional views. About these shortcomings we are in full agreement; but I hope to provide, be-

low, an alternative with none of those shortcomings.

2004

Mind, Vol. 116 . 463 . July 2007

2. Why a new theory of objective chance?

Why have a philosophical theory of objective chance at all, for that mat-

ter? It certainly seems that the vast majority of scientists using non-sub-

jective probabilities overtly or covertly in their research feel little need

to spell out what they take objective probabilities to be. It would seem

that one can get by leaving the notion unde

ﬁ

approach may be happy to characterize the probabilities we

1 Sober (

The Third Way on Objective Probability 551

culties of propensity and long-run frequency views are well

enough known not to require much rehearsal here. 2 My own view of

these problems is that the hypothetical frequency interpretation is met-

aphysically and epistemologically hopeless unless it includes some

account of what grounds the facts about hypothetical frequencies . (Such

an account tends to end up turning the interpretation into one of the

other standard views: actual frequency, subjective degree of belief, or

propensity.) And propensity views, while still actively pursued by many

philosophers, add a very peculiar new sort of entity, property, or type of

causation to the world. 3 One can argue at length about whether or not

this makes propensities metaphysically suspect. I think it more clear

that propensities are epistemologically hopeless (i.e. one can only claim

that statistics are a reliable guide to propensities via arguments that are

all, in the end, ine

a closely

related problem for propensity views of chance will be discussed: their

inability to justify Lewis’s Principal Principle. For now I will just regis-

ter my dissatisfaction with both hypothetical frequency and propensity

views of chance; those who share at least some of my worries will hope-

fully agree with me that a third way obviating at least some of their

problems would be worth spelling out.

Of course, a third way not su

ﬀ

ective — usually, circular). In section

ering from any of the problems alluded

to above is already available: the actual frequency interpretation (some-

times called ‘

ﬀ

nite frequency’). The defects of this view are usually

vastly overestimated, and its virtues underappreciated. Indeed, the

actual frequency interpretation is the only natural starting point for an

empiricist — or sceptical — approach to objective chance. Both Lewis’s

(

ﬁ

) theory and the theory sketched below are in a sense ‘sophistica-

tions’ of the actual frequency approach. They try to

t better with com-

mon sense, with certain uses of probability in sciences such as quantum

mechanics and statistical mechanics, and with classical gambling

devices. But the grounding of all objective chance in matters of actual

(non-modal, non-mysterious) fact is shared by all three approaches.

The goal of this paper is thus to develop and defend a ‘third way’

(di

ﬁ

erent from Lewis’s and from standard actual frequentism) among

‘third way’ approaches (neither propensity- nor hypothetical-fre-

quency-based). The chances to be described here exist — whether or not

determinism is true, and whether or not there exist such things as

primitive propensities or probabilistic causal capacities in nature. The

3 Mellor 1995 contains an extended and thorough exposition and defense of a theory of causa-

tion based on a propensity view of objective chance.

2003

Mind, Vol. 116 . 463 . July 2007

The di

1994

ﬀ

2 See Hájek

552 Carl Hoefer

rst, its coherence with the main uses of the notion

of objective chance, both in science and in other contexts; and second,

its ability to justify the Principal Principle.

ﬁ

3. Correcting the Subjectivist’s Guide: Lewis’s programme,

1980–1994

Because Lewis’s approach to objective chance is well-known, it is per-

haps best to introduce his view, and work toward the proper sceptical/

Humean view by correcting Lewis’s at several important places. 4

3.1 PP

As noted above, one of the two shared fundamentals of Lewis’s inter-

pretation and mine is the claim that the Principal Principle (PP) tells us

most of what we know about objective chance. PP can be written:

(PP) Cr(A|XE) = x

Here ‘ Cr ’ stands for ‘credence’, that is, a subjective probability or degree

of belief function. A is any proposition you like, in the domain of the

objective chance or objective probability function Pr. X is the proposi-

tion stating that the objective chance of A being the case is x , that is, X =

‘ Pr(A) = x ’. F i n a l l y , E is any ‘admissible’ evidence or knowledge held by

the agent whose subjective probability is Cr . 5 The idea contained in PP,

an utterly compelling idea, is this: if all you know about whether A will

occur or not is that A has some objective probability x , you ought to set

your own degree of belief in A ’s occurrence to x . Whatever else we may

say about objective chance, it has to be able to play the PP role: PP cap-

tures, in essence, what objective chances are for , why we want to know

them.

Crucial to the reasonableness of PP is the limitation of E to

‘admissible’ information. What makes a proposition admissible or non-

admissible? Lewis de

ﬁ

ned admissibility completely and correctly in

5 Throughout I will follow Lewis in taking chance as a probability measure over a sub-algebra of

the space of all propositions. Intuitively speaking, the propositions say that a certain outcome oc-

curs in a certain chance setup. Unlike (what many assume about) rational credence, the probabil-

ity measure should not be assumed to extend over all, or even most, of this whole proposition

space. Here we need only assume that the domain of Cr includes at least the domain of Pr and

enough other stu

2004

to serve as suitable X s and E s.

Mind, Vol. 116 . 463 . July 2007

interpretation can thus be defended without making any contentious

metaphysical assumptions. The positive arguments for the view will

turn on two points:

4 For a clear recent exposition and defense of Lewis’s approach, see Loewer

ﬀ

The Third Way on Objective Probability 553

1980

— though he considered this merely a vague,

ﬁ

rst-approximation

nition:

Admissible propositions are the sort of information whose impact on cre-

dence about outcomes comes entirely by way of credence about the chances

of those outcomes. (Lewis

)

This is almost exactly right. When is it rational to make one’s subjective

credence in A exactly equal to (what one takes to be) the objective

chance of A ? When one simply has no information tending to make it

reasonable to think A true or false, except by way of making it reasona-

ble to think that the objective chance of A has a certain value. If E has

any such indirect information about A , that is, information relevant to

the objective chance of A , such information is cancelled out by X , since

X gives A ’s objective chance outright. Here is a slightly more precise def-

inition of

1980

, p.

Admissibility: Propositions that are admissible with respect to

outcome-specifying propositions A i contain only the sort of infor-

mation whose impact on reasonable credence about outcomes A i , if

any, comes entirely by way of impact on credence about the chances

of those outcomes.

This de

ﬁ

nition of admissibility, there is no mention of

past or future, complete histories of the world at a given time, or any of

the other apparatus developed in Lewis

ﬁ

1980

to substitute a precise-

nition of admissibility in place of the correct one. We will

look at some of that apparatus below, but it is important to stress here

that none of it is needed to understand admissibility completely. Lewis’s

substitution of a precise ‘working characterization’ of admissibility in

place of the correct de

ﬁ

nition seems to be behind two important

aspects of his view of objective chance that I will reject below:

ﬁ

rst, the

alleged ‘time dependence’ of objective chance; second, the alleged

incompatibility of chance and determinism. 6

ﬁ

6 It also created mischief in other ways. For example, in the context of his ‘reformulated’ PP,

which we will see below, it caused Lewis to believe for a long time that the true objective chances in

a world had to be necessary , that is, never to have had any chance of not being the case. This mis-

conception helped delay his achievement of his

ﬁ

nal view by well over a decade.

Mind, Vol. 116 . 463 . July 2007

ﬁ

nition of admissibility is clearly consonant with PP’s expres-

sion of what chance is for, namely guiding credence when no better

guide is available. The admissibility clause in PP is there precisely to

exclude the presence of any such ‘better guide’.

Notice that in this de

looking de

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