*First published Mon Oct 21, 2002; substantive revision Mon Dec 19, 2011*

Probability theory was a relative latecomer in intellectual
history. It was inspired by games of chance in 17^{th} century
France and inaugurated by the Fermat-Pascal correspondence. However,
its axiomatization had to wait until Kolmogorov's
classic *Foundations of the Theory of Probability* (1933). Let
Ω be a non-empty set (‘the universal
set’). A *field* (or *algebra*) on Ω is a
set
**F** of subsets of Ω that has Ω as a member,
and that is closed under complementation (with respect to Ω) and
union. Let *P* be a function from **F** to the real
numbers obeying:

- (Non-negativity)
*P*(*A*) ≥ 0, for all*A*∈**F**. - (Normalization)
*P*(Ω) = 1. - (Finite additivity)
*P*(*A*∪*B*) =*P*(*A*) +*P*(*B*) for all*A*,*B*∈**F**such that*A*∩*B*= ∅.

Call *P* a *probability function*, and (Ω,
**F**, *P*) a *probability space*.

The assumption that *P* is defined on a field guarantees that
these axioms are non-vacuously instantiated, as are the various
theorems that follow from them. The non-negativity and normalization
axioms are largely matters of convention, although it is non-trivial
that probability functions take at least the two values 0 and 1, and
that they have a maximal value (unlike various other measures, such as
length, volume, and so on, which are unbounded). We will return to
finite additivity at a number of points below. We may now apply the
theory to various familiar cases. For example, we may represent the
results of tossing a single die once by the set Ω={1, 2, 3, 4, 5,
6}, and we could let **F** be the set of all subsets of
Ω. Under the natural assignment of probabilities to members of
**F**, we obtain such welcome results as *P*({1}) =
1/6, *P*(even) = *P*({2} ∪ {4} ∪ {6}) = 3/6,
*P*(odd or less than 4) = *P*(odd) + *P*(less than
4) − *P*(odd ∩ less than 4) = 1/2 + 1/2 − 2/6 =
4/6, and so on.

We could instead attach probabilities to members of a collection
**S** of *sentences* of a formal language, closed
under (countable) truth-functional combinations, with the following
counterpart axiomatization:

*P*(*A*) ≥ 0 for all*A*∈**S**.- If
*T*is a logical truth (in classical logic), then*P*(*T*) = 1. *P*(*A*∨*B*) =*P*(*A*) +*P*(*B*) for all*A*∈**S**and*B*∈**S**such that*A*and*B*are logically incompatible.

The bearers of probabilities are sometimes also called “events” or “outcomes”, but the underlying formalism remains the same.

Now let us strengthen our closure assumptions regarding
**F**, requiring it to be closed under complementation and
*countable* union; it is then called a *sigma field* (or
*sigma algebra)* on Ω. It is controversial whether we
should strengthen finite additivity, as Kolmogorov does:

3′. (Countable additivity) IfA,_{1}A,_{2}A… is a countably infinite sequence of (pairwise) disjoint sets, each of which is an element of_{3}F, then

P(∞

∪

n=1A_{n})= ∞

∑

n=1P(A_{n})

Kolmogorov comments that infinite probability spaces are idealized models of real random processes, and that he limits himself arbitrarily to only those models that satisfy countable additivity. This axiom is the cornerstone of the assimilation of probability theory to measure theory.

*The conditional probability of A given B* is then given by
the ratio of unconditional probabilities:

P(A|B)= , provided P(B) > 0.

This is often taken to be the *definition* of conditional
probability, although it should be emphasized that this is a technical
usage of the term that may not align perfectly with a pretheoretical
concept that we might have (see Hájek, 2003). We recognize it
in locutions such as “the probability that the die lands 1,
given that it lands odd, is 1/3”, or “the probability that
it will rain tomorrow, given that there are dark clouds in the sky
tomorrow morning, is high”. It is the concept of the probability
of something *given* or *in the light of* some piece of
evidence or information that may be acquired. Indeed, some authors
take conditional probability to be the primitive notion, and
axiomatize it directly (e.g. Popper 1959b, Renyi 1970, van Fraassen
1976, Spohn 1986 and Roeper and Leblanc 1999).

There are other axiomatizations that give up normalization; that give up countable additivity, and even additivity; that allow probabilities to take infinitesimal values (positive, but smaller than every positive real number); that allow probabilities to be imprecise (interval-valued, or more generally represented with sets of numerical values). For now, however, when we speak of ‘the probability calculus’, we will mean Kolmogorov's approach, as is standard.

Given certain probabilities as inputs, the axioms and theorems allow
us to compute various further probabilities. However, apart from the
assignment of 1 to the universal set and 0 to the empty set, they are
silent regarding the initial assignment of probabilities.^{[1]} For guidance
with that, we need to turn to the interpretations of probability.
First, however, let us list some criteria of adequacy for such
interpretations.

## 2. Criteria of adequacy for the interpretations of probability

What criteria are appropriate for assessing the cogency of a
proposed interpretation of probability? Of course, an interpretation
should be precise, unambiguous, non-circular, and use well-understood
primitives. But those are really prescriptions for good philosophizing
generally; what do we want from our interpretations *of
probability*, specifically? We begin by following Salmon (1966,
64), although we will raise some questions about his criteria, and
propose some others. He writes:

Admissibility.We say that an interpretation of a formal system is admissible if the meanings assigned to the primitive terms in the interpretation transform the formal axioms, and consequently all the theorems, into true statements. A fundamental requirement for probability concepts is to satisfy the mathematical relations specified by the calculus of probability…

Ascertainability.This criterion requires that there be some method by which, in principle at least, we can ascertain values of probabilities. It merely expresses the fact that a concept of probability will be useless if it is impossible in principle to find out what the probabilities are…

Applicability.The force of this criterion is best expressed in Bishop Butler's famous aphorism, “Probability is the very guide of life.”…

It might seem that the criterion of admissibility goes without
saying: ‘interpretations’ of the probability calculus that
assigned to *P* the interpretation ‘the number of hairs on
the head of’ or ‘the political persuasion of’ would
obviously not even be in the running, because they would render the
axioms and theorems so obviously false. The word
‘interpretation’ is often used in such a way that
‘admissible interpretation’ is a pleonasm. Yet it turns out
that the criterion is non-trivial, and indeed if taken seriously would
rule out several of the leading interpretations of probability! As we
will see, some of them fail to satisfy countable additivity; for others
(certain propensity interpretations) the status of at least some of the
axioms is unclear. Nevertheless, we regard them as genuine candidates.
It should be remembered, moreover, that Kolmogorov's is just one of
many possible axiomatizations, and there is not universal agreement on
which is ‘best’ (whatever that might mean). Indeed,
Salmon's preferred axiomatization differs from Kolmogorov's.^{[2]} Thus, there
is no such thing as admissibility *tout court*, but rather
admissibility with respect to this or that axiomatization. It would be
unfortunate if, perhaps out of an overdeveloped regard for history, one
felt obliged to reject any interpretation that did not obey the letter
of Kolmogorov's laws and that was thus ‘inadmissible’. In
any case, if we found an inadmissible interpretation that did a
wonderful job of meeting the criteria of ascertainability and
applicability, then we should surely embrace it.

So let us turn to those criteria. It is a little unclear in the
ascertainability criterion just what “in principle” amounts
to, though perhaps some latitude here is all to the good. Understood
charitably, and to avoid trivializing it, it presumably excludes
omniscience. On the other hand, understanding it in a way acceptable to
a strict empiricist or a verificationist may be too restrictive.
‘Probability’ is apparently, among other things, a
*modal* concept, plausibly outrunning that which actually
occurs, let alone that which is actually observed.

Most of the work will be done by the applicability criterion. We
must say more (as Salmon indeed does) about what *sort* of a
guide to life probability is supposed to be. Mass, length, area and
volume are all useful concepts, and they are ‘guides to
life’ in various ways (think how critical distance judgments can
be to survival); moreover, they are admissible and ascertainable, so
presumably it is the applicability criterion that will rule them out.
Perhaps it is best to think of applicability as a cluster of criteria,
each of which is supposed to capture something of probability's
distinctive conceptual roles; moreover, we should not require that all
of them be met by a given interpretation. They include:

Non-triviality:an interpretation should make non-extreme probabilities at least a conceptual possibility. For example, suppose that we interpret ‘P’ as thetruthfunction: it assigns the value 1 to all true sentences, and 0 to all false sentences. Then trivially, all the axioms come out true, so this interpretation is admissible. We would hardly count it as an adequateinterpretation ofprobability, however, and so we need to exclude it. It is essential to probability that, at least in principle, it can takeintermediatevalues. All of the interpretations that we will present meet this criterion, so we will discuss it no more.

Applicability to frequencies:an interpretation should render perspicuous the relationship between probabilities and (long-run) frequencies. Among other things, it should make clear why, by and large, more probable events occur more frequently than less probable events.

Applicability to rational belief:an interpretation should clarify the role that probabilities play in constraining the degrees of belief, orcredences, of rational agents. Among other things, knowing that one event is more probable than another, a rational agent will be more confident about the occurrence of the former event.

Applicability to ampliative inference:an interpretation will score bonus points if it illuminates the distinction between ‘good’ and ‘bad’ ampliative inferences, while explicating why both fall short of deductive inferences.

The next criterion may be redundant, given our list so far, but including it will do no harm:

Applicability to science:an interpretation should illuminate paradigmatic uses of probability in science (for example, in quantum mechanics and statistical mechanics).

Perhaps there are further *metaphysical* desiderata that we
might impose on the interpretations. For example, there appear to be
connections between probability and *modality.* Events with
positive probability *can* happen, even if they don't. Some
authors also insist on the converse condition that *only* events
with positive probability can happen, although this is more
controversial — see our discussion of ‘regularity’ in
Section 4. (Indeed, in uncountable probability spaces this condition
will require the employment of infinitesimals, and will thus take us
beyond the standard Kolmogorov theory — ‘standard’ both in
the sense of being the orthodoxy, and in its employment of standard, as
opposed to ‘non-standard’ real numbers. See Skyrms 1980.)
In any case, our list is already long enough to help in our assessment
of the leading interpretations on the market.

## 3. The Main Interpretations

Broadly speaking, there are arguably three main concepts of probability:

- A quasi-logical concept, which is meant to measure objective
evidential support relations. For example, “in light of the relevant
seismological and geological data, it is
*probable*that California will experience a major earthquake this decade”. - The concept of an agent's degree of confidence, a graded
belief. For example, “I am not sure that it will rain in Canberra this
week, but it
*probably*will.” - An objective concept that applies to various systems in the world,
independently of what anyone thinks. For example, “a particular radium
atom will
*probably*decay within 10,000 years”.

Some philosophers will insist that not all of these concepts are
intelligible; some will insist that one of them is basic, and that the
others are reducible to it. Be that as it may, it will be useful to
keep these concepts in mind. Sections 3.1 and 3.2 discuss analyses of
concept (1), *classical* and *logical* probability; 3.3
discusses analyses of concept (2), *subjective* probability;
3.4, 3.5, and 3.6 discuss three kinds of analysis of concept
(3), *frequentist*, *propensity*,
and *best-system* intepretations.

### 3.1 Classical Probability

The classical interpretation owes its name to its early and august pedigree. Championed by Laplace, and found even in the works of Pascal, Bernoulli, Huygens, and Leibniz, it assigns probabilities in the absence of any evidence, or in the presence of symmetrically balanced evidence. The guiding idea is that in such circumstances, probability is shared equally among all the possible outcomes, so that the classical probability of an event is simply the fraction of the total number of possibilities in which the event occurs. It seems especially well suited to those games of chance that by their very design create such circumstances — for example, the classical probability of a fair die landing with an even number showing up is 3/6. It is often presupposed (usually tacitly) in textbook probability puzzles.

Here is a classic statement by Laplace:

The theory of chance consists in reducing all the events of the same kind to a certain number of cases equally possible, that is to say, to such as we may be equally undecided about in regard to their existence, and in determining the number of cases favorable to the event whose probability is sought. The ratio of this number to that of all the cases possible is the measure of this probability, which is thus simply a fraction whose numerator is the number of favorable cases and whose denominator is the number of all the cases possible. (1814, 1951 6–7)

There are numerous questions to be asked about this formulation. When are events of the same kind? Intuitively, ‘heads’ and ‘tails’ are equally likely outcomes of tossing a fair coin; but if their kind is ‘ways the coin could land’, then ‘edge’ should presumably be counted alongside them. The “certain number of cases” and “that of all the cases possible” are presumably finite numbers. What, then, of probabilities in infinite spaces? Apparently, irrational-valued probabilities such as 1/√2 are automatically eliminated, and thus theories such as quantum mechanics that posit them cannot be accommodated. (We will shortly see, however, that Laplace's theory has been refined to handle infinite spaces.)

Who are “we”, who “may be equally
undecided”? Different people may be equally undecided about
different things, which suggests that Laplace is offering a
subjectivist interpretation in which probabilities vary from person to
person depending on contingent differences in their evidence. This is
not his intention. He means to characterize the objective probability
assignment of a rational agent in an epistemically neutral position
with respect to a set of “equally possible” cases. But then
the proposal risks sounding empty: for what is it for an agent to
*be* “equally undecided” about a set of cases, other
than assigning them equal probability?

This brings us to one of the key objections to Laplace's account.
The notion of “equally possible” cases faces the charge of
either being a category mistake (for ‘possibility’ does not
come in degrees), or circular (for what is meant is really
‘equally probable’). The notion is finessed by the
so-called ‘principle of indifference’, a coinage due to
Keynes. It states that whenever there is no evidence favoring one
possibility over another, they have the same probability. Thus, it is
claimed, there is no circularity in the classical definition after all.
However, this move may only postpone the problem, for there is still a
threat of circularity, albeit at a lower level. We have two cases here:
outcomes for which we have *no evidence at all*, and outcomes
for which we have *symmetrically balanced evidence*. There is no
circularity in the first case unless the notion of
‘evidence’ is itself probabilistic; but artificial examples
aside, it is doubtful that the case ever arises. For example, we have a
considerable fund of evidence on coin tossing from the results of our
own experiments, the testimony of others, our knowledge of some of the
relevant physics, and so on. In the second case, the threat of
circularity is more apparent, for it seems that some sort of
*weighing* of the evidence in favor of each outcome is required,
and it is not obvious that this can be done without reference to
probability. Indeed, the most obvious characterization of symmetrically
balanced evidence is in terms of equality of conditional probabilities:
given evidence *E* and possible outcomes *O*_{1},
*O*_{2}, …, *O _{n}*, the evidence
is symmetrically balanced iff

*P*(

*O*

_{1}|

*E*) =

*P*(

*O*

_{2}|

*E*) = … =

*P*(

*O*

_{n}|

*E*). Then it seems that probabilities reside at the base of the interpretation after all. Still, it would be an achievement if all probabilities could be reduced to cases of equal probability.

As we have seen, Laplace's classical theory is restricted to finite
spaces, one for which there are only finitely many possible
outcomes. When the spaces are countably infinite, the spirit of the
classical theory may be upheld by appealing to the
information-theoretic principle of *maximum entropy*, a
generalization of the principle of indifference championed by Jaynes
(1968). Entropy is a measure of the lack of
‘informativeness’ of a probability function. The more
concentrated is the function, the less is its entropy; the more
diffuse it is, the greater is its entropy. For a discrete assignment
of probabilities
*P* = (*p*_{1}, *p*_{2},
…), the entropy of *P* is defined as:

−∑_{i}p_{i}logp_{i}

The principle of maximum entropy enjoins us to select from the family
of all probability functions consistent with our background knowledge the
function that maximizes this quantity. In the special case of
choosing the most uninformative prior over a finite set of possible
outcomes, this is just the familiar ‘flat’ classical
assignment discussed previously. Things get more complicated in the
infinite case, since there cannot be a flat assignment over
denumerably many outcomes, on pain of violating the standard
probability calculus (with countable additivity). Rather, the best we
can have are sequences of progressively flatter assignments, none of
which is truly flat. We must then impose some *further*
constraint that narrows the field to a smaller family in which there
*is* an assignment of maximum entropy.^{[3]} This constraint has to
be imposed from outside as background knowledge, but there is no
general theory of which external constraint should be applied when.

Let us turn now to uncountably infinite spaces. It is easy — all too easy — to assign equal probabilities to the points in such a space: each gets probability 0. Non-trivial probabilities arise when uncountably many of the points are clumped together in larger sets. If there are finitely many clumps, Laplace's classical theory may be appealed to again: if the evidence bears symmetrically on these clumps, each gets the same share of probability.

Enter Bertrand's paradoxes. They all arise in uncountable spaces and
turn on alternative parametrizations of a given problem that are
non-linearly related to each other. Some presentations are needlessly
arcane; length and area suffice to make the point. The following
example (adapted from van Fraassen 1989) nicely illustrates how
Bertrand-style paradoxes work. A factory produces cubes with
side-length between 0 and 1 foot; what is the probability that a
randomly chosen cube has side-length between 0 and 1/2 a foot? The
tempting answer is 1/2, as we imagine a process of production that is
uniformly distributed over side-length. But the question could have
been given an equivalent restatement: A factory produces cubes with
face-area between 0 and 1 square-feet; what is the probability that a
randomly chosen cube has face-area between 0 and 1/4 square-feet? Now
the tempting answer is 1/4, as we imagine a process of production that
is uniformly distributed over face-area. This is already disastrous, as
we cannot allow the same event to have two different probabilities
(especially if this interpretation is to be admissible!). But there is
worse to come, for the problem could have been restated equivalently
again: A factory produces cubes with volume between 0 and 1 cubic feet;
what is the probability that a randomly chosen cube has volume between
0 and 1/8 cubic-feet? Now the tempting answer is 1/8, as we imagine a
process of production that is uniformly distributed over volume. And so
on for all of the infinitely many equivalent reformulations of the
problem (in terms of the fourth, fifth, … power of the length,
and indeed in terms of every non-zero real-valued exponent of the
length). What, then, is *the* probability of the event in
question?

The paradox arises because the principle of indifference can be used in incompatible ways. We have no evidence that favors the side-length lying in the interval [0, 1/2] over its lying in [1/2, 1], or vice versa, so the principle requires us to give probability 1/2 to each. Unfortunately, we also have no evidence that favors the face-area lying in any of the four intervals [0, 1/4], [1/4, 1/2], [1/2, 3/4], and [3/4, 1] over any of the others, so we must give probability 1/4 to each. The event ‘the side-length lies in [0, 1/2]’, receives a different probability when merely redescribed. And so it goes, for all the other reformulations of the problem. We cannot meet any pair of these constraints simultaneously, let alone all of them.

Jaynes attempts to save the principle of indifference and to extend
the principle of maximum entropy to the continuous case, with
his *invariance condition*: in two problems where we have the
same knowledge, we should assign the same probabilities. He regards
this as a consistency requirement. For any problem, we have a group of
admissible transformations, those that change the problem into an
equivalent form. Various details are left unspecified in the problem;
equivalent formulations of it fill in the details in different
ways. Jaynes' invariance condition bids us to assign equal
probabilities to equivalent propositions, reformulations of one
another that are arrived at by such admissible transformations of our
problem. Any probability assignment that meets this condition is
called an *invariant* assignment. Ideally, our problem will
have a unique invariant assignment. To be sure, things will not always
be ideal; but sometimes they are, in which case this is surely
progress on Bertrand-style problems.

And in any case, for many garden-variety problems such technical machinery will not be needed. Suppose I tell you that a prize is behind one of three doors, and you get to choose a door. This seems to be a paradigm case in which the principle of indifference works well: the probability that you choose the right door is 1/3. It seems implausible that we should worry about some reparametrization of the problem that would yield a different answer. To be sure, Bertrand-style problems caution us that there are limits to the principle of indifference. But arguably we must just be careful not to overstate its applicability.

How does the classical theory of probability fare with respect to our
criteria of adequacy? Let us begin with admissibility. It is claimed
that (Laplacean) classical probabilities are only finitely additive
(see, e.g., de Finetti 1974). It would be more correct to say that
classical probabilities are countably additive, but trivially so. As
we have seen, classical probabilities are only defined on finite
spaces. The statement 3′ of countable additivity, recall, is a
conditional; its antecedent, “{*A _{i}*} is a
countably infinite collection of (pairwise) disjoint sets,” is
never satisfied in such spaces. Thus, the conditional is vacuously
true. Clearly, classical probabilities obey the other axioms, so this
interpretation is admissible.

Classical probabilities are ascertainable, assuming that the space of
possibilities can be determined in principle. They bear a relationship
to the credences of rational agents; the circularity concern, as we
saw above, is that the relationship is vacuous, and that rather than
*constraining* the credences of a rational agent in an
epistemically neutral position, they merely record them.

Without supplementation, the classical theory makes no contact with frequency information. However the coin happens to land in a sequence of trials, the possible outcomes remain the same. Indeed, even if we have strong empirical evidence that the coin is biased towards heads with probability, say, 0.6, it is hard to see how the unadorned classical theory can accommodate this fact — for what now are the ten possibilities, six of which are favorable to heads? Laplace does supplement the theory with his Rule of Succession: “Thus we find that an event having occurred successively any number of times, the probability that it will happen again the next time is equal to this number increased by unity divided by the same number, increased by two units.” (1951, 19) That is:

Pr(success onN+1st trial |Nconsecutive successes)=

Thus, inductive learning is possible — though not by classical
probabilities *per se*, but rather thanks to this further
rule. And must ask whether such learning can be captured once and for
all by such a simple formula, the same for all domains and events. We
will return to this question when we discuss the logical
interpretation below.

Science apparently invokes at various points probabilities that look
classical. Bose-Einstein statistics, Fermi-Dirac statistics, and
Maxwell-Boltzmann statistics each arise by considering the ways in
which particles can be assigned to states, and then applying the
principle of indifference to different subdivisions of the set of
alternatives, Bertrand-style. The trouble is that Bose-Einstein
statistics apply to some particles (e.g. photons) and not to others,
Fermi-Dirac statistics apply to different particles (e.g. electrons),
and Maxwell-Boltzmann statistics do not apply to any known particles.
None of this can be determined *a priori*, as the classical
interpretation would have it. Moreover, the classical theory purports
to yield probability assignments in the face of ignorance. But as Fine
(1973) writes:

If we are truly ignorant about a set of alternatives, then we are also ignorant about combinations of alternatives and about subdivisions of alternatives. However, the principle of indifference when applied to alternatives, or their combinations, or their subdivisions, yields different probability assignments (170).

This brings us to one of the chief points of controversy regarding the classical interpretation. Critics accuse the principle of indifference of extracting information from ignorance. Proponents reply that it rather codifies the way in which such ignorance should be epistemically managed — for anything other than an equal assignment of probabilities would represent the possession of some knowledge. Critics counter-reply that in a state of complete ignorance, it is better to assign imprecise probabilities (perhaps ranging over the entire [0, 1] interval), or to eschew the assignment of probabilities altogether.

### 3.2 Logical probability

Logical theories of probability retain the classical
interpretation's idea that probabilities can be determined a priori by
an examination of the space of possibilities. However, they generalize
it in two important ways: the possibilities may be assigned
*unequal* weights, and probabilities can be computed whatever
the evidence may be, symmetrically balanced or not. Indeed, the logical
interpretation, in its various guises, seeks to encapsulate in full
generality the degree of support or confirmation that a piece of
evidence *E* confers upon a given hypothesis *H*, which
we may write as *c*(*H*, *E*). In doing so, it can
be regarded also as generalizing deductive logic and its notion of
implication, to a complete theory of inference equipped with the notion
of ‘degree of implication’ that relates *E* to
*H*. It is often called the theory of ‘inductive
logic’, although this is a misnomer: there is no requirement that
*E* be in any sense ‘inductive’ evidence for
*H*. ‘Non-deductive logic’ would be a better name,
but this overlooks the fact that deductive logic's relations of
implication and incompatibility are also accommodated as extreme cases
in which the confirmation function takes the values 1 and 0
respectively. Nevertheless, what is significant is that the logical
interpretation provides a framework for induction.

Early proponents of logical probability include Johnson (1921),
Keynes (1921), and Jeffreys (1939). However, by far the most systematic
study of logical probability was by Carnap. His formulation of logical
probability begins with the construction of a formal language. In
(1950) he considers a class of very simple languages consisting of a
finite number of logically independent monadic predicates (naming
properties) applied to countably many individual constants (naming
individuals) or variables, and the usual logical connectives. The
strongest (consistent) statements that can be made in a given language
describe all of the individuals in as much detail as the expressive
power of the language allows. They are conjunctions of complete
descriptions of each individual, each description itself a conjunction
containing exactly one occurrence (negated or unnegated) of each
predicate of the language. Call these strongest statements *state
descriptions*.

Any probability measure *m*(−) over the state
descriptions automatically extends to a measure over all sentences,
since each sentence is equivalent to a disjunction of state
descriptions; m in turn induces a confirmation function
*c*(−, −):

There are obviously infinitely many candidates for *m*, and
hence *c*, even for very simple languages. Carnap argues for his
favored measure “*m**” by insisting that the only
thing that significantly distinguishes individuals from one another is
some qualitative difference, not just a difference in labeling. Call a
*structure description* a maximal set of state descriptions,
each of which can be obtained from another by some permutation of the
individual names. *m** assigns each structure description equal
measure, which in turn is divided equally among their constituent state
descriptions. It gives greater weight to homogenous state descriptions
than to heterogeneous ones, thus ‘rewarding’ uniformity
among the individuals in accordance with putatively reasonable
inductive practice. The induced *c** allows inductive learning
from experience.

Consider, for example, a language that has three names, *a*,
*b* and *c*, for individuals, and one predicate
*F*. For this language, the state descriptions are:

*Fa*&*Fb*&*Fc*- ¬
*Fa*&*Fb*&*Fc* *Fa*& ¬*Fb*&*Fc**Fa*&*Fb*& ¬*Fc*- ¬
*Fa*& ¬*Fb*&*Fc* - ¬
*Fa*&*Fb*& ¬*Fc* *Fa*& ¬*Fb*& ¬*Fc*- ¬
*Fa*& ¬*Fb*& ¬*Fc*

There are four structure descriptions:

{1}, “Everything is F”;{2, 3, 4}, “Two

Fs, one ¬F”;{5, 6, 7}, “One

F, two ¬Fs”; and{8}, “Everything is ¬

F”.

The measure *m** assigns numbers to the state descriptions as
follows: first, every structure description is assigned an equal
weight, 1/4; then, each state description belonging to a given
structure description is assigned an equal part of the weight assigned
to the structure description:

State descriptionStructure descriptionWeightm*1. Fa.Fb.FcI. Everything is F1/4 1/4 2. ¬ Fa.Fb.Fc1/12 3. Fa.¬Fb.FcII. Two Fs, one ¬F1/4 1/12 4. Fa.Fb.¬Fc1/12 5. ¬ Fa.¬Fb.Fc1/12 6. ¬ Fa.Fb.¬FcIII. One F, two ¬Fs1/4 1/12 7. Fa.¬Fb.¬Fc1/12 8. ¬ Fa.¬Fb.¬FcIV. Everything is ¬ F1/4 1/4

Notice that *m** gives greater weight to the homogenous state
descriptions 1 and 8 than to the heterogeneous ones. This will manifest
itself in the inductive support that hypotheses can gain from
appropriate evidence statements. Consider the hypothesis statement
*h* = *Fc*, true in 4 of the 8 state descriptions, with
*a priori* probability *m**(*h*) = 1/2. Suppose we
examine individual “*a*” and find it has property
*F* — call this evidence *e*. Intuitively, *e* is
favorable (albeit weak) inductive evidence for *h*. We have:
*m**(*h* & *e*) = 1/3, *m**(*e*)
= 1/2, and hence

c*(h,e)= = 2/3.

This is greater than the *a priori* probability
*m**(*h*) = 1/2, so the hypothesis has been confirmed. It
can be shown that in general *m** yields a degree of
confirmation *c** that allows learning from experience.

Note, however, that infinitely many confirmation functions, defined
by suitable choices of the initial measure, allow learning from
experience. We do not have yet a reason to think that *c** is
the right choice. Carnap claims nevertheless that *c** stands
out for being simple and natural.

He later generalizes his confirmation function to a continuum of
functions *c*_{λ}. Define a *family* of
predicates to be a set of predicates such that, for each individual,
exactly one member of the set applies, and consider first-order
languages containing a finite number of families. Carnap (1963) focuses
on the special case of a language containing only one-place predicates.
He lays down a host of axioms concerning the confirmation function
*c*, including those induced by the probability calculus itself,
various axioms of symmetry (for example, that *c*(*h*,
*e*) remains unchanged under permutations of individuals, and of
predicates of any family), and axioms that guarantee undogmatic
inductive learning, and long-run convergence to relative frequencies.
They imply that, for a family {*P*_{n}},
*n* = 1, …, *k* (*k* > 2):

c_{λ} |
(individual s + 1 is P_{j},
s_{j} of the first s individuals are
P_{j}) |
= | , |

where λ is a positive real number. The higher the value of
λ, the less impact evidence has: induction from what is
observed becomes progressively more swamped by a classical-style equal
assignment to each of the *k* possibilities regarding
individual *s* + 1.

I turn to various objections to Carnap's program that have been
offered in the literature, noting that this remains an area of lively
debate. (See Maher (2010) for rebuttals of these arguments and for
defenses of Carnap.) Firstly, is there a correct setting of λ,
or said another way, how ‘inductive’ should the
confirmation function be? The concern here is that any particular
setting of λ is arbitrary in a way that compromises Carnap's
claim to be offering a *logical* notion of probability. Also,
it turns out that for any such setting, a universal statement in an
infinite universe always receives zero confirmation, no matter what
the (finite) evidence. Many find this counterintuitive, since laws of
nature with infinitely many instances can apparently be confirmed.
Earman (1992) discusses the prospects for avoiding the unwelcome
result.

Significantly, Carnap's various axioms of symmetry are hardly logical truths. Moreover, Fine (1973, 202) argues that we cannot impose further symmetry constraints that are seemingly just as plausible as Carnap's, on pain of inconsistency. Goodman taught us: that the future will resemble the past in some respect is trivial; that it will resemble the past in all respects is contradictory. And we may continue: that a probability assignment can be made to respect some symmetry is trivial; that one can be made to respect all symmetries is contradictory. This threatens the whole program of logical probability.

Another Goodmanian lesson is that inductive logic must be sensitive to the meanings of predicates, strongly suggesting that a purely syntactic approach such as Carnap's is doomed. Scott and Krauss (1966) use model theory in their formulation of logical probability for richer and more realistic languages than Carnap's. Still, finding a canonical language seems to many to be a pipe dream, at least if we want to analyze the “logical probability” of any argument of real interest — either in science, or in everyday life.

Logical probabilities are admissible. It is easily shown that they satisfy finite additivity, and given that they are defined on finite sets of sentences, the extension to countable additivity is trivial. Given a choice of language, the values of a given confirmation function are ascertainable; thus, if this language is rich enough for a given application, the relevant probabilities are ascertainable. The whole point of the theory of logical probability is to explicate ampliative inference, although given the apparent arbitrariness in the choice of language and in the setting of λ — thus, in the choice of confirmation function — one may wonder how well it achieves this. The problem of arbitrariness of the confirmation function also hampers the extent to which the logical interpretation can truly illuminate the connection between probabilities and frequencies.

The arbitrariness problem, moreover, stymies any compelling
connection between logical probabilities and rational credences. And a
further problem remains even after the confirmation function has been
chosen: if one's credences are to be based on logical probabilities,
they must be relativized to an evidence statement, *e*. But
which is to be? Carnap requires that *e* be one's *total
evidence*, that is, the maximally specific information at one's
disposal, the strongest proposition of which one is certain. However,
when we go beyond toy examples, it is not clear that this is
well-defined. Suppose I have just watched a coin toss, and thus learned
that the coin landed heads. Perhaps ‘the coin landed heads’
is my total evidence? But I also learned a host of other things: as it
might be, that the coin landed at a certain time, bouncing in a certain
way, making a certain noise as it did so … Call this long
conjunction of facts *X*. I also learned a potentially infinite
set of *de se* propositions: ‘I learned that
*X*’, ‘I learned that I learned that
*X*’ and so on. Perhaps, then, my total evidence is the
infinite intersection of all these propositions, although this is still
not obvious — and it is not something that can be represented by a
sentence in one of Carnap's languages, which is finite in length. More
significantly, the total evidence criterion goes hand in hand with
positivism and a foundationalist epistemology according to which there
are such determinate, ultimate deliverances of experience. But perhaps
learning does not come in the form of such ‘bedrock’
propositions, as Jeffrey (1992) has argued — maybe it rather involves
a shift in one's subjective probabilities across a partition, without
any cell of the partition becoming certain. Then it may be the case
that the strongest proposition of which one is certain is expressed by
a tautology *T* — hardly an interesting notion of ‘total evidence’.^{[4]}

In connection with the ‘applicability to science’
criterion, a point due to Lakatos is telling. By Carnap's lights, the
degree of confirmation of a hypothesis depends on the language in which
the hypothesis is stated and over which the confirmation function is
defined. But scientific progress often brings with it a change in
scientific language (for example, the addition of new predicates and
the deletion of old ones), and such a change will bring with it a
change in the corresponding *c*-values. Thus, the growth of
science may overthrow any particular confirmation theory. There is
something of the snake eating its own tail here, since logical
probability was supposed to explicate the confirmation of scientific
theories.

We have seen that the later Carnap relaxed his earlier aspiration
to find a *unique* confirmation function, allowing a continuum
of such functions displaying a wide range of inductive
cautiousness. Various critics of logical probabilities believe that he
did not go far enough - that even his later systems constrain
inductive learning beyond what is rationally required. This recalls
the classic debate earlier in the 20^{th} century between
Keynes, a famous proponent of logical probabilities, and Ramsey, an
equally famous opponent. Ramsey was skeptical of there being any
non-trivial relations of logical probability: he said that he could
not discern them himself, and that others disagree about them. This
skepticism led him to formulate his own, enormously
influential *subjective* interpretation of probability.

### 3.3 Subjective probability

#### 3.3.1 Probability as degree of belief

We may characterize *subjectivism* (also known as
*personalism* and *subjective Bayesianism*) with the
slogan: ‘Probability is degree of belief’. We identify
probabilities with degrees of confidence, or credences, or
“partial” beliefs of suitable agents. Thus, we really have
*many* interpretations of probability here, as many as there
are doxastic states of suitable agents: we have Aaron's degrees of
belief, Abel's degrees of belief, Abigail's degrees of
belief, … , or better still, Aaron's degrees of
belief-at-time-*t*_{1}, Aaron's degrees of
belief-at-time-*t*_{2}, Abel's degrees of
belief-at-time-*t*_{1}, … . Of course,
we must ask what makes an agent ‘suitable’. What we might
call *unconstrained subjectivism* places no constraints on the
agents — anyone goes, and hence anything goes. Various studies
by psychologists (see, e.g., several articles in Kahneman et al. 1982)
are taken to show that people commonly violate the usual probability
calculus in spectacular ways. We clearly do not have here an
admissible interpretation (with respect to any probability calculus),
since there is no limit to what agents might assign. Unconstrained
subjectivism is not a serious proposal.

More interesting, however, is the claim that the suitable agents
must be, in a strong sense, *rational*. Beginning with Ramsey
(1926), various subjectivists have wanted to assimilate probability to
logic by portraying probability as the logic of partial belief. A
rational agent is required to be logically consistent, now taken in a
broad sense. These subjectivists argue that this implies that the agent
obeys the axioms of probability (although perhaps with only finite
additivity), and that subjectivism is thus (to this extent) admissible.
Before we can present this argument, we must say more about what
degrees of belief are.

#### 3.3.2 The betting analysis and the Dutch Book argument

Subjective probabilities are traditionally analyzed in terms of betting behavior. Here is a classic statement by de Finetti (1980):

Let us suppose that an individual is obliged to evaluate the ratepat which he would be ready to exchange the possession of an arbitrary sumS(positive or negative) dependent on the occurrence of a given eventE, for the possession of the sumpS; we will say by definition that this numberpis the measure of the degree of probability attributed by the individual considered to the eventE, or, more simply, thatpis the probability ofE(according to the individual considered; this specification can be implicit if there is no ambiguity). (62)

This boils down to the following analysis:

Your degree of belief inEispiffpunits of utility is the price at which you would buy or sell a bet that pays 1 unit of utility ifE, 0 if notE.

The analysis presupposes that, for any *E*, there is exactly
one such price — let's call this the agent's *fair price*
for the bet on *E*. This presupposition may fail. There may be
no such price — you may refuse to bet on *E* at all
(perhaps unless coerced, in which case your genuine opinion
about *E* may not be revealed), or your selling price may
differ from your buying price, as may occur if your probability
for *E* is imprecise. There may be more than one fair price
— you may find a range of such prices acceptable, as may also
occur if your probability for *E* is vague. For now, however,
let us waive these concerns, and turn to an important argument, again
originating with Ramsey, that uses the betting analysis purportedly to
show that rational degrees of belief must conform to the probability
calculus (with at least finite additivity).

A *Dutch book* (against an agent) is a series of bets, each
acceptable to the agent, but which collectively guarantee her loss,
however the world turns out. Ramsey notes, and it can be easily proven
(e.g., Skyrms 1984), that if your subjective probabilities violate the
probability calculus, then you are susceptible to a Dutch book. For
example, suppose that you violate the additivity axiom by assigning
*P*(*A* ∪ *B*) < *P*(*A*) +
*P*(*B*), where *A* and *B* are mutually
exclusive. Then a cunning bettor could buy from you a bet on *A*
∪ *B* for *P*(*A* ∪ *B*) units, and
sell you bets on *A* and *B* individually for
*P*(*A*) and *P*(*B*) units respectively.
He pockets an initial profit of *P*(*A*) +
*P*(*B*) − *P*(*A* ∪ *B*),
and retains it whatever happens. Ramsey offers the following
influential gloss: “If anyone's mental condition violated these
laws [of the probability calculus], his choice would depend on the
precise form in which the options were offered him, which would be
absurd.” (1980, 41)

Equally important, and often neglected, is the converse theorem that
establishes how you can avoid such a predicament. If your subjective
probabilities conform to the probability calculus, then no Dutch book
can be made against you (Kemeny 1955); your probability assignments are
then said to be *coherent*. In a nutshell, conformity to the
probability calculus is necessary and sufficient for coherence.^{[5]}

But let us return to the betting analysis of credences. It is an
attempt to make good on Ramsey's idea that probability “is a
measurement of belief *qua* basis of action” (34). While
he regards the method of measuring an agent's credences by her betting
behavior as “fundamentally sound” (34), he recognizes that
it has its limitations.

The betting analysis gives an operational definition of subjective probability, and indeed it inherits some of the difficulties of operationalism in general, and of behaviorism in particular. For example, you may have reason to misrepresent your true opinion, or to feign having opinions that in fact you lack, by making the relevant bets (perhaps to exploit an incoherence in someone else's betting prices). Moreover, as Ramsey points out, placing the very bet may alter your state of opinion. Trivially, it does so regarding matters involving the bet itself (e.g., you suddenly increase your probability that you have just placed a bet). Less trivially, placing the bet may change the world, and hence your opinions, in other ways (betting at high stakes on the proposition ‘I will sleep well tonight’ may suddenly turn you into an insomniac). And then the bet may concern an event such that, were it to occur, you would no longer value the pay-off the same way. (During the August 11, 1999 solar eclipse in the UK, a man placed a bet that would have paid a million pounds if the world came to an end.)

These problems stem largely from taking literally the notion of
entering into a bet on *E*, with its corresponding payoffs. The
problems may be avoided by identifying your degree of belief in a
proposition with the betting price you regard as fair, whether or not
you enter into such a bet; it corresponds to the betting odds that you
believe confer no advantage or disadvantage to either side of the bet
(Howson and Urbach 1993). There is something of the Rawlsian
‘veil of ignorance’ reasoning here: imagine that you are to
set the price for the bet, but you do not yet know which side of the
bet you are to take. At your fair price, you should be indifferent
between taking either side.^{[6]}

de Finetti speaks of “an arbitrary sum” as the prize of
the bet on *E*. The sum had better be potentially infinitely
divisible, or else probability measurements will be precise only up to
the level of ‘grain’ of the potential prizes. For example,
a sum that can be divided into only 100 parts will leave probability
measurements imprecise beyond the second decimal place, conflating
probabilities that should be distinguished (e.g., those of a logical
contradiction and of ‘a fair coin lands heads 8 times in a
row’). More significantly, if utility is not a linear function of
such sums, then the size of the prize will make a difference to the
putative probability: winning a dollar means more to a pauper more than
it does to Bill Gates, and this may be reflected in their betting
behaviors in ways that have nothing to do with their genuine
probability assignments. de Finetti responds to this problem by
suggesting that the prizes be kept small; that, however, only creates
the opposite problem that agents may be reluctant to bother about
trifles, as Ramsey points out.

Better, then, to let the prizes be measured in utilities: after all, utility is infinitely divisible, and utility is a linear function of utility. While we're at it, we should adopt a more liberal notion of betting. After all, there is a sense in which every decision is a bet, as Ramsey observed.

#### 3.3.3 Probabilities and utilities

Utilities (desirabilities) of outcomes, their probabilities, and
rational preferences are all intimately linked. The *Port Royal
Logic* (Arnauld, 1662) showed how utilities and probabilities
together determine rational preferences; de Finetti's betting
analysis derives probabilities from utilities and rational
preferences; von Neumann and Morgenstern (1944) derive utilities from
probabilities and rational preferences. And most remarkably, Ramsey
(1926) (and later, Savage 1954 and Jeffrey 1966) derives *both*
probabilities *and* utilities from rational preferences
alone.

First, he defines a proposition to be *ethically neutral* —
relative to an agent — if the agent is indifferent
between having that outcome when the proposition is true and when it is
false. The idea is that the agent doesn't care about the ethically
neutral proposition as such — it is a means to an end that he might
care about, but it has no intrinsic value. Now, there is a simple test
for determining whether, for a given agent, an ethically neutral
proposition *N* has probability 1/2. Suppose that the agent
prefers *A* to *B*. Then *N* has probability 1/2
iff the agent is indifferent between the gambles:

AifN,Bif not

BifN,Aif not.

Ramsey assumes that it does not matter what the candidates for
*A* and *B* are. We may assign arbitrarily to *A*
and *B* any two real numbers *u*(*A*) and
*u*(*B*) such that *u*(*A*) >
*u*(*B*), thought of as the desirabilities of *A*
and *B* respectively. Having done this for the one arbitrarily
chosen pair *A* and *B*, the utilities of all other
propositions are determined.

Given various assumptions about the richness of the preference
space, and certain ‘consistency assumptions’, he can define
a real-valued utility function of the outcomes *A*, *B*,
etc — in fact, various such functions will represent the agent's
preferences. He is then able to define equality of differences in
utility for any outcomes over which the agent has preferences. It turns
out that ratios of utility-differences are invariant — the same
whichever representative utility function we choose. This fact allows
Ramsey to define degrees of belief as ratios of such differences. For
example, suppose the agent is indifferent between *A*, and the
gamble “*B* if *X*, *C* otherwise.”
Then it follows from considerations of expected utility that her degree
of belief in *X*, *P*(*X*), is given by:

P(X)=

Ramsey shows that degrees of belief so derived obey the probability calculus (with finite additivity). He calls what results “the logic of partial belief,” and indeed he opens his essay with the words “In this essay the Theory of Probability is taken as a branch of logic….”

Ramsey avoids some of the objections to the betting analysis, but not
all of them. Notably, the essential appeal to gambles again raises the
concern that the wrong quantities are being measured — an
inveterate gambler might overvalue, and a puritan might undervalue, a
gamble compared to what their true credences would indicate. And his
account has new difficulties. It is unclear what facts about agents
fix their preference rankings. These rankings cannot simply be read
off their behaviors. For example, the coach of a football team might
ostentatiously bet at an inordinately high price on his team winning,
in a public display of support that reveals nothing about his honest
opinion. It is also dubious that
*consistency* requires one to have a set of preferences as rich
as Ramsey requires, or that one can find ethically neutral propositions
of probability 1/2. This in turn casts some doubt on Ramsey's claim to
assimilate probability theory to logic.

Savage (1954) likewise derives probabilities and utilities from
preferences among options that are constrained by certain putative
‘consistency’ principles. For a given set of such preferences, he
generates a class of utility functions, each a positive linear
transformation of the other (i.e. of the form *U*_{1} =
*aU*_{2} + *b*, where *a* > 0), and a
unique probability function. Together these are said to
‘represent’ the agent's preferences. Jeffrey (1966) refines
the method further. The result is theory of decision according to which
rational choice maximizes ‘expected utility’, a certain
probability-weighted average of utilities. Some of the difficulties
with the behavioristic betting analysis of degrees of belief can now be
resolved by moving to an analysis of degrees of belief that is
functionalist in spirit. According to Lewis (1986a, 1994a), an agent's
degrees of belief are represented by the probability function belonging
to a utility function/probability function pair that best rationalizes
her behavioral dispositions, rationality being given a
decision-theoretic analysis.

There is a deep issue that underlies all of these accounts of subjective probability. They all presuppose the existence of necessary connections between desire-like states and belief-like states, rendered explicit in the connections between preferences and probabilities. In response, one might insist that such connections are at best contingent, and indeed can be imagined to be absent. Think of an idealized Zen Buddhist monk, devoid of any preferences, who dispassionately surveys the world before him, forming beliefs but no desires. It could be replied that such an agent is not so easily imagined after all — even if the monk does not value worldly goods, he will still prefer some things to others (e.g., truth to falsehood).

Once desires enter the picture, they may also have unwanted consequences. Again, how does one separate an agent's enjoyment or disdain for gambling from the value she places on the gamble itself? Ironically, a remark that Ramsey makes in his critique of the betting analysis seems apposite here: “The difficulty is like that of separating two different co-operating forces” (1980, 35). See Eriksson and Hájek (2007) for further criticism of preference-based accounts of credence.

The betting analysis makes subjective probabilities ascertainable to the extent that an agent's betting dispositions are ascertainable. The derivation of them from preferences makes them ascertainable to the extent that his or her preferences are known. However, it is unclear that an agent's full set of preferences is ascertainable even to himself or herself. Here a lot of weight may need to be placed on the ‘in principle’ qualification in the ascertainability criterion. The expected utility representation makes it virtually analytic that an agent should be guided by probabilities — after all, the probabilities are her own, and they are fed into the formula for expected utility in order to determine what it is rational for her to do.

#### 3.3.4 Orthodox Bayesianism, and further constraints on rational credences

But do they function as a *good* guide? Here it is useful to
distinguish different versions of subjectivism. *Orthodox
Bayesians* in the style of de Finetti recognize no rational
constraints on subjective probabilities beyond:

- conformity to the probability calculus, and
- a rule for updating probabilities in the face of new evidence,
known as
*conditioning*. An agent with probability function*P*_{1}, who becomes certain of a piece of evidence*E*(and nothing stronger), should shift to a new probability function*P*_{2}related to*P*_{1}by:(Conditioning)

*P*_{2}(*X*) =*P*_{1}(*X*|*E*) (provided*P*_{1}(*E*) > 0).

This is a permissive epistemology, licensing doxastic states that we
would normally call crazy. Thus, you could assign probability 1 to this
sentence ruling the universe, while upholding such extreme subjectivism
— provided, of course, that you assign probability 0 to this sentence
*not* ruling the universe, and that your other probability
assignments all conform to the probability calculus.

Some otherwise extreme subjectivists impose the further rationality
requirement of *regularity:* only *a priori* falsehoods
get assigned probability 0. This is sometimes also called ‘strict
coherence’, and it is advocated by authors such as Kemeny (1955),
Jeffreys (1961), Edwards et al. (1963), Shimony (1970), and Stalnaker
(1970). It is meant to capture a form of open-mindedness and
responsiveness to evidence. But then, perhaps unintuitively, someone
who assigns probability 0.999 to this sentence ruling the universe can
be judged rational, while someone who assigns it probability 0 is
judged irrational. Note also that the requirement of regularity seems
to afford a new argument for the non-existence of God as traditionally
conceived: an omniscient agent, who gives probability 1 to all truths,
would be convicted of irrationality. Thus regularity seems to require
ignorance, or false modesty. See, e.g., Levi (1978) for further
opposition to regularity.

Probabilistic coherence plays much the same role for degrees of
belief that *consistency* plays for ordinary, all-or-nothing
beliefs. What an extreme subjectivist, even one who demands regularity,
lacks is an analogue of *truth*, some yardstick for
distinguishing the ‘veridical’ probability assignments from
the rest (such as the 0.999 one above), some way in which probability
assignments are answerable to the world. It seems, then, that the
subjectivist needs something more.

And various subjectivists offer more. Having isolated the
“logic” of partial belief as conformity to the probability
calculus, Ramsey goes on to discuss what makes a degree of belief in a
proposition *reasonable*. After canvassing several possible
answers, he settles upon one that focuses on *habits* of opinion
formation — “e.g. the habit of proceeding from the opinion that
a toadstool is yellow to the opinion that it is unwholesome”
(50). He then asks, for a person with this habit, what probability it
would be best for him to have that a given yellow toadstool is
unwholesome, and he answers that “it will in general be equal to
the proportion of yellow toadstools which are in fact
unwholesome” (50). This resonates with more recent proposals
(e.g., van Fraassen 1984, Shimony 1988) for evaluating degrees of
belief according to how closely they match the corresponding relative
frequencies — in the jargon, how well *calibrated* they are.
Since relative frequencies obey the axioms of probability (up to finite
additivity), it is thought that rational credences, which strive to
track them, should do so also.^{[7]}

However, rational credences may strive to track various things. For example, we are often guided by the opinions of experts. We consult our doctors on medical matters, our weather forecasters on meteorological matters, and so on. Gaifman (1988) coins the terms “expert assignment” and “expert probability” for a probability assignment that a given agent strives to track: “The mere knowledge of the [expert] assignment will make the agent adopt it as his subjective probability” (193). This idea may be codified as follows:

(Expert)P(A|pr(A) =x) =x,

for allxwhere this is defined.

where ‘*P*’ is the agent's subjective probability
function, and ‘*pr*(*A*)’ is the assignment
that the agent regards as expert. For example, if you regard the local
weather forecaster as an expert on your local weather, and she assigns
probability 0.1 to it raining tomorrow, then you may well follow
suit:

P(rain|pr(rain) = 0.1) = 0.1

More generally, we might speak of an entire probability function as
being such a guide for an agent over a specified set of propositions.
van Fraassen (1989, 198) gives us this definition: “If *P*
is my personal probability function, then *q* is an *expert
function for me concerning* family *F* of propositions
exactly if *P*(*A* |
*q*(*A*) = *x*)
= *x* for all propositions *A* in family
*F*.”

Let us define a *universal expert function* *for* a
given rational agent as one that would guide *all* of that
agent's probability assignments in this way: an expert function for the
agent concerning all propositions. van Fraassen (1984, 1995a),
following Goldstein (1983), argues that an agent's *future
probability functions* are universal expert functions for that
agent. He enshrines this idea in his *Reflection Principle*, where
*P** _{t}* is the agent's probability function at
time

*t*, and

*P*

_{t+Δ}is her function at a later time

*t*+Δ:

P_{t}(A|P_{t+Δ}(A) =x) =x,

for allAand for allxwhere this is defined.

The principle encapsulates a certain demand for ‘diachronic coherence’ imposed by rationality. van Fraassen defends it with a ‘diachronic’ Dutch Book argument (one that considers bets placed at different times), and by analogizing violations of it to the sort of pragmatic inconsistency that one finds in Moore's paradox.

We may go still further. There may be universal expert functions for
large classes of rational agents, and perhaps all of
them. The *Principle of Direct Probability* regards the
*relative frequency* function as a universal expert function for all rational agents; we
have already seen the importance that proponents of calibration place
on it. Let *A* be an event-type, and let
*relfreq*(*A*) be the relative frequency of *A*
(in some suitable reference class). Then for any rational agent with
probability function *P*, we have (cf. Hacking 1965):

P(A|relfreq(A) =x) =x,

for allAand for allxwhere this is defined.

Lewis (1980) posits a similar expert role for the *objective chance
function, ch*, for all rational *initial* credences in
his *Principal Principle* (here
simplified^{[8]}):

C(A|ch(A) =x) =x,

for allAand for allxwhere this is defined.

‘*C*’ denotes the ‘ur’ credence
function of an agent at the beginning of enquiry. This is an
idealization that ensures that the agent does not have any
“inadmissible” evidence that bears on *A* without bearing on
the chance of *A*. For example, a rational agent who somehow
knows that a particular coin toss lands heads is surely *not*
required to assign

C(heads |ch(heads) = 1/2) = 1/2.

Rather, this conditional probability should be 1, since she has information relevant to the outcome ‘heads’ that trumps its chance. The other expert principles surely need to be suitably qualified - otherwise they face analogous counterexamples. Yet strangely, the Principal Principle is the only expert principle about which concerns about inadmissible evidence have been raised in the literature.

I will say more about relative frequencies and chance shortly.

The ultimate expert, presumably, is the *truth* function —
the function that assigns 1 to all the true propositions and 0 to all
the false ones. Knowledge of its values should surely trump knowledge
of the values assigned by human experts (including one's future
selves), frequencies, or chances. Note that for any putative expert
*q*,

P(A|q(A) =x∩A) = 1,

for allAand for allxwhere this is defined.

— the truth of *A* overrides anything the expert might say.
So all of the proposed expert probabilities above should really be
regarded as defeasible. Joyce (1998) portrays the rational agent as
estimating truth values, seeking to minimize a measure of distance
between them and her probability assignments. He argues that for any
measure of distance that satisfies certain intuitive properties, any
agent who violates the probability axioms could serve this epistemic
goal better by obeying them instead, however the world turns out.

There are some unifying themes in these putative constraints on subjective probability. An agent's degrees of belief determine her estimates of certain quantities: the values of bets, or the desirabilities of gambles more generally, or the probability assignments of various ‘experts’ — humans, relative frequencies, objective chances, or truth values. The laws of probability then are claimed to be constraints on these estimates: putative necessary conditions for minimizing her ‘losses’ in a broad sense, be they monetary, or measured by distances from the assignments of these experts.

### 3.4 Frequency Interpretations

Gamblers, actuaries and scientists have long understood that
relative frequencies bear an intimate relationship to probabilities.
Frequency interpretations posit the most intimate relationship of all:
identity. Thus, we might identify the probability of
‘heads’ on a certain coin with the frequency of heads in a
suitable sequence of tosses of the coin, divided by the total number of
tosses. A simple version of frequentism, which we will call *finite
frequentism*, attaches probabilities to events or attributes in a
finite reference class in such a straightforward manner:

the probability of an attribute A in a finite reference class B is the relative frequency of actual occurrences of A within B.

Thus, finite frequentism bears certain structural similarities to the
classical interpretation, insofar as it gives equal weight to each
member of a set of events, simply counting how many of them are
‘favorable’ as a proportion of the total. The crucial
difference, however, is that where the classical interpretation
counted all the *possible* outcomes of a given experiment,
finite frequentism counts *actual* outcomes. It is thus
congenial to those with empiricist scruples. It was developed by Venn
(1876), who in his discussion of the proportion of births of males and
females, concludes: “probability *is* nothing but that
proportion” (p. 84, his emphasis). Finite frequentism remains
the dominant view of probability in statistics, and in the sciences
more generally.

Finite frequentism gives an operational definition of probability, and
its problems begin there. For example, just as we want to allow that
our thermometers could be ill-calibrated, and could thus give
misleading measurements of temperature, so we want to allow that our
‘measurements’ of probabilities via frequencies could be
misleading, as when a fair coin lands heads 9 out of 10 times. More
than that, it seems to be built into the very notion of probability
that such misleading results can arise. Indeed, in many cases,
misleading results are guaranteed. Starting with a degenerate case:
according to the finite frequentist, a coin that is never tossed, and
that thus yields no actual outcomes whatsoever, lacks a probability
for heads altogether; yet a coin that is never measured does not
thereby lack a diameter. Perhaps even more troubling, a coin that is
tossed exactly once yields a relative frequency of heads of either 0
or 1, whatever its bias. Or we can imagine a unique radiocative atom
whose probabilities of decaying at various times obey a continuous law
(e.g. exponential); yet according to finite frequentism, with
probability 1 it decays at the exact time that it *actually*
does, for its relative frequency of doing so is 1/1. Famous enough to
merit a name of its own, these are instances of the so-called
‘problem of the single case’. In fact, many events are
most naturally regarded as not merely unrepeated, but in a strong
sense *unrepeatable* — the 2000 presidential election,
the final game of the 2001 NBA play-offs, the Civil War, Kennedy's
assassination, certain events in the very early history of the
universe. Nonetheless, it seems natural to think of non-extreme
probabilities attaching to some, and perhaps all, of them. Worse
still, some cosmologists regard it as a genuinely chancy matter
whether our universe is open or closed (apparently certain quantum
fluctuations could, in principle, tip it one way or the other), yet
whatever it is, it is ‘single-case’ in the strongest
possible sense.

The problem of the single case is particularly striking, but we
really have a sequence of related problems: ‘the problem of the
double case’, ‘the problem of the triple case’
… Every coin that is tossed exactly twice can yield only the
relative frequencies 0, 1/2 and 1, whatever its bias… A finite
reference class of size *n*, however large *n* is, can
only produce relative frequencies at a certain level of
‘grain’, namely 1/*n*. Among other things, this
rules out irrational probabilities; yet our best physical theories say
otherwise. Furthermore, there is a sense in which any of these problems
can be transformed into the problem of the single case. Suppose that we
toss a coin a thousand times. We can regard this as a *single*
trial of a thousand-tosses-of-the-coin experiment. Yet we do not want
to be committed to saying that *that* experiment yields its
actual result with probability 1.

The problem of the single case is that the finite frequentist fails
to see intermediate probabilities in various places where others do.
There is also the converse problem: the frequentist sees intermediate
probabilities in various places where others do not. Our world has
myriad different entities, with myriad different attributes. We can
group them into still more sets of objects, and then ask with which
relative frequencies various attributes occur in these sets. Many such
relative frequencies will be intermediate; the finite frequentist
automatically identifies them with intermediate probabilities. But it
would seem that whether or not they are genuine *probabilities*,
as opposed to mere tallies, depends on the case at hand. Bare ratios of
attributes among sets of disparate objects may lack the sort of modal
force that one might expect from probabilities. I belong to the
reference class consisting of myself, the Eiffel Tower, the
southernmost sandcastle on Santa Monica Beach, and Mt Everest. Two of
these four objects are less than 7 ft. tall, a relative frequency of
1/2; moreover, we could easily extend this class, preserving this
relative frequency (or, equally easily, not). Yet it would be odd to
say that my *probability* of being less than 7 ft. tall,
relative to this reference class, is 1/2, even though it is perfectly
acceptable (if uninteresting) to say that 1/2 of the objects in the
reference class are less than 7 ft. tall.

Some frequentists (notably Venn 1876, Reichenbach 1949, and von
Mises 1957 among others), partly in response to some of the problems
above, have gone on to consider *infinite* reference classes,
identifying probabilities with *limiting* relative frequencies
of events or attributes therein. Thus, we require an infinite sequence
of trials in order to define such probabilities. But what if the actual
world does not provide an infinite sequence of trials of a given
experiment? Indeed, that appears to be the norm, and perhaps even the
rule. In that case, we are to identify probability with a
*hypothetical* or *counterfactual* limiting relative
frequency. We are to imagine hypothetical infinite extensions of an
actual sequence of trials; probabilities are then what the limiting
relative frequencies *would be* if the sequence were so
extended. We might thus call this interpretation *hypothetical
frequentism*.

Note that at this point we have left empiricism behind. A modal element has been injected into frequentism with this invocation of a counterfactual; moreover, the counterfactual may involve a radical departure from the way things actually are, one that may even require the breaking of laws of nature. (Think what it would take for the coin in my pocket, which has only been tossed once, to be tossed infinitely many times — never wearing out, and never running short of people willing to toss it!) One may wonder, moreover, whether there is always — or ever — a fact of the matter of what such counterfactual relative frequencies are.

Limiting relative frequencies, we have seen, must be relativized to
a sequence of trials. Herein lies another difficulty. Consider an
infinite sequence of the results of tossing a coin, as it might be H,
T, H, H, H, T, H, T, T, … Suppose for definiteness that the
corresponding relative frequency sequence for heads, which begins 1/1,
1/2, 2/3, 3/4, 4/5, 4/6, 5/7, 5/8, 5/9, …, converges to 1/2. By
suitably reordering these results, we can make the sequence converge to
any value in [0, 1] that we like. (If this is not obvious, consider how
the relative frequency of even numbers among positive integers, which
intuitively ‘should’ converge to 1/2, can instead be made
to converge to 1/4 by reordering the integers with the even numbers in
every fourth place, as follows: 1, 3, 5, 2, 7, 9, 11, 4, 13, 15, 17, 6,
…) To be sure, there may be something natural about the ordering
of the tosses as given — for example, it may be their
*temporal* ordering. But there may be more than one natural
ordering. Imagine the tosses taking place on a train that shunts
backwards and forwards on tracks that are oriented west-east. Then the
*spatial* ordering of the results from west to east could look
very different. Why should one ordering be privileged over others?

A well-known objection to any version of frequentism is that
*relative* frequencies must be *relativised* to a
reference class. Consider a probability concerning myself that I care
about — say, my probability of living to age 80. I belong to the class
of males, the class of non-smokers, the class of philosophy professors
who have two vowels in their surname, … Presumably the relative
frequency of those who live to age 80 varies across (most of) these
reference classes. What, then, is my probability of living to age 80?
It seems that there is no single frequentist answer. Instead, there is
my probability-qua-male, my probability-qua-non-smoker, my
probability-qua-male-non-smoker, and so on. This is an example of the
so-called *reference class problem* for frequentism (although it
can be argued that analogues of the problem arise for the other
interpretations as well^{[9]}). And as we have seen in the previous
paragraph, the problem is only compounded for limiting relative
frequencies: probabilities must be relativized not merely to a
reference class, but to a sequence within the reference class. We might
call this the *reference sequence problem.*

The beginnings of a solution to this problem would be to restrict
our attention to sequences of a certain kind, those with certain
desirable properties. For example, there are sequences for which the
limiting relative frequency of a given attribute does not exist;
Reichenbach thus excludes such sequences. Von Mises (1957) gives us a
more thoroughgoing restriction to what he calls *collectives* —
hypothetical infinite sequences of attributes (possible outcomes) of
specified experiments that meet certain requirements. Call a
*place-selection* an effectively specifiable method of selecting
indices of members of the sequence, such that the selection or not of
the index *i* depends at most on the first *i* − 1
attributes. The axioms are:

Axiom of Convergence:the limiting relative frequency of any attribute exists.

Axiom of Randomness:the limiting relative frequency of each attribute in a collective ω is the same in any infinite subsequence of ω which is determined by a place selection.

The probability of an attribute *A*, relative to a collective
ω, is then defined as the limiting relative frequency of
*A* in ω. Note that a constant sequence such as H, H, H,
…, in which the limiting relative frequency is the same in
*any* infinite subsequence, trivially satisfies the axiom of
randomness. This puts some strain on the terminology — offhand, such
sequences appear to be as *non*-random as they come — although
to be sure it is desirable that probabilities be assigned even in such
sequences. Be that as it may, there is a parallel between the role of
the axiom of randomness in von Mises' theory and the principle of
maximum entropy in the classical theory: both attempt to capture a
certain notion of disorder.

Collectives are abstract mathematical objects that are not empirically
instantiated, but that are nonetheless posited by von Mises to explain
the stabilities of relative frequencies in the behavior of actual
sequences of outcomes of a repeatable random experiment. Church (1940)
renders precise the notion of a place selection as a recursive
function. Nevertheless, the reference sequence problem remains:
probabilities must always be relativized to a collective, and for a
given attribute such as ‘heads’ there are infinitely many.
Von Mises embraces this consequence, insisting that the notion of
probability only makes sense relative to a collective. In particular,
he regards single case probabilities as nonsense: “We can say
nothing about the probability of death of an individual even if we
know his condition of life and health in detail. The phrase
‘probability of death’, when it refers to a single person,
has no meaning at all for us” (11). Some critics believe that
rather than solving the problem of the single case, this merely
ignores it. And note that von Mises drastically understates the
commitments of his theory: by his lights, the phrase
‘probability of death’ also has no meaning at all when it
refers to a million people, or a billion, or any finite number —
after all, collectives are *infinite*. More generally, it seems
that von Mises' theory has the unwelcome consequence that probability
statements never have meaning in the real world, for apparently all
sequences of attributes are finite. He introduced the notion of a
collective because he believed that the regularities in the behavior
of certain actual sequences of outcomes are best explained by the
hypothesis that those sequences are initial segments of
collectives. But this is curious: we *know* for any actual
sequence of outcomes that they are *not* initial segments of
collectives, since we know that they are not initial segments of
infinite sequences.

Let us see how the frequentist interpretations fare according to our
criteria of adequacy. Finite relative frequencies of course satisfy
finite additivity. In a finite reference class, only finitely many
events can occur, so only finitely many events can have positive
relative frequency. In that case, countable additivity is satisfied
somewhat trivially: all but finitely many terms in the infinite sum
will be 0. Limiting relative frequencies violate countable additivity
(de Finetti 1972, §5.22). Indeed, the domain of definition of
limiting relative frequency is not even a field, let alone a sigma
field (de Finetti 1972, §5.8). So such relative frequencies do not
provide an admissible interpretation of Kolmogorov's axioms. Finite
frequentism has no trouble meeting the ascertainability criterion, as
finite relative frequencies are in principle easily determined. The
same cannot be said of limiting relative frequencies. On the contrary,
any finite sequence of trials (which, after all, is all we ever see)
puts literally no constraint on the limit of an infinite sequence;
still less does an *actual* finite sequence put any constraint
on the limit of an infinite *hypothetical* sequence, however
fast and loose we play with the notion of ‘in principle’ in
the ascertainability criterion.

It might seem that the frequentist interpretations resoundingly meet
the applicability to frequencies criterion. Finite frequentism meets it
all too well, while hypothetical frequentism meets it in the wrong
way. If anything, finite frequentism makes the connection between
probabilities and frequencies *too* tight, as we have already
observed. A fair coin that is tossed a million times is very
*unlikely* to land heads *exactly* half the time; one
that is tossed a million and one times is even less likely to do so!
Facts about finite relative frequencies should serve as evidence, but
not *conclusive* evidence, for the relevant probability
assignments. Hypothetical frequentism fails to connect probabilities
with finite frequencies. It connects them with limiting relative
frequencies, of course, but again too tightly: for even in infinite
sequences, the two can come apart. (A fair coin could land heads
forever, even if it is highly unlikely to do so.) To be sure, science
has much interest in finite frequencies, and indeed working with them
is much of the business of statistics. Whether it has any interest in
highly idealized, hypothetical extensions of actual sequences, and
relative frequencies therein, is another matter. The applicability to
rational opinion goes much the same way: it is clear that such opinion
is guided by finite frequency information, unclear that it is guided
by information about limits of hypothetical frequencies. For much more
extensive critiques of finite frequentism and hypothetical
frequentism, see Hájek 1997 and Hájek 2009
respectively.

### 3.5 Propensity Interpretations

Like the frequency interpretations, *propensity*
interpretations locate probability ‘in the world’ rather
than in our heads or in logical abstractions. Probability is thought of
as a physical propensity, or disposition, or tendency of a given type
of physical situation to yield an outcome of a certain kind, or to
yield a long run relative frequency of such an outcome.

While Popper is often credited as being the pioneer of propensity
interpretations, we already find the key idea in the writings of
Peirce (1910, 79-80): “I am, then, to define the meaning of the
statement that the *probability*, that if a die be thrown from
a dice box it will turn up a number divisible by three, is one-third.
The statement means that the die has a certain ‘would-be’;
and to say that the die has a ‘would-be’ is to say that it
has a property, quite analogous to any *habit* that a man might
have.” A man's habit is a paradigmatic example of a disposition;
according to Peirce the die's probability of landing 3 or 6 is an
analogous disposition. We might question whether the analogy is
apt—the modal flavour of a habit is more one
of *necessity* than possibility. The die's landing 3 or 6 is
more like a man's *ability* to do something at which he
succeeds a third of the time. But then one might also question
Peirce's talk of probability as a ‘would-be’. Rather, it
seems more like a graded ‘*might-be*’.

Be that as it may, Peirce continues: “Now in order that the full effect of the die's ‘would-be’ may find expression, it is necessary that the die should undergo an endless series of throws from the dice box”, and he imagines the relative frequency of the event-type in question oscilating from one side of 1/3 to another. This again anticipates Popper's view. But an important difference is that Peirce regards the propensity as a property of the die itself, whereas Popper attributes the propensity to the entire chance set-up of throwing the die.

Popper (1957) is motivated by the desire to make sense of single-case
probability attributions that one finds in quantum mechanics—for
example “the probability that this radium atom decays in 1600
years is 1/2”. He develops the theory further in (1959a). For
him, a probability *p* of an outcome of a certain type is a
propensity of a repeatable experiment to produce outcomes of that type
with limiting relative frequency *p*. For instance, when we say
that a coin has probability 1/2 of landing heads when tossed, we mean
that we have a repeatable experimental set-up — the tossing
set-up — that has a propensity to produce a sequence of outcomes
in which the limiting relative frequency of heads is 1/2. With its
heavy reliance on limiting relative frequency, this position risks
collapsing into von Mises-style frequentism according to some
critics. Giere (1973), on the other hand, explicitly allows
single-case propensities, with no mention of frequencies: probability
is just a propensity of a repeatable experimental set-up to produce
sequences of outcomes. This, however, creates the opposite problem to
Popper's: how, then, do we get the desired connection between
probabilities and frequencies?

It is thus useful to follow Gillies (2000a) in distinguishing
*long-run* propensity theories and *single-case*
propensity theories:

A long-run propensity theory is one in which propensities are associated with repeatable conditions, and are regarded as propensities to produce in a long series of repetitions of these conditions frequencies which are approximately equal to the probabilities. A single-case propensity theory is one in which propensities are regarded as propensities to produce a particular result on a specific occasion (822).

Hacking (1965) and Gillies offer long-run (though not infinitely
long-run) propensity theories; Fetzer (1982, 1983) and Miller (1994)
offer single-case propensity theories. Note that
‘propensities’ are categorically different things depending
on which sort of theory we are considering. According to the long-run
theories, propensities are tendencies to produce relative frequencies
with particular values, but the propensities are not measured by the probability
values themselves; according to the single-case theories, the
propensities *are* measured by the probability values. According to Popper,
for example, a fair die has a propensity — an *extremely
strong* tendency — to land ‘3’ with long-run relative
frequency 1/6. The small value of 1/6 does *not* measure this
tendency. According to Giere, on the other hand, the die has a
*weak* tendency to land ‘3’. The value of 1/6
*does* measure this tendency.

It seems that those theories that tie propensities to frequencies do
not provide an admissible interpretation of the (full) probability
calculus, for the same reasons that relative frequencies do not. It is
*prima facie* unclear whether single-case propensity theories
obey the probability calculus or not. To be sure, one can
*stipulate* that they do so, perhaps using that stipulation as
part of the implicit definition of propensities. Still, it remains to
be shown that there really are such things — stipulating what a witch
is does not suffice to show that witches exist. Indeed, to claim, as
Popper does, that an experimental arrangement has a tendency to produce
a given limiting relative frequency of a particular outcome,
presupposes a kind of stability or uniformity in the workings of that
arrangement (for the limit would not exist in a suitably
*unstable* arrangement). But this is the sort of
‘uniformity of nature’ presupposition that Hume argued
could not be known either *a priori*, or empirically. Now,
appeals can be made to limit theorems — so called ‘laws of large
numbers’ — whose content is roughly that under suitable
conditions, such limiting relative frequencies almost certainly exist,
and equal the single case propensities. Still, these theorems make
assumptions (e.g., that the trials are independent and identically
distributed) whose truth again cannot be known, and must merely be
postulated.

Part of the problem here, say critics, is that we do not know enough
about what propensities are to adjudicate these issues. There is
*some* property of this coin tossing arrangement such that this
coin would land heads with a certain long-run frequency, say. But as
Hitchcock (2002) points out, “calling this property a
‘propensity’ of a certain strength does little to indicate
just what this property is.” Said another way, propensity
accounts are accused of giving empty accounts of probability, à
la Molière's ‘dormative virtue’ (Sober 2000, 64).
Similarly, Gillies objects to single-case propensities on the grounds
that statements about them are untestable, and that they are
“metaphysical rather than scientific” (825). Some might
level the same charge even against long-run propensities, which are
supposedly *distinct* *from* the testable relative
frequencies.

This suggests that the propensity account has difficulty meeting the applicability to science criterion. Some propensity theorists (e.g., Giere) liken propensities to physical magnitudes such as electrical charge that are the province of science. But Hitchcock observes that the analogy is misleading. We can only determine the general properties of charge — that it comes in two varieties, that like charges repel, and so on — by empirical investigation. What investigation, however, could tell us whether or not propensities are non-negative, normalized and additive?

More promising, perhaps, is the idea that propensities are to play
certain theoretical roles, and that these place constraints on the way
they must behave, and hence what they could be (in the style of the
Ramsey/Lewis/‘Canberra plan’ approach to theoretical terms
— see Lewis 1970 or Jackson 2000). The trouble here is that these
roles may pull in opposite directions, *overconstraining* the
problem. The first role, according to some, constrains them to obey the
probability calculus (with finite additivity); the second role,
according to others, constrains them to violate it.

On the one hand, propensities are said to constrain the degrees of
belief, or *credences*, of a rational agent. I will have more
to say in the next section about what credences are and what makes them
rational, but for now recall the ‘applicability to rational
belief’ criterion: an interpretation should clarify the role that
probabilities play in constraining the credences of rational agents.
One such putative role for propensities is codified by Lewis’s
‘Principal Principle’. (See section 3.3.) The
Principal Principle underpins an argument (Lewis 1980) that whatever
they are, propensities must obey the usual probability calculus (with
finite additivity). After all, it is argued, rational credences, which
are guided by them, do.

On the other hand, Humphreys (1985) gives an influential argument
that propensities do *not* obey Kolmogorov's probability
calculus. The idea is that the probability calculus implies *Bayes'
theorem*, which allows us to reverse a conditional
probability:

P(A|B)=

Yet propensities seem to be measures of ‘causal
tendencies’, and much as the causal relation is asymmetric, so
these propensities supposedly do not reverse. Suppose that we have a
test for an illness that occasionally gives false positives and false
negatives. A given sick patient may have a (non-trivial) propensity to
give a positive test result, but it apparently makes no sense to say
that a given positive test result has a (non-trivial) propensity to
have come from a sick patient. Thus, we have an argument that whatever
they are, propensities must *not* obey the usual probability
calculus. ‘Humphreys' paradox’, as it is known, is really
an argument against any formal account of propensities that has as a
theorem:

(*) if the probability ofB, givenAexists, then the probability ofA, givenBexists,

however one understands these conditional probabilities. The
argument has prompted Fetzer and Nute (in Fetzer 1981) to offer a
“probabilistic causal calculus” that looks quite different
from Kolmogorov's calculus.^{[10]} But one could respond more conservatively.
For example, Popper's axiomatization of primitive conditional
probabilities does not have (*) as a theorem, and thus propensities may
conform to it despite Humphreys' argument.^{[11]} At least to that extent
they may still deserve to be called ‘probabilities'.

Or perhaps all this shows that the notion of ‘propensity’ bifurcates: on the one hand, there are propensities that bear an intimate connection to relative frequencies and rational credences, and that obey the usual probability calculus (with finite additivity); on the other hand, there are causal propensities that behave rather differently. In that case, there would be still more interpretations of probability than have previously been recognized.

### 3.6 Best-System Interpretations

Traditionally, philosophers of probability have recognized five
leading interpretations of probability—classical, logical,
subjectivist, frequentist, and propensity. But recently,
so-called *best-system* interpretations of chance have become
increasingly popular and important. While they bear some similarities
to frequentist accounts, they avoid some of frequentism's major
failings; and while they are sometimes assimilated to propensity
accounts, they are really quite distinct. So they deserve separate
treatment.

The best-system approach was pioneered by Lewis (1994b). His analysis
of chance is based on his account of *laws of nature* (1973),
which in turn refines an account due to Ramsey (1928/1990). According
to Lewis, the laws of nature are the theorems of the *best
systematization* of the universe—the *true* theory
that best combines the theoretical virtues of *simplicity
and *strength. These virtues trade off. It is easy for a theory to
be simple but not strong, by saying very little; it is easy for a
theory to be strong but not simple, by conjoining lots of disparate
facts. The best theory balances simplicity and strength
optimally—in short, it is the most economical true theory.

So far, there is no mention of chances. Now, we allow probabilistic
theories to enter the competition. We are not yet in a position to
speak of such theories as being true. Instead, let us introduce
another theoretical virtue: *fit*. The more probable the actual
history of the universe is by the lights of the theory, the better it
fits that history. Now the theories compete according to how well they
combine simplicity, strength, and fit. The theorems of the winning
theory are the laws of nature. Some of these laws may be
probabilistic. The chances are the probabilities that are determined
by these probabilistic laws.

According to Lewis (1986b), intermediate chances are incompatible with
determinism. Loewer (2004) agrees that
intermediate *propensities* are incompatible with determinism,
understanding those to be essentially *dynamical*: “they
specify the degree to which one state has a tendency to cause
another” (15). But he argues that *chances* are best
understood along Lewisian best-system lines, and that there is no
reason to limit them to dynamical chances. In particular, best-system
chances may also attach to *initial conditions*: adding to the
dynamical laws a probability assignment, or *distribution*,
over initial conditions may provide a substantial gain in strength
with relatively little cost in simplicity. Science furnishes important
examples of deterministic theories with such initial-condition
probabilities. Adding the so-called micro-canonical distribution to
Newton's laws (and the assumption that the distant past had low
entropy) yields all of statistical mechanics; adding the so-called
quantum equilibrium distribution to Bohm's dynamical laws yields
standard quantum mechanics. Indeed, this contact with actual science
is one of the selling points of best-system analyses.

This approach solves, or at least eases, some of frequentism's problems. Progress can be made on the problem of the single case. The chances of a rare atom decaying in various time intervals may be determined by a more pervasive functional law, in which decay chances are given for a far wider range of atoms by plugging in a range of settings of some other magnitude (e.g., atomic number). And simplicity may militate in favour of this functional law being continuous, so even irrational-valued probabilities may be assigned. Moreover, bare ratios of attributes among sets of disparate objects will not qualify as chances if they are not pervasive enough, for then a theory that assigns them probabilities will lose too much simplicity without sufficient gain in strength.

However, some other problems for frequentism remain, and some new ones emerge, beginning with more basic problems for the Lewisian account of lawhood itself. Some of them are partly a matter of Lewis's specific formulation. Critics (e.g. van Fraassen 1989) question the rather nebulous notion of “balancing” simplicity and strength, which are themselves somewhat sketchy. But arguably some technical story (e.g. information-theoretic) could be offered to precisify them. Lewis himself worries that the exchange rate for such balancing may depend partly on our psychology, in which case there is the threat the laws themselves depend on our psychology, an unpalatable idealism about them. But he maintains that this threat is not serious as long as “nature is kind”, and one theory is so robustly the front-runner that it remains so under any reasonable standards for balancing. And again, perhaps technical tools can offer some objectivity here. (See section 4 for a gesture at such tools.)

More telling is the concern that simplicity is language-relative, and
indeed that any theory can be given the simplest specification
possible: simply abbreviate it as *T*! Lewis replies that a
theory's simplicity must be judged according to its specification in a
canonical language, in which all of the predicates correspond
to *natural* properties. Thus, ‘green’ may well be
eligible, but ‘grue’ surely is not. Our abbreviation,
then, has to be unpacked in terms of such a language, in which its
true complexity will be revealed. But this now involves a substantial
metaphysical commitment to a distinction between natural and unnatural
properties, one that various empiricists (e.g. van Fraassen 1989) find
objectionable.

Further problems arise with the refinement to handle probabilistic
laws. Again, some of them may be due to Lewis's particular
formulation. Elga (2004) observes that Lewis's notion of fit is
problematic in various infinite universes—think of an infinite
sequence of tosses of a coin. Offhand, it seems that the particular
infinite sequence that is actualized will be assigned
probability *zero* by any plausible candidate theory that
regards the probability of heads as intermediate and the trials as
independent. Elga argues, moreover, that there are technical
difficulties with addressing this problem with infinitesimal
probabilities. However, perhaps we merely need a different
understanding of ‘fit’—perhaps understood as
‘typicality’ (Elga), or perhaps one closer to that
employed by statisticians with ‘chi-squared’ tests of
goodness of fit (Schwarz forthcoming).

Hoefer (2007) modifies Lewis's best-system account in light of some of these problems. Hoefer understands “best” as “best for us”, covering regularities that are of interest to us, using the language both of science and of daily life, without any special privilege bestowed upon natural properties. Moreover, the “best system” is now one of chances directly, rather than of laws. Thus, there may be chances associated with the punctuality of trains, for example, without any presumption that there are any associated laws. Hoefer follows Elga in understanding ‘fit’ as ‘typicality’. Strength is a matter of the size of the overall domain of the best system's probability functions. Simplicity is to be understood in terms of elegant unification, and user-friendliness to beings like us. As a result, Hoefer embraces the agent-centric nature of chances in his sense, regarding as essential the credence-guiding role for them that is captured by the Principal Principle.

However, some other problems for Lewis's account may run deeper,
threatening best-system analyses more generally, and symptomatic of
the ghost of frequentism that still hovers behind such analyses. One
problem for frequentism that we saw strikes at the heart of any
attempt to reduce chances to properties of patterns of outcomes. Such
outcomes may be highly misleading regarding the true
chances, *because of* their probabilistic nature. This is most
vivid for events that are single-case by any reasonable
typing. Whether or our universe turns out to be open or closed,
plausibly that outcome is compatible with any underlying intermediate
chance. The point generalizes, however pervasive the probabilistic
pattern might be. Plausibly, a coin's landing 9 heads out of 10 tosses
is compatible with any underlying intermediate chance for heads; and
so on. The pattern of outcomes that is instantiated may be a poor
guide to the true chance. (See Hájek 2009 for further arguments
against frequentism that carry over to best-system accounts.)

Another way of putting the concern is that best-system accounts
mistake an idealized epistemology of chance for its metaphysics
(though see Lewis' insistence that this is not the case). Such
accounts single out three theoretical virtues—and one may wonder
why *just* those three—and reifies the probabilities of a
theory that displays the virtues to the highest degree. But a
probabilistic world may be recalcitrant to even the best theorizing:
nature may be unkind.

## 4. Conclusion: Future Prospects?

It should be clear from the foregoing that there is still much work to be done regarding the interpretations of probability. Each interpretation that we have canvassed seems to capture some crucial insight into a concept of it, yet falls short of doing complete justice to this concept. Perhaps the full story about probability is something of a patchwork, with partially overlapping pieces. In that sense, the above interpretations might be regarded as complementary, although to be sure each may need some further refinement. My bet, for what it is worth, is that we will retain at least three distinct notions of probability: one quasi-logical, one objective (chance), and one subjective.

There are signs of the rehabilitation of classical and logical probability, and in particular the principle of indifference and the principle of maximum entropy, by authors such as Paris and Vencovská (1997), Maher (2000, 2001), and Bartha and Johns (2001). Relevant here may also be advances in information theory and complexity theory (see Li and Vitanyi 1997). These theories have already proved to be fruitful in the study of randomness (Kolmogorov 1965, Martin-Löf 1966), which obviously is intimately related to the notion of probability.

Refinements of our understanding of randomness, in turn, should have a bearing on the frequency interpretations (recall von Mises' appeal to randomness in his definition of ‘collective’), and on propensity accounts (especially those that make explicit ties to frequencies). Given the apparent connection between propensities and causation adumbrated in Section 3.5, powerful causal modeling techniques by authors such as Spirtes, Glymour and Scheines (1993) and Pearl (2000), and recent work on causation more generally (e.g., Hall 2003, Woodward 2003) should also prove fruitful here.

Johns (2002) offers a causal theory of chance; roughly, the chance of
an event is the idealised epistemic probability of the event, given a
maximal specification of its (possible) causes. Regarding best-system
interpretations of chance, I noted that it is somewhat unclear exactly
what ‘simplicity’ and ‘strength’ consist in,
and exactly how they are to be balanced. Perhaps insights from
statistics and computer science may be helpful here: approaches to
statistical model selection, and in particular the
‘curve-fitting’ problem, that attempt to characterize
simplicity, and its trade-off with strength — e.g., the Akaike
Information Criterion (see Forster and Sober 1994), the Bayesian
Information Criterion (see Kieseppä 2001), Minimum Description
Length theory (see Rissanen 1999) and Minimum Message Length theory
(see Wallace and Dowe 1999). Another growth area is the study
of *non-fundamental* objective probabilities, as one finds in
statistical mechanics and evolutionary theory. Are they genuine
chances? Do they show that chance is compatible with determinism? See
Lyon (2011) for a discussion of some of these issues. Relatedly, an
important approach to objective probability that is gaining in
popularity involves the so-called *method of arbitrary
functions*. Originating with Poincaré (1896), it is a
mathematical technique for determining probability functions for
certain systems with chaotic dynamical laws mapping input conditions
to outcomes. Roughly speaking, the probabilities for the outcomes are
relatively insensitive to the probabilities over the various initial
conditions — think of how the probabilities of outcomes of spins
of a roulette wheel apparently do not depend on who is spinning the
wheel, sometimes vigorously, sometimes feebly. See Strevens (2003) for
a detailed treatment of this approach.

The subjectivist theory of probability is also thriving. Developments
in the last decade or so include Schervish, Seidenfeld and Kadane's
(2003) research on degrees of incoherence (measuring the extent of
departures from obedience to the probability calculus). I foresee
related attempts to ‘humanize’ Bayesianism—for
example, the further study of imprecise probability and imprecise
decision theory, in which credences need not be precise numbers. (See
http://www.sipta.org/ for up-to-date research in this area.) And a
recently burgeoning area of research has concerned
the *contents* of subjective probability assignments, the
objects to which such assignments attach— whether they should be
to propositions (sets of possible worlds), or to more
fine-grained *self-locating* propositions (sets of centered
worlds — see Lewis 1979), or to something else. Thus, an agent
may not assign credences simply to propositions concerning the way the
world is, but to more specific propositions concerning who she is,
where she is, or what time it is. This in turn has ramifications for
updating rules, in particular calling into question the
appropriateness of conditionalization. The so-called Sleeping Beauty
problem (Elga 2000) has generated much discussion in this
regard. These promise to be fertile areas of future research. We may
expect that further criteria of adequacy for subjective probabilities
will be developed — perhaps refinements of ‘scoring
rules’ (Winkler 1996), and more generally, candidates for
playing a role for subjective probability analogous to the role that
truth plays for belief.

Here we come full circle. For belief is apparently answerable both
to logic and to objective facts. A refined account of
degrees-of-belief may be answerable both to a refined quasi-logical
notion and a refined notion of chance. Indeed, various cottage
industries are springing up involving the interrelations among the
different concepts of probability. A notable recent trend concerns the
putative connections between objective chance and subjective
probability, along the lines of the Principal Principle. Can the
principle be justified? Does *it* need refining? How should we
understand “inadmissible” evidence? See Hall (1994, 2004)
and Schwarz (forthcoming) for further discussion.

Well may we say that probability is a guide to life; but the task of understanding exactly how and why it is has still to be completed, and will surely be a guide to future theorizing about it.

### Suggested Further Reading

Kyburg (1970) contains a vast bibliography of the literature on
probability and induction pre-1970. Also useful for references before
1967 is the bibliography for “Probability” in the Macmillan
*Encyclopedia of Philosophy*. Earman (1992) and Howson and
Urbach (1993) have more recent bibliographies, and give detailed
presentations of the Bayesian program. Skyrms (2000) is an excellent
introduction to the philosophy of probability. von Plato (1994) is
more technically demanding and more historically oriented, with
another extensive bibliography that has references to many landmarks
in the development of probability theory in the last century. Fine
(1973) is still a highly sophisticated survey of and contribution to
various foundational issues in probability, with an emphasis on
interpretations. More recent philosophical studies of the leading
interpretations include Gillies (2000b), Galavotti (2005), and Mellor
(2005). Eagle (2010) is a valuable anthology of many significant
papers in the philosophy of probability. Billingsley (1995) and Feller
(1968) are classic textbooks on the mathematical theory of
probability.