An Objective Justification of Bayesianism I: Measuring Inaccuracy


Philosophy of Science, 77 (April 2010) pp. 201–235. 0031-8248/2010/7702-0002$10.00
Copyright 2010 by the Philosophy of Science Association. All rights reserved.

201

An Objective Justification of
Bayesianism I: Measuring Inaccuracy*

Hannes Leitgeb and Richard Pettigrew†‡

In this article and its sequel, we derive Bayesianism from the following norm: Accu-
racy—an agent ought to minimize the inaccuracy of her partial beliefs. In this article,
we make this norm mathematically precise. We describe epistemic dilemmas an agent
might face if she attempts to follow Accuracy and show that the only measures of
inaccuracy that do not create these dilemmas are the quadratic inaccuracy measures.
In the sequel, we derive Bayesianism from Accuracy and show that Jeffrey Condi-
tionalization violates Accuracy unless Rigidity is assumed. We describe the alternative
updating rule that Accuracy mandates in the absence of Rigidity.

1. Introduction. One of the fundamental problems of epistemology is to
say when the evidence in an agent’s possession justifies the beliefs she
holds and, when it does, how it does this and to what extent. In this
article and its sequel (Leitgeb and Pettigrew 2010), we defend the Bayesian
solution to this problem for those cases in which the set of possible worlds
about which the agent holds an opinion is finite. If W is such a set of
possible worlds, let be the power set of W, and let be theP(W ) Bel(W )
set of functions . We regard each function in as a�b : P(W ) r � Bel(W )0
(potential) belief function on the power set of W. Indeed, one of the
distinctive presuppositions of Bayesianism is that, if W is the set of possible
worlds about which an agent holds an opinion, then that agent’s epistemic

*Received July 2009; revised September 2009.

†To contact the authors, please write to: Department of Philosophy, University of Bristol,
9 Woodland Road, Bristol BS8 1BX, United Kingdom; e-mail: hannes.leitgeb@bris.ac.uk;
richard.pettigrew@bris.ac.uk.

‡We would like to thank F. Arntzenius, F. Dietrich, K. Easwaran, B. Fitelson (and his
Berkeley reading group), A. Hájek, L. Horsten, F. Huber, J. Joyce, T. Kuipers, W. Myr-
vold, S. Okasha, G. Schurz, T. Seidenfeld, R. Williams, J. Williamson, and B. van Fraassen
for their comments on earlier versions of this article. Hannes Leitgeb would like to thank
the Leverhulme Trust and the Alexander von Humboldt Foundation for their generous
support of this work. Richard Pettigrew would like to thank the British Academy with
whom he was a postdoctoral fellow during work on this article.



202 HANNES LEITGEB AND RICHARD PETTIGREW

state at a given time t may be represented by a belief function b �t
that takes each proposition A, represented as a subset of W, toBel(W )

a real number that measures the degree of credence the agent assignsb (A)t
to A. Thus, to solve the fundamental problem of epistemology, the Bayes-
ian must say, for a given body of evidence, which belief functions it would
be rational for an agent to have at a time when she is in possession of
that evidence.

At the core of Bayesianism lie two claims that go some way to solving
the fundamental problem of epistemology as set out above: they are Prob-
abilism and Conditionalization. Probabilism is a synchronic norm: that
is, it concerns the intrinsic properties of an agent’s belief function at
particular moments in her epistemic life. In particular, it demands that
an agent’s belief function should be a probability function at any such
moment. Conditionalization, however, is a diachronic norm: that is, it
concerns the relation between an agent’s belief functions at different times.
In particular, it demands that an agent who learns the truth of a prop-
osition E between times t and t′ (and nothing stronger) ought to update
her belief function by conditionalizing on E.1

These central claims have been extended in various ways by various
philosophers. Two such extensions will be of particular interest to us.
First, the synchronic claim that characterizes (a version of) Objectivist
Bayesianism: if an agent has opinions only about finitely many possible
worlds and if E is the strongest proposition given to her by her evidence
so far, then her belief function ought to be the uniform probability dis-
tribution over the worlds in E: we will call this Uniform Distribution.
And, second, Richard Jeffrey proposed a diachronic norm, which we will
call Jeffrey Conditionalization, that is meant to cover those instances of
updating in which the evidence learned does not come in the form of a
proposition learned with certainty, as in Conditionalization, but rather in
the form of some weaker side constraints on the agent’s belief function.

In this article and in its sequel (Leitgeb and Pettigrew 2010), we defend
the two core tenets of Bayesianism (and, to a much lesser extent, the
additional Objectivist Bayesian tenet) by appealing to the following fun-
damental norm:

Accuracy. An epistemic agent ought to approximate the truth. In
other words, she ought to minimize her inaccuracy.2

1. As Jon Williamson reminded us, Conditionalization must be qualified: it holds only
when plain factual evidence about the world is involved; cf. Williamson, forthcoming.

2. Why not maximize your accuracy instead? Because we like to think of inaccuracy as
being given by a distance: the lesser the distance from the truth, the lesser the inaccuracy;
the greater the distance from the truth, the greater the inaccuracy. Since distances from
the truth are bounded from below, that is, by the zero distance, they can be minimized,



MEASURING INACCURACY 203

Also, we use this norm to criticize Jeffrey’s updating rule, and we defend
an alternative to Jeffrey Conditionalization that applies to the same type
of situations. We consider all of this to be an objective manner of justifying
Bayesianism, which is based on theoretical considerations on how to get
to the truth rather than on practical considerations on how to make
prudent decisions. Indeed, for us an agent’s degree of belief in a propo-
sition A is such that the agent ought to minimize its distance from the
truth value of A; for all epistemological purposes, this feature is in fact
constitutive of the notion of degree of belief. Although the chosen type
of justification is objective in this sense, it should be kept in mind that
what gets justified in this way is still mainly just standard subjective Bayes-
ianism.

We agree with Jim Joyce (1998, 2009) that the relevant notion of ac-
curacy here is what he calls gradational accuracy, wherein gradational
accuracy depends only on the truth values of propositions at worlds and
on the agent’s belief function.3 We will make this feature of accuracy more
precise in the Local Normality and Dominance postulate in section 5,
when we will give the notion a mathematical analysis, as promised. The
quantitative notion of accuracy that interests us differs from Popper’s
(1972) comparative concept of verisimilitude, according to which some
sets of statements are closer to the truth than other sets of statements.
This concept was proven inadequate by Miller’s (1974) and Tichý’s (1974)
triviality results.4

Given this understanding of the notion of accuracy, it is the purpose
of this article to make the Accuracy norm precise. In the sequel, we will
then investigate the consequences of this norm.

2. The Basic Concepts and the Argument in Brief. Our argument is long,
and it involves a number of distinctions. Thus, for the sake of clarity, we

and that is what we are asking for in Accuracy. Using some means of transformation,
these properties might perhaps be captured just as well by accuracy or ‘inverse distance’,
but employing the notion of inaccuracy directly seems to give us a much more appealing
way of stating our central epistemic goal.

3. Indeed, Joyce’s original article has been a major inspiration for the project in this
article and its sequel. We discuss Joyce’s own account in detail in the sequel of this article.

4. In the meantime, refined theories of verisimilitude have been introduced that do not
suffer from triviality results; some of them do resemble our theory in important respects.
In particular, Niiniluoto’s (1987) theory of estimated truthlikeness defines a relational
notion of truthlikeness in terms of an expected value of quantitative truthlikeness that
bears some similarity to the expected inaccuracy of propositions that we will be interested
in. We leave it to another paper to work out the details of this correspondence that was
pointed out to us by Theo Kuipers.



204 HANNES LEITGEB AND RICHARD PETTIGREW

Figure 1.

present an overview of its underlying concepts and its structure before
we give it in a fully detailed form.5

We begin by drawing a distinction between local and global measures
of inaccuracy. A local inaccuracy measure is a mathematical function that
takes a proposition , a world , and a nonnegative realA P W w � W
number x and gives a measure of the local inaccuracy of havingI(A, w, x)
degree of credence x in proposition A at world w. So measuresI(A, w, x)
the distance of x from the truth value of A at w, where the truthx (w)A
values are represented by the real numbers 0 and 1, as shown in figure
1. Intuitively, will be greater, the more x differs from the truthI(A, w, x)
value of A in w.

However, a global inaccuracy measure is a mathematical function that
takes a belief function b and a world w and gives a measure ofG(w, b)
the global inaccuracy of having belief function b at world w. So G(w, b)
measures the distance of b from the world w, where both belief function
and world will be represented geometrically in terms of vectors, as we
will explain in detail in section 3.2 and as shown in figure 2. Again in-
tuitively, will be greater, the more the degree of belief assignmentG(w, b)
that is determined by b differs from the truth value assignment that is
determined by w. Obviously, not any such function I or G will do; rather,
we will have to restrict ourselves to sensible or legitimate choices of such
functions, which will be achieved later by formulating postulates on what
such legitimate I s or Gs are like.

Thus, our first attempt to make Accuracy precise results in the norm
bifurcating:

Accuracy (Local). An agent ought to minimize the local inaccuracy
of her degrees of credence in all propositions relative to aA P W
legitimate measure of local inaccuracy.
Accuracy (Global). An agent ought to minimize the global inaccuracy

5. Bas van Fraassen’s comments were instrumental in making this section clearer.



MEASURING INACCURACY 205

Figure 2.

of her belief function relative to a legitimate measure of global in-
accuracy.

Note that, while we will be interested throughout in versions of both of
these norms, Joyce (1998, 2009) is concerned solely with a version of
Accuracy (Global).

Thus, we have two norms to which Accuracy gives rise. However, as
formulated above, the norms are still incomplete: both local and global
inaccuracy are only defined relative to a world, but we have not specified
yet relative to which world or set of worlds the inaccuracies in questions
are to be calculated. The obvious answer at this point would seem to be
“relative to the actual world.” But from an internalist point of view on
justification—which we are going to adopt, as we will explain at greater
length below (sec. 3.3)—this will not do, since we should not presuppose
that the agent knows which world w in W is the actual world. Instead,
the agent should take into account inaccuracies with respect to all and
only the worlds that are epistemically possible for her; if this set is taken
to be epistemically accessible to her, then we do not violate internalism
about justification by demanding that she assesses her overall inaccuracy
in terms of it. Hence, we focus on the measures of expected local and



206 HANNES LEITGEB AND RICHARD PETTIGREW

global inaccuracy to which any pair of legitimate local and global inac-
curacy measures will give rise, and we evaluate these expected inaccuracy
measures over the set E of epistemically possible worlds.

The expected local inaccuracy of a degree of credence is defined, as one
would expect, as the sum of its inaccuracies at various worlds weighted
by the degree of belief assigned to each of those worlds (or rather to their
singleton sets). Thus, to determine the expected local inaccuracy of a degree
of credence x in a proposition A, we must specify three parameters:

i) The belief function that gives the degree of belief assigned to each
of the worlds over which the sum is taken. That is, the belief
function we use to weight the inaccuracies that we sum to give the
expected local inaccuracy.

ii) The set of worlds over which the sum is taken. This is the set of
worlds that are epistemically possible for the agent.

iii) The local inaccuracy measure that gives the inaccuracies of the
degree of credence at the various worlds over which the sum is
taken.

Thus, we have the following definition:

Definition 1 (Expected local inaccuracy). Given a local inaccuracy
measure I, a belief function b, a degree of credence x, and propositions

, we define the expected local inaccuracy of x in propositionA, E P W
A by the lights of b, with respect to I, and over the set E of episte-
mically possible worlds as follows:

LExp (I, A, E, x) p b({w})I(A, w, x).�b
w�E

Expected global inaccuracy requires us to fix the same parameters. It is
defined as follows:

Definition 2 (Expected global inaccuracy). Given a global inaccuracy
measure G, belief functions b and b′, and a proposition , weE P W
define the expected global inaccuracy of b′ by the lights of b, with
respect to G, and over the set E of epistemically possible worlds as
follows:

′ ′GExp (G, E, b ) p b({w})G(w, b ).�b
w�E

Thus, our second attempt to make Accuracy precise gives:

Accuracy (Expected local). An agent ought to minimize the expected
local inaccuracy of her degrees of credence in all propositions A P

relative to a legitimate measure of local inaccuracy.W



MEASURING INACCURACY 207

Accuracy (Expected global). An agent ought to minimize the expected
global inaccuracy of her belief function relative to a legitimate mea-
sure of global inaccuracy.

However, neither of these proposals is fully specified in its current form
either, for we have not said by the lights of which belief function an agent
ought to assess her expected local or global inaccuracy or over which set
of epistemically possible worlds. Specifying the belief function by the lights
of which we assess expected local or global inaccuracy leads us to a further
distinction: the distinction between synchronic and diachronic versions of
both Accuracy (Expected local) and Accuracy (Expected global). Here
are the synchronic versions of Accuracy (Expected local) and Accuracy
(Expected global):

Accuracy (Synchronic expected local). An agent ought to minimize
the expected local inaccuracy of her degrees of credence in all prop-
ositions by the lights of her current belief function, relativeA P W
to a legitimate local inaccuracy measure and over the set of worlds
that are currently epistemically possible for her.
Accuracy (Synchronic expected global). An agent ought to minimize
the expected global inaccuracy of her current belief function by the
lights of her current belief function, relative to a legitimate global
inaccuracy measure and over the set of worlds that are currently
epistemically possible for her.

And here are the diachronic versions of Accuracy (Expected local) and
Accuracy (Expected global), where an agent has learned evidence between
time t and time t′ that imposes constraints C on her belief function atb ′t
time t′ or on the set E of worlds that are epistemically possible for her
at t′ or both:

Accuracy (Diachronic expected local). At time t′, such an agent ought
to have a belief function that satisfies constraints C and is minimal
among belief functions thus constrained with respect to the expected
local inaccuracy of the degrees of credence it assigns to each proposi-
tion by the lights of her belief function at time t, relative toA P W
a legitimate local inaccuracy measure and over the set of worlds that
are epistemically possible for her at time t′ given the constraints C.
Accuracy (Diachronic expected global). At time t′, such an agent ought
to have a belief function that satisfies constraints C and is minimal
among belief functions thus constrained with respect to expected global
inaccuracy by the lights of her belief function at time t, relative to a
legitimate global inaccuracy measure and over the set of worlds that
are epistemically possible for her at time t′ given the constraints C.



208 HANNES LEITGEB AND RICHARD PETTIGREW

These are very nearly our final versions of Accuracy. All that remains
is to specify the legitimate local and global inaccuracy measures. This is
the work of section 5, where we argue for the main thesis of this article:
the only legitimate local and global inaccuracy measures are quadratic
inaccuracy measures, which are known as Brier scores in the literature on
scoring rules (Brier 1950). We call these characterizations Local Inaccu-
racy Measures and Global Inaccuracy Measures, respectively. By substi-
tuting them into the appropriate norms above, we obtain the four math-
ematically precise versions of Accuracy that are the aim of this article.
In Leitgeb and Pettigrew (2010), we investigate the consequences of these
norms.

Of course, as with any set of norms, it may turn out that some or all
of these norms are simply unsatisfiable. Before investigation, there is no
reason to think that there are such minimally inaccurate belief functions
in the senses required by these norms. However, as we will show in the
sequel to this article, such a situation arises only for certain instances of
Accuracy (Diachronic expected local); for each of the other norms and
indeed for many instances of this norm, there are belief functions that
satisfy them, and, moreover, they are the belief functions that the Bayesian
demands. Furthermore, in the situations in which Accuracy (Diachronic
expected local) cannot be satisfied, its global analogue Accuracy (Dia-
chronic expected global) can be. Thus, our approach does make a demand
in these situations. We will discuss this further in the second article.

Note that and will always be zero if′LExp (I, A, E, x) GExp (G, E, b )b b
the global belief function b is identical to the constant zero function on
singletons of members of E. So if an agent has ruled out all worlds in E
by means of b, any degree of belief whatsoever may be assigned to A in
order to minimize expected local inaccuracy, and any belief function b′

may be chosen to minimize global inaccuracy. In order to do better, we
would have to generalize our framework and aim at a justification of
Popper functions (see Popper [1968], for one reference among many)
rather than standard absolute probability measures. We leave this topic
to a different paper.

In section 3, we discuss in greater detail the transition from Accuracy
to the four norms just listed, making explicit all the formal and philo-
sophical presuppositions of our theory. We are not going to justify these
presuppositions in any substantial manner, but at least we will make sure
we put all of our cards on the table, and we will formulate explicitly the
main questions that will remain open. In section 4, we give a brief sketch
of the argument in favor of quadratic inaccuracy measures, both local
and global, and in section 5, we give this argument in full detail.



MEASURING INACCURACY 209

3. The Presuppositions of Our Argument.

3.1. The Ought-Can Principle. We wish to make the following norm
precise:

Accuracy. An epistemic agent ought to approximate the truth. In
other words, she ought to minimize her inaccuracy.

In doing so, we will be guided at a number of points by the following
version of a well-known normative or metanormative principle, the func-
tion of which is to constrain our choice of normative systems:

Ought-Can. A norm should not demand anything of an agent that
is beyond her epistemic reach.

This is just a variant of the classical Ought-Can principle applied in
the present context. We leave open the exact character of the possibility
modality that is implicit in ‘Can’ and ‘reach(ability)’. However, as will
become clear from our applications of Ought-Can, it is certainly stronger
than mere logical possibility and not too far from a notion of realistic
achievability in the epistemic domain.

One important consequence of Ought-Can is that one should not de-
mand of an agent that she draw distinctions that she is conceptually unable
to draw. Here is one way in which this is relevant to our argument: in
the following sections, we presuppose that our agents hold opinions about
a finite and nonempty set W of possible worlds of which they assume the
actual world to be one. There is nothing particularly philosophical about
our decision to stick to the case of finitely many worlds in this article;
we simply assume this to be so and postpone the discussion of the infinite
case to another time. However, already this seemingly harmless suppo-
sition may appear to have drastic—and even drastically wrong—conse-
quences: since there are, presumably, infinitely many possible worlds, every
choice of a finite set of ‘possible worlds’ must consist in carving the actual
set of worlds into finitely many chunks, which are then presupposed to
figure as the chosen ‘(pseudo)possible worlds’ of the framework. Worse,
if a uniform probability measure has been defined over the resulting finite
set W, as demanded by Objectivist Bayesianism (Uniform Distribution),
it is unclear whether this measure is also uniform over the actual set of
worlds and, if so, in what sense. So are we buying into a presupposition
that makes our approach appear highly questionable from the start? Not
necessarily, in light of the Ought-Can principle. Assume that we are solely
concerned with agents whose conceptual resources cut the space of logical
possibilities into finitely many pieces—the members of W. Indeed, we may
assume these agents to be epistemically unable to distinguish between any
two of the possible worlds that belong to one and the same partition set



210 HANNES LEITGEB AND RICHARD PETTIGREW

and to be incapable also of altering their conceptual framework on any
rational grounds for the time of our investigation. From the viewpoint
of any of these agents, at the time of our investigation, there is thus no
way of having any sort of epistemic access to the ‘actual’ infinite set of
possible worlds. Demanding of such agents that they transcend their con-
ceptual boundaries would mean demanding that they go beyond their
epistemic reach, in conflict with Ought-Can. Therefore, whenever we refer
to agents in this article, let them be as just described, and Ought-Can will
allow us to proceed without any further worries. For many practical
purposes, maybe even we can be taken to be such finitely constrained
agents. In all other contexts, that is, whenever we are dealing with full-
fledged human agents and their (maybe) infinitary conceptual capacities,
the arguments of this article do not apply. But hopefully, even in the latter
case, our arguments will be interesting in themselves. Ought-Can will
prove even more relevant in section 3.3, where we derive from it an in-
ternalist view of justification that forces us to move from Accuracy (Local)
and Accuracy (Global) to Accuracy (Expected local) and Accuracy (Ex-
pected global), as we mentioned above.

And finally, the principle will play a crucial role in section 5 as well,
where we argue for Local and Global Inaccuracy Measures, which specify
the legitimate local and global measures of inaccuracy: it follows from
Ought-Can that a normative system should not be such that it may lead
an agent who obeys the norms of this system into an epistemic dilemma,
that is, into a situation in which she ought to change her epistemic state
in two or more ways that are jointly impossible. We show that an agent
who employs an inaccuracy measure that is not permitted by Local and
Global Inaccuracy Measures may face an epistemic dilemma akin to the
so-called discursive dilemma in the theory of judgment aggregation. From
this conclusion and Ought-Can, our characterizations of the legitimate
inaccuracy measures as being the quadratic ones will follow.

3.2. The Geometrical Framework. So much for Ought-Can; we will re-
turn to it below. In the meantime, we turn to the other background as-
sumptions of our approach. Broadly speaking, we are taking a geometrical
approach; to coin a slogan, we are interested in the ‘geometry of reason’.
Of course, it is common procedure to go back and forth between prob-
ability measures as characterized by the Kolmogorov axioms and prob-
ability measures viewed as points in geometrical space, so it might seem
that putting forward a geometrical account of belief dynamics is com-
pletely unproblematic. However, our task is not so much to presuppose
probability theory and to exploit its mathematical features in applications
of probabilistic reasoning but rather to justify probabilistic reasoning in



MEASURING INACCURACY 211

a context in which probability theory is not yet in place. So we will have
to take things more slowly.

Since W is finite, there is an n such that . We start by posi-FWF p n
tioning the members of W in the n-dimensional Euclidean space byn�
‘identifying’ each world with the ith unit vectorw � W p {w , . . . , w }i 1 n
in , that is, with the vector , where if andn� (d , . . . , d ) d p 1 j p ii,1 i,n i, j

if . If the real number 1 is taken to represent truth and thed p 0 j ( ii, j
real number 0 is taken to stand for falsity, then each such vector

can be considered as an assignment of the geometrical coun-(d , . . . , d )i,1 i,n
terparts of truth and falsity to the singleton propositions ,{w }, . . . , {w }1 n
such that the jth coordinate of the vector is identical to the(d , . . . , d )i,1 i,n
real number 1 if and only if the proposition is true in the world that{w }j
is represented by this vector. Given this mapping from worlds to geo-
metrical points or vectors, we can measure distances or ‘closeness’ between
worlds with respect to each other in terms of the Euclidean distance
between their geometric counterparts. Accordingly, regarding truth values

as points , respectively, in the one-dimensional Euclidean space,1, 0 1, 0
that is, , allows us to measure distances between truth values geomet-�
rically. Now, there is certainly not just one geometrically ‘natural’ notion
of distance in Euclidean space. In fact, there are lots of geometrically
plausible metrics on and that are distinct from the Euclidean metric,n� �
and indeed we are going to argue later that inaccuracy has to be measured
in terms of one of them. However, we will demand that every such notion
of the geometrical distance between two points supervenes on—is func-
tionally dependent on—the Euclidean distance between these points. In
this sense, measuring closeness will always amount to a geometrical, and
indeed Euclidean, procedure in the context of this article.

The next step is to locate degrees of belief within and to place belief�
functions into . As pointed out at the beginning of this article, wen�
consider belief functions to be mappings of the form , so�b : P(W ) r � 0
degrees of belief are assumed to be quantitative objects from the start.6

Following Accuracy from above, we regard rational agents to be aiming
at distributing their degrees of belief in such a way that every such degree

approximates the truth value of the proposition A (although it is notb(A)
yet clear exactly in what sense). Hence, a rational agent’s degree of belief
for a proposition is nothing but the agent’s best possible estimate or
‘simulation’ of the truth value of that proposition, given her present ep-
istemic situation. Since truth and falsity have been represented by real

6. We could have regarded belief functions as mappings into rather than ; some�� �0
parts of our argumentation are in fact not going to hang on this. Our reason for not
doing so from the start is quite simply that some readers might find negative degrees of
belief odd.



212 HANNES LEITGEB AND RICHARD PETTIGREW

numbers, too, degrees of belief and truth values are comparable—they
occupy the same quantitative or geometrical scale. So, for example, as-
signing a degree of belief 1 to a proposition A would mean that the agent
believes that A is true rather than false, since the degree of belief 1 is
closer—in fact, identical—to the real number 1 that represents truth than
it is to the real number 0 representing falsity. In this way, closeness of a
degree of belief to the truth can be measured again according to a metric
on the one-dimensional Euclidean space.

In order to see how belief functions determine points in the n-dimen-
sional Euclidean space, by which it then becomes possible to measure
their distances from the point vectors that represent possible worlds, it
is useful to introduce the following terminology: call any function

a global belief function, and let be the set of all�b : W r � Bel (W )glo 0 glo
global belief functions. Obviously, every such global belief function may
be regarded as a vector with forn(a , . . . , a ) � � a p b (w ) j p1 n j glo j

. Since every belief function b induces a global belief function1, . . . , n
by means of , we can position b in through itsnb b (w ) p b({w }) �glo glo j j

corresponding global belief function. Apart from supplying belief func-
tions with a geometrical interpretation, the latter equation also yields a
way of interpreting global belief functions: we call them global belief
functions as they may be taken to summarize an agent’s attitude toward
all the worlds w in W, that is, toward all of the singleton propositions

for . For example, a global belief function with the vector{w} w � W
represents an agent who is—in the “degrees of belief being(1, 0, . . . , 0)

best possible estimates or ‘simulations’ of truth values” sense mentioned
above—certain of the truth of the proposition and certain that every{w }1
other singleton proposition is false. Geometrically, the distance between
the vector that belongs to this global belief function and the vector that
corresponds to the world is 0, which is exactly what we want to bew1
the case in such a situation. So, (global) belief functions and worlds have
become comparable as well, again by means of their geometrical repre-
sentations.7

7. We are not suggesting that this is the only ‘natural’ way of identifying belief functions
with vectors or that this type of identification is free of presuppositions. In particular,
representing belief functions geometrically in such a manner corresponds in some sense
to a ‘bias’ toward worlds. It might well be that if a different form of representation were
chosen, then we would not end up justifying Bayesianism but some different account of
belief dynamics (say, the Dempster-Shafer approach or something else). At the same time,
it is not as if this ‘bias’ toward worlds gives us anything like the Bayesian tenets in any
obvious and immediate manner: it will need elaborate arguments, substantial postulates,
and mathematical proofs until we will have finally derived these tenets (as we will in the
sequel to this article). As we will point out later, there are various parameters in our
theory that we set in a particular way and for which we investigate the consequences of



MEASURING INACCURACY 213

We have seen how belief functions determine corresponding global be-
lief functions. Is there also a way of inverting this procedure? That is,
given a global belief function , is there a similarly salient way of de-bglo
termining a belief function b that assigns degrees of belief not just to
singleton propositions (or worlds) but to all propositions whatsoever? If
Bayesianism were taken for granted, the answer would of course be yes,
by iterated application of finite additivity. But Bayesianism is exactly what
is at issue here, so without further argument there does not seem to be
any obvious way of determining a unique belief function from a given
global belief function justifiedly or, for that matter, of determining jus-
tifiedly any belief function from a given global belief function at all. For
this reason, presupposing that an agent’s epistemic state at a time involves
the acceptance of a global belief function is at least prima facie a weaker
presupposition than assuming that the agent’s epistemic state at that time
involves the acceptance of a belief function.

3.3. Internalism, Expected Inaccuracy, and Ought-Can Again. In this
section, we return to Ought-Can. Above, we noted that this principle
forces the shift from Accuracy (Local) and Accuracy (Global) to Accuracy
(Expected local) and Accuracy (Expected global). Here we explain how.

In section 5, we will argue that the only legitimate inaccuracy measures
are quadratic inaccuracy measures. However, while this conclusion gives
us an important auxiliary notion of inaccuracy, it does not yet by itself
yield a notion of inaccuracy that an agent can actually make use of in
order to determine the local inaccuracy of her degree of credence in a
proposition or the global inaccuracy of her belief function. The reason,
as we observed above, is simply that the agent cannot be assumed to
know at which world she is evaluating inaccuracies. So, by Ought-Can,
we require a conception of the minimization of inaccuracy toward which
the agent is epistemically capable of aiming. On such a conception, an
agent’s assessment of the local inaccuracy of a degree of belief in a prop-
osition will be bound to take into account the local inaccuracies of that
degree of belief at all possible worlds that are not excluded by the evidence
available to her, that is, all worlds that are epistemically possible for her.
Similarly, on this conception, an agent’s assessment of the global inac-
curacy of a belief function will be bound to take into account the global
inaccuracies of that belief function at all epistemically possible worlds.
Thus, the Ought-Can principle gives rise to a form of internalism about
justification, and this internalism leads in turn to the notions of expected
local and global inaccuracy defined above (definitions 1 and 2) and to

setting them as such. But we would be equally interested in studying alternative ways of
setting these parameters and in determining the consequences of these alternative settings.



214 HANNES LEITGEB AND RICHARD PETTIGREW

versions of the Accuracy norm that demand that an agent should minimize
not her actual local or global inaccuracy but her expected local or global
inaccuracy instead. And, as we described above, this move results in a
further bifurcation of the norms: Accuracy (Expected local) splits to give
Accuracy (Synchronic expected local) and Accuracy (Diachronic expected
local), while Accuracy (Expected global) splits to give Accuracy (Syn-
chronic expected global) and Accuracy (Diachronic expected global). The
synchronic norms constrain the intrinsic nature of an agent’s belief func-
tion at particular times in her epistemic life: at any such time t, the agent’s
belief function bt must have minimal expected local (respectively, global)
inaccuracy by the lights of bt itself. And the diachronic norms constrain
the relation between an agent’s belief function at times t and t′ between
which the agent obtains evidence that places constraints C on the legit-
imate belief functions at t′ or on the epistemically possible worlds at t′ or
on both: must satisfy C and must be minimal, among those beliefb ′t
functions that satisfy C, with respect to expected local (respectively, global)
inaccuracy by the lights of bt and over the possible worlds that are epi-
stemically possible at t′ and that satisfy the constraint C.

We should say a little more about the mathematical notion of expec-
tation in this context. Usually, expectations are defined only for proba-
bility measures. However, in the absence of Probabilism—a claim that we
wish to establish, not presuppose—there is no reason for thinking that
belief functions are probability measures. However, while probability the-
ory is the usual context in which expectations are defined, there is no
objection in principle to extending the definition to cover the case of belief
functions that may not be probability measures. Of course, if the belief
function is not additive, we will not be able to prove the equivalence of
the definitions we have given (see definitions 1 and 2) with two alternative
definitions that arise from definitions of expected values that are standard
in probability theory: namely,

LExp (I, A, E, x) p r # b({w � W : I(A, w, x) p r}),�b
r�Ran(I )

′GExp (G, E, b ) p r # b({w � W : G(w, b) p r}).�b
r�Ran(G)

But we will never use the alternative definitions, and we take the definitions
that we have given before to be the conceptually basic ones, so we do
not regard this as a problem. It would be great to support this view by
having to hand a general and abstract theory of the concept of expected
value for which one could actually prove that, given the presuppositions
of our approach, the resulting notion of expected value must be the one



MEASURING INACCURACY 215

that is employed in definitions 1 and 2.8 We will have to leave any in-
vestigations into that topic for a different occasion.

By using what we take to be the basic definitions of expectation, we
avoid the objection that Joyce (1998, 589) raises against Rosenkrantz’s
(1981) appeal to expected inaccuracy. Like us, Rosenkrantz favors qua-
dratic global inaccuracy measures. He lists constraints on global inac-
curacy measures and conjectures that they are satisfied uniquely by the
quadratic functions. However, he gives no proof of this claim, and, as
Joyce notes, he gives no noncircular arguments in favor of his constraints.
In particular, Rosenkrantz demands of an agent that she minimize her
expected inaccuracy calculated over every possible partition of the space
simultaneously. As Joyce points out, unless her belief function is already
assumed to be a probability function, this will not be possible. Our precise
versions of the Accuracy norms are not vulnerable to this objection since
we only ever appeal to the most fine-grained partition of W that is con-
ceptually available to the agent, that is, the set of singletons of worlds in
W.

3.4. The Status of Our Presuppositions. Now that the underlying formal
framework of our theory has been made more explicit, how are we going
to argue for it? Short answer: we do not. Not that there is nothing at all
to say in favor of it: among other things, one could point out that some
aspects of our geometrical framework are purely conventional and thus
do not need any further justification at all. For example, we could have
represented truth by, say, the real number 2 and, accordingly, worlds by
vectors that are like unit vectors but where the coordinate 1 is replaced
by 2 and so forth.9 In other words, the absolute position of truth values,
worlds, and belief functions in the Euclidean plane is arbitrary. However,
fortunately, none of our results depend on it.

Furthermore, if that Kantian move helped at all, the choice of a Eu-
clidean geometry rather than a non-Euclidean one could perhaps be
grounded in our intuitions about space and distance in general—although
the ‘geometry of reason’ we are after can hardly be called Kantian, and
in a context in which information is represented geometrically, often met-
rics other than the Euclidean one are used. In fact, one would be com-
pletely justified in wondering why degrees of belief should “live” in a
Euclidean space at all—after all, physical space turned out to be non-
Euclidean, and even from our internalist perspective it is perfectly rea-

8. We thank Franz Dietrich for highlighting this in a personal communication.

9. Franz Huber (2006, 2) cites a corresponding worry raised by Colin Howson in an
unpublished manuscript.



216 HANNES LEITGEB AND RICHARD PETTIGREW

sonable to assume that there might be agents whose “cognitive spaces”
are non-Euclidean.10

Other aspects of the framework do not even have a conventional or
intuitive character at all: for example, why is the notion of expected local
inaccuracy given by taking the sum of weighted local inaccuracies rather
than their product or their maximum or whatever other function comes
to mind? Again, one could surely do better than just leaving the discussion
at that point: indeed, one should be able to defend some necessary con-
ditions on expected inaccuracy on the basis of assuming Ought-Can and
the rest of the geometrical framework; it might also be possible to prove
representation theorems by which our expected local inaccuracy functions
would turn out to have an equivalent, and intrinsically plausible, quali-
tative or comparative formulation and so forth. But it is hard to see how
any such justification would be ultimate. Similarly, why does each world
have the same distance from each other world according to their geometric
representations? Is this because we have renormalized the scales of our
given coordinate system in a way that leads to this result trivially, or do
we commit ourselves to a substantial assumption that is at least implicitly
pointing toward Objectivist Bayesianism? We will have to say just slightly
more about this last point in the sequel to this article, but otherwise we
simply leave the status of the framework untouched; that is, we take this
geometrical framework as a presupposition of our justification of (Ob-
jectivist) Bayesianism without giving it any further defense.

One final remark, though: we do regard the question of just how much
work in our argumentation is done by its geometrical Euclidean back-
ground framework as a very important one; eventually, this article ought
to be complemented by one that abstracts from the epistemic-geometrical
models that we presuppose all and only the essential axioms that are
needed in order for our arguments to go through, and only then will it
be possible to see how much weight is carried by our Euclidean presup-
positions. We will return to this as an open question to be formulated in
the final section of this article. When we determine the quadratic inac-
curacy measures as the legitimate ones in section 5, this ought to be
understood in the way that they are the legitimate ones relative to the
chosen Euclidean framework; other inaccuracy measures might be legit-
imate if given a different framework.11

10. We are grateful to Alan Hájek and Kenny Easwaran for highlighting this as an
important open problem.

11. We thank Branden Fitelson for making this point in a discussion of our article.



MEASURING INACCURACY 217

TABLE 1. ARGUMENTS.

First
Argument

Second
Argument

Third
Argument

Shared premises (sec. 5.1):
Local Normality and Dominance X X X
Global Normality and Dominance X X X
Local and Global Comparability X X X
Local and Global Minimum Inaccuracy X X X

Dilemma 1 premise (sec. 5.2.1): Agreement on
Inaccuracy X

Dilemma 2 premise (sec. 5.2.2): Separability of
Global Inaccuracy X

Dilemma 3 premises (sec. 5.2.3): Continuous
Differentiability and Agreement on Di-
rected Urgency

X

Conclusion:
Local Inaccuracy Measures X X X
Global Inaccuracy Measures X X X

4. The Argument for Local and Global Inaccuracy Measures: An Over-
view. Section 5 is devoted to justifying two claims: Local Inaccuracy Mea-
sures and Global Inaccuracy Measures. The former says that the legiti-
mate local inaccuracy measures are the quadratic scoring rules. The latter
says that the legitimate global inaccuracy measures are the quadratic
functions of the Euclidean metric on . In this section, we give an over-n�
view of the argument.

In fact, we will state three arguments for identifying these characteri-
zations of the legitimate local and global inaccuracy measures. Each begins
in the same way by imposing four conditions that restrict the class of
legitimate inaccuracy measures (both local and global) on the basis of the
principles highlighted in section 3. And each continues by noting that
these restrictions fail to exclude inaccuracy measures that give rise to a
dilemma for the agent, that is, a situation in which the epistemic norms
of the previous section, combined with these inaccuracy measures, entail
two mutually exclusive prescriptions for the agent. In each case, when we
restrict the class of legitimate inaccuracy measures to exclude those that
lead to the dilemma in question, we are left only with the quadratic
inaccuracy measures, and local and global inaccuracy measures follow
immediately.

Thus, in section 5, we give three separate arguments, each of which
shares its first four premises with the others. We set them out in table 1
for ease of reference. Each of the premises will be supported by our
presuppositions—some strictly, some defeasibly. Furthermore, each of the
arguments mentioned in this section will be seen to be strictly valid, that
is, deductively valid given mathematics, as shown by the proofs given in
the appendix.



218 HANNES LEITGEB AND RICHARD PETTIGREW

The conclusions of these three arguments will be our promised two
characterizations of legitimate inaccuracy measures that we want to es-
tablish in section 5:

Local Inaccuracy Measures. If is a legiti-� �I : P(W ) # W # � r �0 0
mate measure of the local inaccuracy of a degree of credence x in a
proposition A at a possible world w, then there is such thatl � � 10

2I(A, w, x) p l[x (w) � x] ,A

where is the characteristic function of the propositionx : W r {0, 1}A
A.
Global Inaccuracy Measures. If is a legitimate�G : W # Bel(W ) r � 0
measure of the global inaccuracy of a belief function b at a possible
world w, then there is such thatl � � 10

2G(w, b) p lFFw � b FF ,glo

where w and are represented by vectors as in section 3.2 andbglo
is the Euclidean distance between vectors u and : that is,FFu � vFF v

2 2�FFu � vFF p (u � v ) � . . . � (u � v ) .1 1 n n

The claim that quadratic inaccuracy measures yield the only legitimate
scoring rules is similar to Selten’s central claim (1998). Thus, it might
seem that we could easily adapt Selten’s ingenious argument to establish
Local Inaccuracy Measures. However, at a number of points in his proof,
Selten relies on the assumption that belief functions are probability func-
tions. For his avowed purpose, this is perfectly legitimate. However, for
the purposes of this article, we could not avail ourselves of a result pre-
mised on this assumption to establish Local Inaccuracy Measures, which
we then in turn wish to use to derive Probabilism, among other things
(and, to the best of our knowledge, similar points can be made about
much of the excellent and highly evolved literature on scoring rules and
decision theory). Thus, we must beat our own path to our conclusion.

As pointed out, we beat three paths, which begin together and diverge
only at the final premises of the arguments. Thus, we begin with the shared
premises in section 5.1; then, in section 5.2, we consider the three dilemmas
that motivate the three different final premises. All three arguments turn
on mathematical theorems; their proofs are annexed in the appendix.
Among the three arguments, we consider the final one (sec. 5.2.3) to be
the strongest and most convincing, but the first two arguments (secs. 5.2.1
and 5.2.2) are easier to state, which is why we will turn to them before
we give the third, and philosophically central, argument.



MEASURING INACCURACY 219

5. Measuring Inaccuracy.

5.1. The Shared Premises. The first premise of each of our arguments
combines local analogues of Joyce’s Normality and Dominance conditions
(1998, 596 and 593, respectively): the local version of Joyce’s Normality
condition says that the inaccuracy of degree of credence x in proposition
A at world w ought to depend only on the difference between x and the
value of the characteristic function of A at w (i.e., the truth value of A
at w); the local analogue of Dominance merely states that local inaccuracy
increases as this difference increases.

Local Normality and Dominance. If I is a legitimate inaccuracy mea-
sure, then there is a strictly increasing function such that,� �f : � r �0 0
for any , , and ,�A P W w � W x � � 0

I(A, w, x) p f (Fx (w) � xF).A

Note that this also implies that distances from the truth ( )x (w) p 1A
and distances from falsity ( ) are measured in the same way,x (w) p 0A
which is entailed by our geometrical take on truth and falsity as points
in a space. It is clear that once a Euclidean framework such as ours is in
place, a condition analogous to Local Normality and Dominance ought
to hold for global inaccuracy measures as well. Local Normality and
Dominance asserts that the local inaccuracy of a degree of credence x in
proposition A at world w ought to be a strictly increasing function only
of the difference (i.e., the Euclidean distance) between x and . Itsx (w)A
analogue, Global Normality and Dominance asserts that the global in-
accuracy of a global belief function b at a world w ought to be a strictly
increasing function only of the Euclidean distance between the vector
representation of b and the vector representation of w. That is,

Global Normality and Dominance. If G is a legitimate global inac-
curacy measure, there is a strictly increasing function � �g : � r �0 0
such that, for all worlds w and belief functions ,b � Bel(W )

G(w, b) p g(FFw � b FF).glo

Global Normality and Dominance is a consequence of taking seriously
the talk of inaccuracy as ‘distance’ from the truth, and it endorses the
geometrical picture provided by Euclidean n-space as the correct clarifi-
cation of this notion. As explained in section 3.2, the assumption of this
geometrical picture is one of the presuppositions of our account, and we
do not have much to offer in its defense, except for stressing that we
would be equally interested in studying the consequences of minimizing



220 HANNES LEITGEB AND RICHARD PETTIGREW

expected inaccuracy in a non-Euclidean framework. But without a doubt,
starting with the Euclidean case is a natural thing to do.

The third premise that is shared by each of our arguments for local
and global inaccuracy measures says that any function on the real numbers
that gives rise to a legitimate local inaccuracy measure also gives rise to
a legitimate global inaccuracy measure and vice versa. Local and Global
Comparability is as follows:

i) If is a legitimate local inaccuracy mea-I(A, w, x) p f(Fx (w) � xF)A
sure, then is a legitimate global inaccuracyG(w, b) p f (FFw � b FF)glo
measure.

ii) If is a legitimate global inaccuracy mea-G(w, b) p g(FFw � b FF)glo
sure, then is a legitimate local inaccu-I(A, w, x) p g(Fx (w) � xF)A
racy measure.

Again, this is a consequence of our geometrical interpretation of accuracy:
we interpret inaccuracy as distance from the truth, and we interpret dis-
tance as being given by a strictly increasing function of the Euclidean
metric. Since distances are independent of dimension, it should always be
possible to use legitimate local inaccuracy measures in order to determine
their global counterparts and also the other way round; it is simply not
relevant on which dimensions Euclidean distances are measured.

The final premise shared by each of our three arguments for local and
global inaccuracy measures does nothing more than to lay down a con-
vention: we will permit only inaccuracy functions that take value zero
when the distance between truth value and degree of belief or between
world and global belief function is zero. Minimum Inaccuracy is as fol-
lows:

i) If is a legitimate local inaccuracy mea-I (A, w, x) p f(Fx (w) � xF)A
sure, then .f (0) p 0

ii) If is a legitimate global inaccuracy mea-G(w, b) p g(FFw � b FF)glo
sure, then .g(0) p 0

In the presence of Local and Global Comparability, we can derive i from
ii and ii from i. Thus, we need only impose one of these conditions.
However, we state both, lest the reader be given the mistaken impression
that one or the other is more fundamental.

Together, Local Normality and Dominance, Global Normality and
Dominance, Local and Global Comparability, and Minimum Inaccuracy
restrict the class of legitimate inaccuracy measures. However, as we shall
see in the next section, they do not restrict them enough. There are func-
tions that satisfy these restrictions but which have undesirable properties.
We call attention to one particular sort of undesirable property, that is,
the property of giving rise to a dilemma for an epistemic agent. In each



MEASURING INACCURACY 221

case, the dilemmas in question concern possible discrepancies between
measuring inaccuracy in a local and in a global fashion. We show that,
when we exclude the inaccuracy measures that give rise to these dilemmas,
we are left with only the quadratic inaccuracy measures. This will complete
our argument for local and global inaccuracy measures. Quadratic in-
accuracy measures will be the ones that allow the local and the global
perspective on belief functions to be compatible with each other.

5.2. Excluding Dilemmas: Completing the Three Arguments. An inac-
curacy measure gives rise to a dilemma for an agent if the prescription
to be as accurate as possible with respect to that inaccuracy measure
entails two prescriptions for the agent such that she cannot satisfy both
together. In sections 5.2.1–5.2.3, we consider three dilemmas to which an
inaccuracy measure may give rise. In each case, we introduce a principle
to exclude such inaccuracy measures and show that local and global in-
accuracy measures follow from this stipulation, along with the four con-
ditions enumerated in section 5.1. The respective dilemmas are serious in
the following sense: (i) they are about minimizing inaccuracy, the central
goal of our epistemic agents; (ii) they involve an agent’s having to choose
to follow either of two options or norms; (iii) there does not seem to be
any principled way of ranking the two options or norms, such that one
would become epistemically prior or superior to the other. Since the di-
lemmas below are serious in this sense, they have to be avoided, and the
only manner in which this can be done is by making sure that the two
options or norms never lead to different epistemic recommendations to
the agent.

This—defeasible—argumentation in favor of principles by which the
dilemmas may be avoided can only be defeated in either of two ways:
first of all, by attacking iii, that is, by showing that for each of the three
dilemmas there is in fact a way of ranking one option or norm over the
other. For example, if someone were to put forward a sufficiently strong
argument in favor of a form of epistemic holism according to which
considerations of global inaccuracy always overrule considerations of lo-
cal inaccuracy, then this would defeat the seriousness of each of our three
dilemmas. Second, it can be defeated by presenting yet another serious
dilemma in the sense of i–iii into which an agent is led by opting for our
quadratic inaccuracy measures. In that case, no inaccuracy measure what-
soever could protect an agent from being confronted with some serious
dilemma, and the best one could hope for would be a multitude of mu-
tually exclusive and partially defective choices of inaccuracy measures,
such that each one of them would avoid some epistemic dilemmas, but
none of them would avoid all. We hope that at least as things stand, our
arguments are undefeated as yet.



222 HANNES LEITGEB AND RICHARD PETTIGREW

5.2.1. Agreement on Inaccuracy. By Local and Global Comparability
from section 5.1, if the function f gives rise to a legitimate local inaccuracy
measure, then it gives rise to a legitimate global inaccuracy measure as
well, and vice versa. However, the four conditions enumerated in section
5.1 can be shown not to exclude functions f such that (1) the global
inaccuracy measure determines the inaccuracy of a belieff (FFw � b FF)glo
function at a world, (2) its counterpart local inaccuracy measure

yields an indirect way of also determining the inaccuracyf (Fx (w) � b(A)F)A
of a belief function at a world by summing up the local inaccuracies of
degrees of belief assigned to world propositions in the expected manner,
and yet the outcomes of the two determination procedures differ. Such a
disagreement would give rise to a dilemma: the agent who uses both the
global inaccuracy measure and its local counterpart will come to two
conflicting conclusions concerning the inaccuracy of her beliefs. Of course,
it might be the case that despite the numerical disagreement, some formal
properties are still shared by the globally and the indirectly locally de-
termined inaccuracies of belief functions at a world—for instance, the
ordering of belief functions according to their inaccuracies. And searching
for the belief function that minimizes expected inaccuracy first in a glob-
ally and then in a locally induced way might still yield one and the same
output, even when the globally and the locally determined inaccuracies
diverge in value for some or even all arguments. But the only way for
the agent to have a guarantee that the global and the local procedure will
never lead to any conflict whatsoever—and thus that the global and the
local procedure always lead to the same epistemic recommendations, in-
dependently of how sensitive the agent is to the exact numerical inaccuracy
values—is to postulate a convergence between the global and local way
of determining the inaccuracy of any belief function at any world.

Otherwise the agent’s situation would be analogous to that of a group
of individuals faced with making a collective judgment that is vulnerable
to the paradoxes of judgment aggregation. For instance, consider the stock
example of the so-called discursive dilemma, the most vivid of these par-
adoxes (see, e.g., Pettit 2001).

Discursive Dilemma. Three judges must decide whether to convict a
defendant. By law, the defendant may be convicted if and only if
propositions P and Q hold. The judges’ judgments on P and Q and
the consequence for conviction are recorded in the following table,
along with the majority judgment in each case.



MEASURING INACCURACY 223

P Q Conviction

Judge 1 True True Yes
Judge 2 True False No
Judge 3 False True No
Majority True True Yes/no

Thus, while the majority of the individual judgments concerning P
and Q leads to a conviction, the majority of the consequences of
those judgments for conviction leads to acquittal. Which conviction
consequence reflects the aggregate of the judges’ judgments? This is
the discursive dilemma.

In the discursive dilemma, there is a tension between two algorithms by
which to aggregate individual judgments on two propositions by three
individuals into a single judgment. The first algorithm begins by deriving
the conviction consequences from the individual judgments on P and Q
and then takes the majority verdict; the other begins by taking the majority
verdict on each of P and Q and then derives the conviction consequence.
They lead to conflicting results, and it is not clear at all how to tell between
them.

The agent who uses global and local inaccuracy measures that give rise
to different values for the inaccuracy of her belief function faces a similar
problem. She faces the problem of aggregating her various degrees of
belief in various propositions into a value for the inaccuracy of this belief
function as a whole. She has at her disposal two obvious measures by
which to obtain this value: one is just given by applying the global in-
accuracy function itself, and the other by summing up the relevant local
inaccuracies. If they disagree, the agent faces an irresolvable dilemma,
analogous to that faced by the judges in the discursive dilemma.12

We exclude the possibility that gives rise to this dilemma by imposing
the following condition on legitimate inaccuracy measures that yields a
principled way of relating global and local inaccuracy judgments:

Agreement on Inaccuracy. Suppose I is a legitimate local inaccuracy
measure. Then, by Local Normality and Dominance, there is a strictly
increasing function such that� �f : � r � I(A, w, x) p f (Fx (w) �0 0 A

. Further, by Local and Global Comparability,xF) G(w, b) p f (FFw �
is a legitimate global inaccuracy measure. Then, the followingb FF)glo

12. It would be great if we had some results that would show that even if the agent did
not aggregate local inaccuracies by summing them up but rather by applying some other
numerical operation to them that would satisfy certain natural constraints, then a theorem
similar to the one stated below could be derived. Unfortunately, we do not have anything
like that to offer at this point, so we have to leave this for future work.



224 HANNES LEITGEB AND RICHARD PETTIGREW

must hold: if b is a belief function and is a world,wi
n

G(w , b) p I({w }, w , b({w })).�i j i j
jp1

That is,
n

f (FFw � b FF) p f (Fx (w ) � b({w })F).�i glo {w } i jj
jp1

From this, along with the four conditions stated in section 5.1, local and
global inaccuracy measures follow by the following theorem:

Theorem 3. The following two propositions are equivalent:

i) Function f is strictly increasing, and, for all belief functions, b,
and worlds ,wi

n

f(kw � b k) p f (Fx (w ) � b({w })F).�i glo {w } i jj
jp1

ii) There is such that, for all , .� 2l � � x � � f (x) p lx10 0

This theorem is proved in the appendix.

5.2.2. Separability of Global Inaccuracy. In the previous section, we
drew attention to a possible dilemma that results from using the legitimate
local and global inaccuracy measures given by the same function f. And
we ruled out the possibility by introducing Agreement on Inaccuracy. In
this section, we describe another way in which an inaccuracy measure
could give rise to conflicting values for the inaccuracy of a belief function
at a world.

To state the problem, we introduce the following terminology: if 1 ≤
and , then is the projection ofnj ≤ n (a , . . . , a ) � � proj ((a , . . . , a ))1 n 0 j 1 n

onto the linear subspace that is spanned by the unit vectors(a , . . . , a )1 n
that represent the worlds in : that is,W � {w }j

proj ((a , . . . , a )) p (a , . . . , a , 0, a , a ).j 1 n 1 j�1 j�1 n

Hence, for , , whereas .i ( j proj (w ) p w proj (w ) p (0, . . . , 0)j i i j j
Now, as in the previous section, suppose that f is a function that gives

rise to a local inaccuracy measure I and a global inaccuracy measure G.
And suppose further that our epistemic agent’s global belief function is
represented by the vector . Then, given a world wi, there seem(a , . . . , a )1 n
to be two ways to measure the inaccuracy of the agent’s belief function
at wi that arise from combining I and G:



MEASURING INACCURACY 225

1. One might simply use G: that is, the inaccuracy of at(a , . . . , a )1 n
wi is G(w , (a , . . . , a )).i 1 n

2. Or, for any world wj with , one might take the inaccuracy ofi ( j
at wi to be(a , . . . , a )1 n

I({w }, w , a ) � G(proj (w ), proj ((a , . . . , a ))).j i j j i j 1 n

That is, one might take the local inaccuracy of the degree of cre-
dence in proposition at world wi and add it to the global{w }j
inaccuracy at wi of the ‘remainder’ of when world wj(a , . . . , a )1 n
is not considered: that is, geometrically speaking, one adds the
global inaccuracy at world wi of the belief function represented by
the projection of onto the subspace spanned by(a , . . . , a )1 n

.W � {w }j
As in the previous section, the conditions listed in section 5.1 do not rule
out the possibility that these two ways of measuring the inaccuracy of
the agent’s belief function at wi disagree, and, if they do, a dilemma might
arise for the agent. As before, we rule out the functions f that give rise
to this dilemma by laying down a further principle:

Separability of Global Inaccuracy. Suppose I is a legitimate local in-
accuracy measure. Then, by Local Normality and Dominance, there
is a strictly increasing function, such that� �f : � r � I(A, w, x) p0 0

. Further, by Local and Global Comparability,f (Fx (w) � xF) G(w,A
is a legitimate global inaccuracy measure. Then,b) p f (FFw � b FF)glo

what follows must hold: for all with ,w , w � W i ( ji j

G(w , (a , . . . , a )) p f (FFw � (a , . . . , a )FF)i 1 n i 1 n

p f (Fx (w ) � a F) � f (FFproj (w ) � proj ((a , . . . , a ))FF).{w } i j j i j 1 nj

As in the case of Agreement on Inaccuracy, this condition may be justified
by noting that, if it were to fail, two legitimate ways by which an agent
may determine her inaccuracy would lead to different results in at least
one situation; such disagreement would lead to a situation analogous to
that described in the discursive dilemma.

From Separability of Global Inaccuracy along with the four conditions
stated in section 5.1, local and global inaccuracy measures follow by the
following theorem:

Theorem 4. Separability of Global Inaccuracy and Minimum Inac-
curacy entail Agreement on Inaccuracy (which, in combination with
the assumptions of sec. 5.1, yields local and global inaccuracy mea-
sures, by theorem 3).



226 HANNES LEITGEB AND RICHARD PETTIGREW

The theorem is proved in the appendix again.

5.2.3. Agreement on Directed Urgency. Before we can state our final
dilemma—saving the best for last—we must restrict the class of legitimate
accuracy measures a little more than is done by the four conditions of
section 5.1. In particular, we demand Continuous Differentiability:

i) If is a legitimate local inaccuracy mea-I(A, w, x) p f(Fx (w) � xF)A
sure, then f is continuously differentiable on .�� 0

ii) If is a legitimate global inaccuracy mea-G(w, b) p g(FFw � b FF)glo
sure, then g is continuously differentiable on .�� 0

Again, in the presence of Local and Global Comparability, we can derive
i from ii and ii from i. However, again, we state both conditions to avoid
the mistaken impression that one is more fundamental than the other.
Having said this, we will give our argument for Continuous Differentia-
bility only in terms of i. This is not because we derive ii by inferring it
from i, rather, it is because the argument for ii is exactly analogous and
may be easily reconstructed from the argument for i.

Suppose that f is a function that gives rise to a local inaccuracy measure.
By Local Normality and Dominance, f is an increasing function. But it
is clear that should be also a continuous function of x, andf (Fx (w) � xF)A
thus f should be a continuous function. After all, if weref (Fx (w) � xF)A
discontinuous as a function of x at some particular , an agent’s�x � �0 0
accuracy could improve or deteriorate dramatically by an arbitrarily small
change to her degree of credence in the neighborhood of x0. Thus, f must
be continuous.

However, Continuous Differentiability demands something further. It
demands that f be continuously differentiable on . To justify this claim,�� 0
consider again the notion of expected local inaccuracy, which we intro-
duced in section 2. Given a belief function b, propositions , aA, E P W
degree of credence x in A, and a local inaccuracy measure I, we have
interpreted as the expected value of the inaccuracy of xLExp (I, A, E, x)b
by the lights of b, with respect to I and over the epistemically possible
worlds .w � E

Now, if I were a legitimate local inaccuracy function, the function
would provide not only a measure of the expected in-LExp (I, A, E, x)b

accuracy of x in A by the lights of b, with respect to I and over E. It
would provide also a means by which to measure the urgency and direc-
tion—in short, the directed urgency—with which an agent for whom E is
the set of epistemically possible worlds ought to change her degree of
credence in A by the lights of some belief function b. Clearly, this measure
would be provided by the derivative of with respect toLExp (I, A, E, x)b
x, were this derivative to exist. Wherever it is defined, the absolute value



MEASURING INACCURACY 227

of the function measures the rate at which the ex-d/dx LExp (I, A, E, x)b
pected inaccuracy of x by the lights of b is changing (the slope of the
tangent): thus, by the lights of b, it would be more urgent to change the
degree of credence r in proposition A than to change the degree of
credence s in the same proposition, just in case the absolute value of

evaluated at r were greater than the absolute valued/dx LExp (I, A, E, x)b
of the same derivative evaluated at s.13 Furthermore, if the sign of

evaluated at r differed from the sign of the samed/dx LExp (I, A, E, x)b
derivative evaluated at s, then the degree of credence r in proposition A
ought to be increased when the degree of credence s in the same prop-
osition ought to be decreased, or vice versa. Indeed, only the derivative
of could supply the agent with this sort of informa-LExp (I, A, E, x)b
tion. Thus, if I is to be a legitimate local inaccuracy measure, then

should be defined on since there ought to be a�d/dx LExp (I, A, E, x) �b 0
measure of directed urgency that an agent can use to determine a local
recommendation of where to go epistemically and, as it were, how quickly
she should move. If Ought-Can by itself does not support this claim
sufficiently, then one hopes it does so in conjunction with the geometrical
framework that we presuppose. In any case, it is straightforward to
show that, if is defined on , for every belief�d/dx LExp (I, A, E, x) �b 0
function b, propositions , and degree of credence x, thenA, E P W

must be differentiable on this domain as well. Thus, ifI(A, w, x)
, then f must be differentiable on . Of course,�I(A, w, x) p f (Fx (w) � xF) �A 0

this argument requires some amount of idealization since for all “real
world” cases in which an agent ought to determine the directed urgency
of a belief change, computing small real-valued differences rather than
infinitesimal ones should be sufficient.14 But if the agent wants to be certain
about this, whatever the level of precision required, then f ought to be
differentiable.

What’s more, just as we wish our local inaccuracy measure to be a
continuous function of the degree of credence whose inaccuracy it is mea-
suring, we would like our directed urgency measure to be a continuous
function of the degree of credence of the directed urgency of the change
to which it is measuring: we would not wish the urgency and direction
by which an agent should change her degree of credence to change dras-
tically after an arbitrarily small shift in her degree of credence. From this,

13. Gibbard (2008), too, interprets the derivative of the inaccuracy measure as a measure
of the urgency of updating.

14. Alan Hájek and Kenny Easwaran pointed this out to us.



228 HANNES LEITGEB AND RICHARD PETTIGREW

the local part of Continuous Differentiability—namely, i from above—
follows.15

As we mentioned above, our argument in favor of the global part of
Continuous Differentiability—namely, ii—is analogous to the argument
just given in favor of the local part. Focusing just on propositions A,
which are singleton propositions of the form for , we{w } j p 1, . . . , nj
say that if is a legitimate global inaccuracy mea-G(w, b) p g(FFw � b FF)glo
sure, then it also ought to give rise to a measure of the urgency with
which an agent must change her degree of belief in such a singleton
proposition. Given a belief function b, then the urgency to change the
degree of belief x in proposition for an agent with belief function b′{w }j
by the lights of b would be given by the absolute value of

d ′ ′ ′ ′GExp (G, E, (b ({w }), . . . , b ({w }), x, b ({w }), . . . , b ({w })),b 1 j�1 j�1 ndx

if this derivative were to exist (and which, given Continuous Differentia-
bility, indeed exists). And, as in the local case, its sign would indicate the
direction in which the change must occur. Thus, we interpret the deriv-
atives of both expected global inaccuracy and expected local inaccuracy
as measures of directed urgency. Granted this, we claim that these two
measures ought to agree on singleton propositions whenever the global
and local inaccuracy measures from which they arise are based on the
same strictly increasing and continuously differentiable function, f. This
is the content of Agreement on Directed Urgency.

Agreement on Directed Urgency. If andI(A, w, x) p f (Fx (w) � xF)A
are legitimate local and global inaccuracyG(w, b) p f (FFw � b FF)glo

measures, respectively, and if f is differentiable, then, for all belief
functions b and b′ and all worlds ,w � Wj

d
LExp (I, {w }, E, x)b jdx

d ′ ′ ′ ′p GExp (G, E, (b ({w }), . . . , b ({w }), x, b ({w }), . . . , b ({w })).b 1 j�1 j�1 ndx

Suppose this condition were not to hold. Then the agent who employed
these measures of inaccuracy, and the measures of directed urgency to
which they give rise, would be left with a dilemma. Where (1) the local

15. If this little transcendental argument is not convincing enough, here is a much more
mundane thought: let us restrict ourselves just to ‘geometrically nice’ local and global
inaccuracy measures. But in order to be ‘geometrically nice’, these measures will have to
be given by continuously differentiable functions.



MEASURING INACCURACY 229

measure of directed urgency for differed from (2) the global measure{w }j
of directed urgency for the coordinate , she would be unable to determinewj
the urgency with which she must update her belief in order to minimize
her expected inaccuracy and perhaps even the direction in which her
update should proceed; the two measures would give conflicting values
between which she could not choose in a principled way.

Together with Continuous Differentiability and the four conditions
stated in section 5.1, Agreement on Directed Urgency entails local and
global inaccuracy measures by means of the following theorem:

Theorem 5. The following two propositions are equivalent:

i) Function is strictly increasing and continuously dif-� �f : � r �0 0
ferentiable, , and, for all belief functions ,f (0) p 0 b � Bel(W )
all , and all , :�w � W a , . . . , a a , . . . , a � �j 1 j�1 j�1 n 0

d
b({w })f (Fx (w ) � xF)� i {w } ijdx w �Wi

d
p b({w })f (FFw � (a , . . . , a , x, a , . . . , a )FF).� i i 1 j�1 j�1 ndx w �Wi

ii) There is , such that, for all , .� 2l � � x � � f (x) p lx10 0

The proof is given in the appendix.

6. A Look Ahead to the Sequel and to Future Work. This concludes our
argument for local and global inaccuracy measures and, with it, our de-
fense of the four mathematically precise versions of the Accuracy norm
introduced in section 2, that is, the synchronic local and global versions
and the diachronic local and global versions, now supplied with the right
inaccuracy measures.

In the sequel to this article (Leitgeb and Pettigrew 2010), we investigate
the consequences of these norms. Before considering some open questions
about our approach, we report the results of that investigation:

1. From the synchronic local version of Accuracy, we derive Prob-
abilism.

2. From the diachronic local version of Accuracy, we derive Condi-
tionalization.

3. From a related, but much stronger norm, we derive Uniform Dis-
tribution.



230 HANNES LEITGEB AND RICHARD PETTIGREW

4. We show that, in the situations normally assumed to be covered
by Jeffrey’s updating rule, there is no updating rule that satisfies
the diachronic local version of Accuracy. However, the diachronic
global version can be satisfied. We show that Jeffrey’s updating
rule sometimes violates this diachronic version of the norm, unless
the so-called postulate of rigidity is not required by fiat, and we
describe the alternative updating rule that satisfies it.

We finish with five open questions that point toward future research:

• How can the approach taken in this article be extended to the case
of an infinite set of worlds; in particular, how can it be extended to
the case of a nondenumerable set of worlds? What role does count-
able additivity play in such extensions?

• Which of our conclusions depend essentially on our geometrical
background machinery being Euclidean? Which conclusions can be
drawn in a non-Euclidean setting?

• How can the theory be translated into a more abstract system of
axiomatic constraints on both belief update and the geometrical
background system (along the lines of Joyce but also Greaves and
Wallace [2006])? How robust are our results if, at the relevant places,
summing up of inaccuracies gets replaced by any numerical oper-
ation that satisfies some set of plausible constraints?

• How does the theory relate to theories of verisimilitude in which
relational truthlikeness is analyzed in terms of the expected degree
of truthlikeness (as in Niiniluoto’s [1987] theory of estimated truth-
likeness)?

• How does our theory of expected inaccuracy work in a framework
that permits partial beliefs in self-locating propositions? Such an
application would cover the well-known Sleeping Beauty problem.
See Kierland and Monton (1999) for a related attempt in which it
is assumed that quadratic inaccuracy measures provide the only le-
gitimate scoring rule.

• Could a variant of our theory of expected inaccuracy justify prob-
abilistic methods of judgment aggregation or amalgamation?

Appendix: Proofs of Theorems 3–5.

Here, we prove the three theorems used to argue for local and global
inaccuracy measures in section 5.



MEASURING INACCURACY 231

Proof of Theorem 3. It will suffice to show that the following two prop-
ositions are equivalent:

i′) Function g is strictly increasing and, for all belief functions, b, and
worlds ,w � Wi

n

2 2g(FFw � b FF ) p g(Fx (w ) � b({w })F ).�i glo {w } i jj
jp1

ii′) There is , such that, for all , .�l � � x � � g(x) p lx10 0

Suppose i′ and ii′ are equivalent. Then, if i, then satisfies i′1/2g(x) p f (x )
and thus ii′, so , which gives ii. Similarly, if ii, then1/2f (x ) p lx g(x) p

satisfies ii′ and thus i′, so f satisfies i. Thus, we will prove the1/2f (x )
equivalence of i′ and ii′.

First, we show that ii′ implies i′. Thus, suppose . Clearly, gg(x) p lx
is strictly increasing. Now suppose that b is a belief function and wi a
world; then,

2 2 2lFFw � b FF p l[b({w }) � . . . � (1 � b({w })) � . . . � b({w })]i glo 1 i n
n

2p l Fx (w ) � b({w })F .� {w } i jj
jp1

Thus, ii′ implies i′.
Now, we show that i′ implies ii′. Our strategy is to show that, from i′,

it follows that, for any , . Since, by i′, g�x, y � � g(x � y) p g(x) � g( y)0
is also strictly increasing, ii′ follows by Cauchy’s classical result that all
monotone additive functions on are linear on .16� �

Thus, suppose . Then let ,�x, y � � b({w }) p . . . p b({w }) p 00 1 n�2
, and . Then, by i′,1/2 1/2b({w }) p x b({w }) p 1 � yn�1 n

2 2 2� � � �g(FFw � (0, . . . , 0, x, 1 � y)FF ) p g(F0 � xF ) � g(F1 � (1 � y)F ).n

From this, we have , as required.g(x � y) p g(x) � g( y)

Proof of Theorem 4. Consider the following statements:

16. In fact, we need a slightly different version, which states that all monotone additive
functions on are linear on . Suppose is additive on . Then define� � � � �� � g : � r � �0 0 0 0 0

as follows: if and � if . Then it is easy to show′ ′g : � r � g (x) p g(x) 0 ≤ x g(�x) x ! 0
that g′ is additive and monotone on if g is additive and monotone on . Thus, g′�� �0
satisfies the hypotheses of Cauchy’s result. For the proof of Cauchy’s result, see Aczél
and Dhombres (1989).



232 HANNES LEITGEB AND RICHARD PETTIGREW

i) Function f is strictly increasing, , and, for all belief func-f (0) p 0
tions b and all worlds and such that ,w w i ( ji j

f (FFw � b FF)i glo

p f (Fx (w ) � b({w })F) � f (FFproj (w ) � proj ((a , . . . , a ))FF).{w } i j j i j 1 nj

ii) Function f is strictly increasing and, for all belief functions, b, and
worlds ,wi

n

f (FFw � b FF) p f (Fx (w ) � b({w })F).�i glo {w } i jj
jp1

We must show that i entails ii. Suppose i holds. Then, by repeatedly
separating the local inaccuracy measure from the global one, we obtain:

f (FFw � b FF) p f (Fx (w ) � b({w })F)[� ]i glo {w } i jj
j(i

� f (FFw � (0, . . . , 0, a , 0, . . . , 0)FF)i i

p f (Fx (w ) � b({w })F) � f (Fx (w ) � b({w })F)[� ]{w } i j {w } i ij i
j(i

n

p f (Fx (w ) � b({w })F).� {w } i jj
jp1

Thus, ii, as required.

Proof of Theorem 5. For reasons analogous to those given in the proof
of theorem 3, it will suffice to show that the following two statements are
equivalent:

i′) Function is strictly increasing and differentiable,� �g : � r �0 0
, and, for all belief functions and :g(0) p 0 b � Bel(W ) w � Wj

d
2b({w })g(Fx (w ) � xF )� i {w } ijdx w �Wi

d
2p b({w })g(FFw � (a , . . . , a , x, a , . . . , a )FF ).� i i 1 j�1 j�1 ndx w �Wi

ii′) There is , such that, for all , .�l � � x � � g(x) p lx10 0

First, we prove that ii′ implies i′. If and , then g isl � � g(x) p lx10
certainly strictly increasing and differentiable: indeed, . Thus, by′g (x) p l



MEASURING INACCURACY 233

straightforward differentiation and direct calculation, if a , . . . , a ,1 j�1
, then�. . . , a � �n 0

d
2b({w })g(Fx (w ) � xF )� i {w } ijdx w �Wi

p b({w })[�2lx (w ) � 2lx]� i {w } ij
w �Wi

d
2p b({w })g(FFw � (a , . . . , a , x, a , . . . , a )FF ),� i i 1 j�1 j�1 ndx w �Wi

as required.
Next, we prove that i′ implies ii′. Our strategy will be to show that i′

implies that g′ is constant. This will suffice since, by i′, g is increasing, so
. We do this in two stages. First, we prove that, on the assumptionl 1 0

of i′, on , and then we prove that, on the assumption′ ′ �g (1 � x) p g (1) � 0
of i′, on .′ ′ �g (x) p g (x � 1) � 0

Thus, suppose i′ holds. We wish to show that on .′ ′ �g (x � 1) p g (1) � 0
So, suppose . Then let , , and .� 1/2a � � j p 1 a p a a p . . . p a p 00 2 3 n
Then, by i′,

d
2b({w })g(Fx (w ) � xF )� i {w } i1dx w �Wi

d
2�p b({w })g(FFw � (x, a, 0, . . . , 0)FF ).� i idx w �Wi

But

d
2b({w })g(Fx (w ) � xF )� i {w } i1dx w �Wi

d
2 2 2p [b({w })g((1 � x) ) � b({w })g(x ) � . . . � b({w })g(x )]1 2 ndx

′ 2 ′ 2 ′ 2p �2(1 � x)b({w })g ((1 � x) ) � 2xb({w })g (x ) � . . . � 2xb({w })g (x ),1 2 n

and

d
2�b({w })g(FFw � (x, a, 0, . . . , 0)FF )� i idx w �Wi

d
2 2 2�p [b({w })g((1 � x) � a) � b({w })g(x � (1 � a) )1 2dx

2 2� b({w })g(x � a � 1) � . . . � b({w })g(x � a � 1)]3 n



234 HANNES LEITGEB AND RICHARD PETTIGREW

′ 2 ′ 2 2�p �2(1 � x)b({w })g ((1 � x) � a) � 2xb({w })g (x � (1 � a) )1 2
′ 2 ′ 2� 2xb({w })g (x � a � 1) � . . . � 2xb({w })g (x � a � 1).3 n

Thus, taking , we havex p 0

′ ′�2b({w })g (1) p �2b({w })g (1 � a),1 1

which gives , as required.′ ′g (1) p g (1 � a)
Now we wish to show that on . Thus, suppose′ ′ �g (x) p g (x � 1) � 0

. Then let b be a belief function in such that�a � � Bel(W ) b({w }) p 10 2
and , and let .b({w }) p b({w }) p . . . p b({w }) p 0 a p . . . p a p 01 3 n 2 n
Then, by i′,

d d
2 2g(Fx (w ) � xF ) p g(FFw � (x, 0, 0, . . . , 0)FF ),{w } 2 21dx dx

which gives for such that , and thus′ 2 ′ 2 �g (x ) p g (x � 1) x � � x ( 00
for such that since x2 is bijective between′ ′ �g (x) p g (x � 1) x � � x ( 00

and . Thus, since g′ is continuous on , for� � � ′ ′� � � g (x) p g (x � 1)0 0 0
. Thus, for all , , so g′ is constant� � ′ ′ ′x � � x � � g (x) p g (x � 1) p g (1)0 0

on . This completes our proof.�� 0

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