The Problem of Measure Sensitivity Redux∗

Peter Brössel
Institut für Philosophie II

Universität Bochum
E-Mail: peter.broessel@rub.de

∗I am very grateful to four exceedingly helpful referees of this journal, especially for spot-
ting a mistake in one of my proofs and encouraging me to discuss, among other issues, the
ramifications of the problem of measure sensitivity in more detail in Section 4. In addition,
I would like to thank Ralf Busse, Vincenzo Crupi, Anna-Maria A. Eder, Branden Fitelson,
Franz Huber, James M. Joyce, and Hannes Leitgeb for their invaluable feedback on various
versions of this paper. I also want to emphasize my gratitude to James M. Joyce for the
permission to cite and quote his unpublished manuscript “On the Plurality of Probabilist
Measures of Evidential Relevance.” Finally, I also would like to thank Robert Lehnert and
Ben Young for proofreading the manuscript.

1



Abstract

Fitelson (1999) demonstrates that the validity of various arguments
within Bayesian confirmation theory depends on which confirmation mea-
sure is adopted. The present paper adds to the results set out in Fitelson
(1999), expanding on them in two principal respects. First, it considers
more confirmation measures. Second, it shows that there are important
arguments within Bayesian confirmation theory and that there is no con-
firmation measure that renders them all valid. Finally, the paper reviews
the ramifications that this ”strengthened problem of measure sensitivity”
has for Bayesian confirmation theory and discusses whether it points at
pluralism about notions of confirmation.

2



Contents

1 Introduction 3

2 The Plurality of Bayesian Measures of Confirmation 4

3 The Problem of Measure Sensitivity 7
3.1 Earmann (1992) and The Problem of Irrelevant Conjunction . . 9
3.2 Horwich (1982) and the Paradox of the Ravens . . . . . . . . . . 11
3.3 Fitelson (2006), Confirmation, and Inductive Logic . . . . . . . . 12
3.4 Huber (2005) and the Assessment of Theories . . . . . . . . . . . 13
3.5 The Strengthened Problem of Measure Sensitivity . . . . . . . . . 15

4 The Problem of Measure Sensitivity as an Argument for Plu-
ralism? 17

5 Summary 18

1 Introduction

A plurality of Bayesian confirmation measures have been proposed as a means
of providing a theory of confirmation. However, this plurality of measures of
(incremental) confirmation is now seen as bringing about one of the biggest
challenges with which Bayesian confirmation theory is faced: the problem of
measure sensitivity. Eells and Fitelson, for example, write that:

A plethora of non-equivalent measures of evidential support (or con-
firmation) have been proposed and defended in the philosophical lit-
erature. This plurality of measures of support is problematic, since it
affects a great many arguments surrounding Bayesian confirmation
theory (and inductive logic, generally). (Eells and Fitelson 2002,
129, amendments in brackets in original)

To date, the most forceful display of the problem of measure sensitivity is one
provided by Fitelson (1999). In his paper, Fitelson shows that the validity
of various central arguments within Bayesian confirmation theory depends on
which Bayesian measure of confirmation is adopted. The present paper adds
to the results set out in Fitelson (1999), expanding on them in two principal
respects. First, this paper considers more confirmation measures; in Fitelson
(1999) only a proper subclass of those confirmation measures considered here is
taken into account. Second, this paper shows that the problem of measure sen-
sitivity actually runs deeper and is far more severe than it is noted in Fitelson
(1999).

The following section introduces the plurality of confirmation measures of Bayesian
confirmation theory. Section 3 introduces the problem of measure sensitivity. In

3



particular, this section considers four prominent arguments in Bayesian confir-
mation theory and shows that there is no confirmation measure which validates
all of them. Subsections 3.1 and 3.2 introduce arguments already considered
in Fitelson (1999), while Subsections 3.3 and 3.4 consider two new arguments
from Huber (2005) and Fitelson (2006). Section 4 discusses the ramifications
that the strengthened problem of measure sensitivity has for Bayesian confir-
mation theory.

2 The Plurality of Bayesian Measures of Con-
firmation

Providing a theory of confirmation is often taken to be one of the pivotal tasks of
Bayesian philosophy of science and epistemology, since providing such a theory
is considered to be central for understanding various aspects of scientific reason-
ing. Intuitively, how worthy of belief certain theories or hypotheses are depends
on how well confirmed they are, and which experiments scientists should conduct
depends on whether or not the possible experimental outcomes would confirm
the theory or hypotheses in question. Others take Bayesian confirmation theory
to govern “the steps in the arguments by which our current opinions are to be
justified by the evidence that we have already assembled” (Lange 1999, 295).
However, whether a Bayesian theory of confirmation can meet the expectations
and hopes placed in it by Bayesian philosophers of science and epistemologists
cannot be investigated until a complete theory of confirmation has been pro-
vided.

Bayesian confirmation theory holds that confirmation is best understood as a
relation between evidence E and hypothesis H, relative to some probability
measure Pr, and background assumptions B. A complete Bayesian theory of
confirmation requires that one provide an interpretation of the probability mea-
sure Pr and say when and to what degree a hypothesis H is confirmed by the
evidence E relative to the background assumptions B (henceforth mention of
the background assumptions is suppressed). The present paper will not address
the former task, i.e. answering the question how one should interpret the prob-
abilities (evidential probabilities, subjective probabilities, logical probabilities,
etc.) relevant in the context of Bayesian confirmation theory. Instead, it focuses
on the latter task of saying when and to what degree a hypothesis is confirmed
by the evidence relative to some probability function. To this day, no confirma-
tion measure is widely enough accepted that it can be considered to be part of
the canon of Bayesian confirmation theory. However, two points are now com-
monly agreed upon. First, almost all Bayesian confirmation theorists argue that
the only adequate (or at least interesting) notion of confirmation is incremental
confirmation. Evidence E is said to incrementally confirm hypothesis H relative
to some background assumptions B and probability function Pr if and only if
Pr(H|E ∩ B) > Pr(H|B). Second, most Bayesian epistemologists lean explic-

4



itly or implicitly towards a monistic position regarding confirmation measures
(Christensen 1999; Eells and Fitelson 2002; Fitelson 2001; Milne 1996). Monism
of confirmation measures is the position that there is only one adequate confir-
mation measure (up to ordinal equivalence). For most Bayesian confirmation
theorists the only question that remains to be answered is which incremental
confirmation measure to accept.

While Bayesian confirmation theorists are in agreement about when a hypothesis
is incrementally confirmed by the evidence, they disagree about how the quan-
titative notion of degree of incremental confirmation provided by some evidence
is to be explicated. To this end, various probabilistic measures of incremental
confirmation have been proposed. Crupi, Tentori, and Gonzalez (2007), Ear-
man (1992, ch. 5), and Fitelson (1999, 2001) provide excellent overviews of the
proposed measures of incremental confirmation and the suggested conditions of
adequacy for an explication of “degree of incremental confirmation.” As already
said, none of these confirmation measures can be considered to be part of the
canon of Bayesian confirmation theory. Nevertheless, there are two requirements
that any Bayesian confirmation theorist agrees with:

Requirement 2.1 (Demarcation). A function c(·, ·) is a measure of (incremen-
tal) confirmation relative to probability function Pr only if there is some (marker
or neutral value) r ∈ R so that:
For all hypotheses H and evidence E if 1 > Pr(H|E) > 0 and Pr(E) > 0, then

c(H,E)



> r, Pr(H|E) > Pr(H)
= r, Pr(H|E) = Pr(H)
< r, Pr(H|E) < Pr(H).

This first requirement on confirmation measures is taken from Fitelson (2001)1

and requires that they should be measures of incremental confirmation. In par-
ticular, it requires that if the evidence does not guarantee that the hypothesis
in question is true or false, then there is a marker or neutral value r such that
the degree of confirmation of hypothesis H by evidence E is higher than the
marker r if and only if H and E are positively correlated with respect to prob-
ability function Pr. The degree of confirmation of hypothesis H by evidence E
is lower than the marker r if and only if H and E are negatively correlated with
respect to probability function Pr. The degree of confirmation of hypothesis H
by evidence E equals the marker r if and only if H and E are probabilistically
independent with respect to probability function Pr.

In addition, confirmation measures should satisfy what Joyce (2003) calls the
Weak Law of Likelihood requirement which “encapsulates a minimal form of
Bayesianism to which all parties can agree” Joyce (2003, § 3).

1The name “Demarcation” has been adopted based on Crupi, Tentori, and Gonzalez (2007).

5



Requirement 2.2 (Weak Law of Likelihood). If Pr(E|H1) > Pr(E|H2) and
Pr(E|¬H1) < Pr(E|¬H2), then c(H1,E) > c(H2,E).

Joyce (2004, § 3) even contends that “the weak likelihood principle must be
an integral part of any account of evidential relevance that deserves the title
‘Bayesian.’ To deny it is to misunderstand the central message of Bayes’ The-
orem for questions of evidence: namely, that hypotheses are confirmed by data
they predict.”2 This second requirement holds that when the evidence E is
more probable in the light of hypothesis H1 than in the light of H2 and if the
evidence E is also less probable under the assumption that H1 is false than it
is under the assumption that H2 is false with respect to probability function Pr
then E confirms H1 more than it confirms H2.

Among the most prominent Bayesian confirmation measures that satisfy both
requirements are the following:

(Carnap 1962)
d(H,E) = Pr(H|E) − Pr(H)

if Pr(E) > 0

(Carnap 1962)
r(H,E) = Pr(H ∩E) − Pr(H) × Pr(E)

(Christensen 1999, Joyce 1999)

S(H,E) = Pr(H|E) − Pr(H|¬E)
2In the light of Joyce’s strong claims about the weak law of likelihood it is interesting to

note that Crupi, Festa, and Buttasi (2010) are able to show that the weak law of likelihood is
logically independent of the demarcation requirement (Requirement 2.1). It is also interesting
to note that minimally strengthening this “integral part of any account of evidential relevance”
leads to a requirement that not all of the measures of confirmation discussed in the literature
are able to satisfy. The strengthened weak law of likelihood I have in mind is this:

Requirement 2.3 (Strengthened Weak Law of Likelihood). If 0 < Pr(E|H1) ≥ Pr(E|H2)
and Pr(E|¬H1) < Pr(E|¬H2), or if Pr(E|H1) > Pr(E|H2) and 0 < Pr(E|¬H1) ≤ Pr(E|¬H2),
then c(H1, E) > c(H2, E).

Joyce’s Weak Law of Likelihood requires that if the evidence E is strictly more probable
in the light of hypothesis H1 than in the light of H2 and if it is strictly less probable in the
light of ¬H1 than in the light of ¬H2, then E confirms H1 more than H2. The Strengthened
Weak Law of Likelihood slightly weakens the antecedent of Joyce’s requirement and requires
that E confirms H1 more than H2, if the evidence E is strictly more probable in the light
of hypothesis H1 than in the light of H2 and if E is strictly less or equally probable in the
light of ¬H1 than ¬H2 or, if the evidence E is strictly more or equally probable in the light
of hypothesis H1 than in the light of H2 and if E is strictly less probable in the light of ¬H1
than ¬H2. This slightly weakened antecedent seems to suffice to underscore the claim that H1
predicts E better than H2 and thus should be confirmed to a higher degree of confirmation.
Confirmation measures discussed in the literature that do not satisfy the strengthened weak
law of likelihood are the Z measure by Crupi, Tentori, and Gonzalez (2007), the r measure
of confirmation defended in Milne (1996), and the M measure of confirmation by Mortimer
(1988). These confirmation measures are introduced in detail below.

6



if 1 > Pr(E) > 0

(Crupi, Tentori, and Gonzalez 2007)

Z(H,E) =




d(H,E)
1−Pr(H) if Pr(H|E) ≥ Pr(H) > 0
d(H,E)

1−Pr(¬H) if Pr(H|E) < Pr(H)
1 if Pr(H) = 0

(Fitelson 2001, Good 1960)

l(H,E) =




log
[

Pr(E|H)
Pr(E|¬H)

]
if Pr(E|H) > 0 and Pr(E|¬H) > 0

∞ if Pr(E) > 0 and Pr(E|¬H) = 0
−∞ if Pr(E) = 0 or Pr(E|H) = 0

(Kemeny and Oppenheim 1952)

F(H,E) =




Pr(E|H) − Pr(E|¬H)
Pr(E|H) + Pr(E|¬H)

if Pr(E|H) > 0 and Pr(E|¬H) > 0

1 if Pr(E) > 0 and Pr(¬H) = 0
−1 if Pr(E) = 0 or Pr(H) = 0

(Milne 1996)

r(H,E) =


log

[
Pr(H|E)
Pr(H)

]
if Pr(H|E) > 0

−∞ if Pr(H|E) = 0

(Mortimer 1988)
M(H,E) = Pr(E|H) − Pr(E)

if Pr(H) > 0

(Nozick 1981)
N(H,E) = Pr(E|H) − Pr(E|¬H)

if 1 > Pr(H) > 0

As already noted, this plurality of Bayesian measures of (incremental) confirma-
tion is generally taken to bring about one of the biggest challenges for Bayesian
confirmation theory: the problem of measure sensitivity. This problem is the
subject of the next section.

3 The Problem of Measure Sensitivity

In Fitelson’s own words, the problem of measure sensitivity is as follows:

7



Below, I will show that seven well-known arguments surrounding
contemporary Bayesian confirmation theory are sensitive to choice of
measure. I will argue that this exposes a weakness in the theoretical
foundation of Bayesian confirmation theory which must be shored-
up. I call this problem the problem of measure sensitivity.

This section of the paper demonstrates how serious the problem is, mainly by
following Fitelson’s lead. The aim is to show that the validity of different but
equally important arguments within confirmation theory depends on which mea-
sure of confirmation is adopted. However, the results displayed here are in two
respects more powerful than those achieved in Fitelson (1999). First, this pa-
per considers more confirmation measures. In Fitelson’s (1999) considerations
only a proper subclass of the confirmation measures introduced in Section 2
is taken into account. In particular, Fitelson (1999) considers only the confir-
mation measures d, r, r, and l. Second, this paper shows that the problem of
measure sensitivity actually runs deeper and is far more severe than is noted in
Fitelson (1999). Fitelson (1999) draws only the following, relatively tentative,
conclusion:

[M]any well-known arguments in quantitative Bayesian confirmation
theory are valid only if the difference measure d is to be preferred
over other relevance measures (at least, in the confirmational con-
texts in question). I have also shown that there are compelling
reasons to prefer d over the log-ratio measure r. Unfortunately, like
Rosenkrantz (1981), I have found no compelling reasons offered in
the literature to prefer d over the log-likelihood ratio measure l.

According to this, it is possible to adopt the difference measures of confirmation
and no argument of confirmation theory considered in Fitelson (1999) would
be affected. The only problem Fitelson has with adopting d is that there is
no compelling argument against the l measure of confirmation. However, the
arguments Fitelson (1999) reviews are central to Bayesian confirmation theory
insofar that almost all Bayesians are at least tempted to argue that validating
these arguments is a desideratum that any adequate measure of confirmation
should satisfy. If only one confirmation measure validates all these arguments,
this is a reason to believe that it is the correct measure of confirmation.

However, the following subsections consider four prominent arguments in Bayesian
confirmation theory and show that there is no confirmation measure which vali-
dates all of them. This strengthens the results achieved in Fitelson (1999) since
it shows that there cannot be a single confirmation measure which validates all
of those prominent arguments. Hence, it is impossible to argue that validating
these four arguments is a desideratum that any adequate measure of confir-
mation should satisfy. Subsections 3.1 and 3.2 introduce arguments already
considered in Fitelson (1999), while Subsections 3.3 and 3.4 consider two new
arguments from Huber (2005) and Fitelson (2006).

8



3.1 Earmann (1992) and The Problem of Irrelevant Con-
junction

First, let us look at what is called the problem of irrelevant conjunction.3 There
are two (closely related) formulations of this problem. The first is the traditional
formulation and the second is Fitelson’s (2002) version. The traditional formu-
lation of the problem is the following:

The Problem of Irrelevant Conjunction 1. If H implies E, then E confirms
H ∩ I for arbitrary I provided Pr(H ∩ I) > 0.
Fitelson (2002) introduces a slightly different formulation of the problem. Ac-
cording to him, the problem of irrelevant conjunction is the following:

The Problem of Irrelevant Conjunction 2. If E confirms H, and I is
confirmationally irrelevant to H, E, and H ∩E, then E also confirms H ∩ I.
Both formulations are statements of the problem of irrelevant conjunction be-
cause one can tack a confirmationally irrelevant conjunct I (the irrelevant con-
junct) to the hypothesis H (the relevant conjunct) and still get the result that
H ∩ I (the irrelevant conjunction) is confirmed.

This first formulation of the problem was first discussed in Hempel (1945) in con-
nection with the hypothetico-deductive method of confirmation. If one follows
Hempel (1945) in adopting the special consequence condition, which requires
that “[i]f an observation report confirms a hypothesis H, then it also confirms
every consequence of H”, one must reject the first formulation of the problem.
The reason is that taken together they imply that the evidence E confirms even
the arbitrarily chosen conjunct I. This would trivialize the notion of confirma-
tion: every piece of evidence E would confirm every hypothesis I.

The standard Bayesian conception of qualitative confirmation validates the two
formulations of the problem of irrelevant conjunction, but it does not validate
Hempel’s special consequence condition. Accordingly, the standard Bayesian
conception of qualitative confirmation does not threaten to trivialize the no-
tion of confirmation. Nevertheless, Bayesian confirmation theorists typically
consider the result that their qualitative notion of confirmation validates both
formulations of the problem to be counterintuitive. In particular, in the litera-
ture one can find examples which seem to strongly support the claim that this
result is counterintuitive. For example, Glymour (1980) writes that:

Nor are we greatly impressed by a theory of evidence which claims,
for instance, that the Stark effect confirms the conjunction of quan-
tum mechanics and the principle that the president of the Church
of Jesus Christ of Latter Day Saints is infallible, all the while reas-
suring us that the Stark effect does not confirm the latter principle
alone. (Glymour 1980, 31)

3I am grateful to Vincenzo Crupi for a helpful discussion about the Problem of Irrelevant
Conjunction in general and Requirements 3.1–3.3 in particular.

9



However, Earman (1992) argues that examples like this are not too bad, as long
as according to any reasonable confirmation measure the irrelevant conjunction
H ∩ I is less confirmed than the relevant conjunct H. This is supposed to
“soften the impact” (as Fitelson (2002) puts it) of this counterintuitive result
regarding the qualitative conception of incremental confirmation. Thus, aiming
at cushioning the impact of the problem in its traditional formulation, Earman
(1992) requires that any reasonable confirmation measure satisfy the following
condition:

Requirement 3.1 (Irrelevant Conjunction I).
If c is a reasonable confirmation measure, H |= E, and I is logically independent
of H, E, and H ∩E, then:

c(H,E) > c(H ∩ I,E).

Hawthorne and Fitelson (2004) suggest an alternative and in their view better-
suited requirement to “cushion” the impact of the problem of irrelevant con-
junction in its formulation by Fitelson (2002).

Requirement 3.2 (Irrelevant Conjunction II).
If c is a reasonable confirmation measure, Pr(H|E) > Pr(H), Pr(E|H ∩ I) =
Pr(E|H), and Pr(I|H) 6= 1, then:

c(H,E) > c(H ∩ I,E).

It is an interesting question which confirmation measures of Section 2 satisfy
these requirements. The following observation provides a list of which confir-
mation measures satisfy Requirements 3.1 and 3.2 and which do not.

Observation 1. Confirmation measures d, r, S, Z, l, F , and N satisfy Re-
quirements 3.1 and 3.2. r and M do not satisfy Requirements 3.1 and 3.2.4

One might be tempted to argue that Observation 1 shows that: a) the ratio mea-
sure r and the Mortimer measure M are not adequate measures of confirmation;
b) all other Bayesian measures of confirmation are candidates for being the one
true measure of confirmation. This is because they “soften the impact” of the
counterintuitive result insofar as they can be used to explain why philosophers
have the pre-explicatory intuition that irrelevant conjunctions are or should not
be confirmed. The explanation is that they are confirmed to such a minimal
degree that pre-explicatory many philosophers and scientists do not have the
intuition that they are confirmed. Thus Bayesian confirmation theory can ex-
plain (away) these intuitions given they reject the ratio measure of confirmation.

However, inferring from Observation 1 that all other Bayesian measures of con-
firmation except r and M are candidates for being the one true measure of
confirmation would be hasty. The above Requirements 3.1 and 3.2 apply only

4Proofs for all observations can be found on the author’s website.

10



in the case of confirmation. Accordingly, it is an interesting question whether
a similar requirement holds for the case of disconfirmation. If one accepts
Hawthorne’s and Fitelson’s Requirement 3.2 for the case of confirmation, one
might be tempted to accept an analogous requirement for the case of disconfir-
mation. More specifically, one might be tempted to argue that it should hold
in general that if E confirms or disconfirms hypothesis H, then E confirms or
disconfirms H to a higher degree than it confirms the conjunction H ∩ I with
the confirmationally irrelevant conjunction I. The following requirement is a
slightly weaker version of a requirement discussed (but ultimately not endorsed)
in Crupi and Tentori (2010).

Requirement 3.3 (Irrelevant Conjunction III5).
If c is a reasonable confirmation measure with marker or neutral value r, Pr(H|E) <
Pr(H), and Pr(E|H ∩ I) = Pr(E|H) > 0 and Pr(I|H) 6= 1, then:

|c(H,E) −r| > |c(H ∩ I,E) −r|.

According to this requirement, “[i]f E has a positive [or negative] confirmatory
impact on H and I is a confirmationally irrelevant conjunct to H with regard
to E, then E will have the same kind of impact (namely, positive [or negative])
on H ∩ I but to a minor extent” (Crupi and Tentori 2010, 4)

Observation 2. Confirmation measures d, r, S, l, F , and N satisfy Require-
ment 3.3. Z, r, and M do not satisfy Requirement 3.3.

It interesting to note that the Z confirmation measure is the only confirmation
measure that satisfies requirement 3.2 but that does not satisfy the analogous
requirement for the case of disconfirmation. The important question is whether
Observation 2 can be taken to show that: a) the Z measure, the ratio measure
r, and the Mortimer measure M are not adequate measures of confirmation;
and b) all other Bayesian measures of confirmation are candidates for being
the one true measure of confirmation. I do not think so. The problem with
this interpretation of the observations is that equally reasonable considerations
exclude other confirmation measures.

3.2 Horwich (1982) and the Paradox of the Ravens

Another requirement on confirmation measures arises from arguments surround-
ing the well-known paradox of the ravens (Hempel 1945). This paradox is fueled
by the following inconsistent intuitions:

Ravens Paradox 1. (i) a non-black non-raven (e.g., a white shoe) does not con-
firm the hypotheses “All Ravens are black.” (ii) a non-black non-raven confirms

5In particular, this requirement is logically weaker than the one considered in Crupi and
Tentori (2010) since it is restricted to cases of (dis-)confirmation where the evidence does not
falsify H and H ∩I, i.e., if Pr(E|H ∩I) = Pr(E|H) > 0 (without this caveat the requirement
would be satisfied only by the measures d, r, S, and N). It is also interesting to note that
this requirement is implied by the strengthened Weak Law of Likelihood (Requirement 2.3)
discussed in Footnote 2 of Section 2.

11



the hypotheses “All non-black things are non-ravens.” (iii) Logically equiva-
lent hypotheses are equally confirmed by a given piece of evidence (Equivalence
Condition).

Now let H1 be the hypothesis “All ravens are black” and let H2 be the hypoth-
esis “All non-black things are non-ravens.” Obviously, H1 and H2 are logically
equivalent. According to (i), non-black non-ravens do not confirm H1. Hence,
by the Equivalence Condition (iii) non-black non-ravens do not confirm H2.
This contradicts intuition (ii), according to which non-black non-ravens should
confirm H2 and black ravens should not confirm it.

This shows that these intuitions are logically inconsistent. The standard Bayesian
solution of the ravens paradox is similar to the solutions suggested for the prob-
lem of irrelevant conjunction. In particular, the idea is that the observation
of a non-black non-raven can confirm the hypothesis “All Ravens are black”
but does so less strongly than the observation of black ravens. This “softens
the impact” of the counterintuitive result that the observation of white shoes
confirms “All Ravens are black” at all. Horwich (1982) relies on the following
requirement to show that Bayesian confirmation theory can soften the impact
of the ravens paradox.

Requirement 3.4 (The Ravens Paradox).
If c is a reasonable confirmation measure and Pr(H|E1) > Pr(H|E2) then:

c(H|E1) > c(H|E2).

Observation 3. Confirmation measures d, Z, l, F, r satisfy Requirement 3.4.
r, S, M, and N do not satisfy Requirement 3.4.

All confirmation measures that are in agreement with Requirement 3.4 can
explain why philosophers have the pre-explicatory intuition that a non-black
non-raven does not confirm “All ravens are black.” The explanation is that
a non-black non-raven confirms this hypothesis to such a minimal degree that
philosophers and scientists do not have the pre-explicatory intuition that it con-
firms it at all. However, the problem of irrelevant conjunction and the paradox
of the ravens together do not show that Bayesians can exclude r, S, r, M, and
N and that all other measures are reasonable confirmation measures. Again,
other equally reasonable considerations exclude other measures.

3.3 Fitelson (2006), Confirmation, and Inductive Logic

Other arguments in Bayesian confirmation theory deal with the question of what
is the purpose of Bayesian confirmation theory. For example, in his 2006 pa-
per “Logical Foundations of Evidential Support,” Fitelson stresses that, since
Carnap (1962), the concept of confirmation has been closely linked to that of
inductive logic. Carnap’s project was to provide a theory of confirmation or in-
ductive logic that can be considered as an extension of deductive logic (Fitelson

12



2006, Crupi, Tentori, and Gonzalez 2007). Fitelson (2006, 501) argues that this
requires that confirmation (i.e., partial entailment, in Carnap’s view) must be
analogous to deductive entailment regarding certain aspects. In particular, the
evaluation of inductive arguments should retain the idea that there is no better
inductive argument for some proposition or hypothesis than a deductive argu-
ment. Similarly, there is no worse argument than an argument with premises
which imply the falsity of the conclusion. In accordance with this consideration,
Fitelson (2006) suggests that any theory of confirmation that is supposed to be
an account of inductive logic has to satisfy the requirement of Logicality.6

Requirement 3.5 (Logicality).
If c is an adequate confirmation measure, then (i) if E implies H, then c(H,E) =
max, and (ii) if E implies ¬H, then c(H,E) = min. (Fitelson 2006)

Observation 4. Confirmation measures Z, l, and F satisfy Requirement 3.5.
d, r, S, r, M, and N do not satisfy Requirement 3.5.

The confirmation measures d, r, S, r, M, and N fail to assign the maximum
value if H is implied by E. r, d, S, M, and N also fail to assign the minimum
value if hypothesis H is refuted by the evidence E.

Again, this observation together with the preceding one can be taken to show
that Z, l, and F are the only candidates for being an adequate confirmation
measure. However, one can find another important argument within Bayesian
confirmation theory which is not valid according to these confirmation measures.

3.4 Huber (2005) and the Assessment of Theories

Huber (2007) focuses on another, more venerable question that arises in the
context of Bayesian confirmation theory and inductive logic: Hume’s problem
of induction. In his 2007 encyclopedia entry, “Confirmation and Induction”,
Huber writes:

Let us grant that Bayesian confirmation theory adequately expli-
cates the concept of confirmation. [. . . ] The question remains what
happened to Hume’s problem of the justification of induction. We
know – by definition – that the conclusion of an inductively strong
argument is well-confirmed by its premises. But does that also jus-
tify our acceptance of that conclusion? Don’t we first have to justify
our definition of confirmation before we can use it to justify our
inductive inferences? (Huber 2007, § 7)

6Crupi, Tentori, and Gonzalez (2007) discuss another interesting requirement whose sat-
isfaction they take to be necessary for considering confirmation and inductive logic as an
extension of deductive logic. However, for present purposes it is sufficient to consider the rel-
atively weak Logicality requirement. In addition, Crupi, Tentori, and Gonzalez (2007) agree
in principle with the latter requirement. Accordingly, it is safe to assume that the Logi-
cality requirement is a minimal requirement that any confirmation measure must satisfy for
considering confirmation theory an extension of deductive logic.

13



In an earlier article from 2005, “What Is the Point of Confirmation?”, Huber
suggests an answer to this question and the answer Huber gives should be good
news for Bayesian confirmation theory:

An examination of two popular measures of incremental confirma-
tion suggests the view that one should stick to incrementally well
confirmed theories, because incremental confirmation takes one to
(the most) informative (among all) true theories. However, incre-
mental confirmation does not further this goal in general. (Huber
2005, 1146)

That some confirmation measures lead one to accept the most informative
among all true theories is truly good news. However, the last sentence adds
to the problem at hand: the plurality of confirmation measures and the prob-
lem of measure sensitivity. Again, one interesting and valuable argument in fa-
vor of Bayesian confirmation theory depends on the choice of the confirmation
measure. Skipping the details, Huber (2005) proposes a requirement dubbed
Continuity on confirmation measures so that they take one to “(the most) in-
formative (among all) true theories”. Continuity requires that “any surplus in
informativeness succeeds, if the difference in plausibility is small enough” (Hu-
ber 2005, 1156). Following philosophers such as Carnap and Bar-Hillel (1952),
Hempel (1960), and Popper (1954), Huber (2005, 2008) suggests measuring the
amount of information provided by a hypothesis with the help of the measure
i(H) = Pr(¬H). The idea of this measure of information is that a hypothe-
sis is the more informative the more logical possibilities it excludes and it is
the more probable the fewer possibilities it excludes. The informativity and
the probabilities of some hypotheses are therefore “conflicting in the sense that
the former is an increasing and the latter a decreasing function of the logical
strength of the theory to be assessed” (Huber 2008, 91). With the help of this
measure of the informatively of confirmation measures at hand, Huber (2005,
2008) suggests the following Continuity requirement (here a minimally weaker
version of Huber’s requirement is considered).

Requirement 3.6 (Continuity).
For all probability functions Pr if c is an adequate confirmation measure defined
in terms of Pr then ∀� > 0 ∃δ� > 0 so that if i(H1) ≥ i(H2) + �, Pr(H1|E) ≥
Pr(H2|E) − δ�, and Pr(H1|E) > 0, then c(H1,E) > c(H2,E).

With the help of the Continuity requirement, Huber (2005) is able to provide a
strong argument for Bayesian confirmation theory. Again skipping the details,
if measures of confirmation satisfy, among other requirements, the Continuity
requirement, they lead a Bayesian agent to “(the most) informative (among all)
true theories.” However, as already noted in the quote from Huber (2005), not
all incremental confirmation measures satisfy the Continuity requirement.

Observation 5. Confirmation measures d, r, S, r, M and N satisfy Require-
ment 3.6. Z, l, and F do not satisfy Requirement 3.6.

14



That the confirmation measures Z, l, and F do not satisfy Requirement 3.5 and
that these confirmation measures do not lead Bayesian agents to “(the most)
informative (among all) true theories,” as Huber puts it, can be considered to
be a powerful argument against these measures of confirmation. If Bayesians
subscribe to the point of view that “inductive procedures [. . . ] lead to the
acceptance of certain empirical hypotheses on the basis of evidence that gives
them more or less strong [. . . ] support” (Hempel 1960, 462), it seems Bayesians
should reject the confirmation measures Z, l, and F. Instead, Bayesians should
choose between those measures that satisfy Requirement 3.6 and that can lead
one “to (the most) informative (among all) true theories.”

3.5 The Strengthened Problem of Measure Sensitivity

The previous subsections successfully demonstrate that the plurality of confir-
mation measures and the problem of the measure sensitivity of various argu-
ments surrounding confirmation theory presents an enormous challenge for the
standard Bayesian conception of confirmation. No single confirmation measure
satisfies all the requirements; hence, no confirmation measure renders all the
above-mentioned arguments in Bayesian confirmation theory valid. Given the
usual assumption that monism of confirmation measure holds true, this strength-
ened problem of measure sensitivity is especially problematic. As already noted,
monism of confirmation measures says that there is only one adequate confir-
mation measure (up to ordinal equivalence). Hence, monism requires providing
plausible desiderata that single out the “one true measure of confirmation,” as
Milne (1996) puts it.

The following table displays whether the different confirmation measures in-
troduced in Section 2 satisfy the requirements presupposed in the context of
various arguments of Bayesian confirmation theory .

d r S Z l F r M N
Requirement 3.1 Yes Yes Yes Yes Yes Yes No No Yes
Requirement 3.2 Yes Yes Yes Yes Yes Yes No No Yes
Requirements 3.3 Yes Yes Yes No Yes Yes No No Yes
Requirement 3.4 Yes No No Yes Yes Yes Yes No No
Requirement 3.5 No No No Yes Yes Yes No No No
Requirement 3.6 Yes Yes Yes No No No Yes Yes Yes
Requirements 3.1–3.6 No No No No No No No No No

Table 1

However, one might object that the preceding considerations do not show that
there is no confirmation measure that validates all of them. The considera-
tions could just demonstrate that we have not found the relevant confirmation
measure yet. More specifically, there might be different confirmation measures
as yet unproposed (and therefore not mentioned in Section 2) that validate all

15



four arguments by satisfying the six requirements 3.1–3.6. However, one can
show that this is not possible. Closer inspection of table 1 reveals that there
are confirmation measures that satisfy the requirements 3.1–3.5, i.e. the confir-
mation measures l and F , and there is one confirmation measure that satisfies
requirements 3.1–3.4 and Requirement 3.6, i.e. the confirmation measure d.
However, there is no confirmation measure that satisfies Requirement 3.5 and
Requirement 3.6. The reason for this is that both requirements are inconsistent.

Observation 6. It is logically impossible to construct a confirmation measure
c that satisfies Requirement 3.5 and Requirement 3.6.

According to Requirement 3.5 (Fitelson’s Logicality requirement), two hypothe-
ses H1 and H2 should be assigned the maximum degree of confirmation if they
are implied by the evidence, independently of any surplus in informativeness
between i(H1) and i(H2) (for the sake of the argument let us assume there is
such a difference in informativeness). However, according to Requirement 3.6
(Huber’s Continuity requirement), any surplus in informativeness between i(H1)
and i(H2) (i.e. Pr(¬H1) and Pr(¬H2)) succeeds if the difference in plausibil-
ity is small enough. Since by assumption both hypotheses are implied by the
evidence, there is no difference in plausibility since Pr(H1|E) = Pr(H1|E) = 1.
Hence, the surplus in informativeness should succeed, but this contradicts what
is required by Logicality. Accordingly, one can give up searching for confirma-
tion measures that satisfy all six requirements.

In conclusion, the strengthened problem of measure sensitivity teaches two im-
portant lessons. First, one cannot use the above requirements as desiderata that
an adequate measure of confirmation must satisfy to single out the one true con-
firmation measure. According to the arguments considered in Fitelson (1999),
it would be possible to adopt the confirmation measure d and all arguments be
valid. Indeed, many Bayesian confirmation theorists would at least be tempted
to argue that validating these arguments is a desideratum that any adequate
measure of confirmation should satisfy. This is not possible with the arguments
and requirements considered here. None of the confirmation measures intro-
duced in Section 2 satisfies them all and it is impossible to construct such a
confirmation measure. This makes the plurality of confirmation measures in
conjunction with the strengthened problem of measure sensitivity a real chest-
nut of confirmation theory.

Second, given monism of confirmation measures, one of the four arguments con-
sidered above is not valid. Either Bayesian confirmation theory cannot be used
to soften the impact of the problem of irrelevant conjunction and the paradox of
the ravens, or one has to give up considering confirmation theory as an inductive
logic, (i.e., as an extension of deductive logic in the sense of Fitelson (2006)), or
one has no argument that it leads one to accept the most informative of all true
theories. This is true because none of the confirmation measures introduced
in Section 2 satisfies all six requirements. Of course, there might be different

16



arguments for similar conclusions with weaker or different requirements. How-
ever, according to the state of the art in Bayesian confirmation theory, monism
requires that at least one of the above requirements is given up and that the
arguments that presuppose these requirements are therefore not sound. As-
suming that the Bayesian monist aims at maximizing the number of arguments
validated, two options present themselves: Option one is to choose measure d,
which softens the impact of the problem of irrelevant conjunction and the para-
dox of the ravens and which leads us to accept the most informative of all true
theories. Option two is to choose measure l (or F), which softens the impact of
the problem of irrelevant conjunction and the paradox of the ravens and which
we can consider an extension of deductive logic in the sense of Fitelson (2006).

4 The Problem of Measure Sensitivity as an Ar-
gument for Pluralism?

However, there is yet another plausible alternative. This alternative is to give
up monism and to embrace the plurality of Bayesian confirmation measures
(Huber 2008; Joyce 1999, 2003; Steel 2007). Then one could argue that indeed
all four arguments considered in Section 3 are valid even though they are not
valid with respect to one notion of confirmation. Instead, one could argue that
there are at least two (and possibly more) diverging notions of confirmation and
for each of the arguments considered in Section 3 there is at least one notion
of confirmation that validates it. For example, one could argue that there is
one notion of confirmation that satisfies Requirements 3.1–3.5 and one notion
of confirmation that satisfies Requirements 3.1–3.4 and Requirement 3.6. Joyce
(ms) labels this position pluralism and describes it as follows:

The various measures of confirmation capture different, but entirely
legitimate, concepts of evidence, concepts that must play a cen-
tral role in any adequate epistemology. [. . . ] We need a theory of
confirmation that is able to characterize a variety of evidential re-
lationships that hold across a wide range of states of background
knowledge and opinion. Only a probabilistic theory that provides
many measures of incremental confirmation is up to the task. (Joyce
ms, 1)

One might be tempted to argue that pluralism must be rejected since it is
not possible to adopt more than one explication for the one term “evidence E
incrementally confirms hypothesis H (with respect to background knowledge B)
to degree r.” However, on this question Joyce has Carnap on his side. In the
2nd edition of The Logical Foundations of Probability Carnap writes:

It has repeatedly occurred in the history of science that a vehement
but futile controversy arose between the proponents of two or more
explicata who shared the erroneous belief that they had the same

17



explicandum; when finally it became clear that they meant different
explicanda, unfortunately designated by the same term, and that the
different explicate were hence compatible and moreover were found
to be equally fruitful scientific concepts, the controversy evaporated
into nothing. (Carnap 1962, 26)7

Accordingly, it might be the case that the various confirmation theorists have
different explicanda in mind, but use the same term “incremental confirmation”
to refer to them. This assumption is the most plausible explanation for the
findings of Section 3. However, proponents of pluralism cannot simply assume
that all the different measures introduced in Section 2 “play a central [and
irreplaceable] role in any adequate epistemology,” as Joyce puts it, or, in Car-
nap’s words, that they are “equally fruitful scientific concepts.” In particular,
they have to show that these (ordinally) different confirmation measures re-
flect equally important but nevertheless different aspects of scientific reasoning.
Hence, proponents of pluralism have to show that an adequate confirmation
theory requires at least two diverging notions of confirmation and therefore
also two confirmation measures.8 In addition, they have to provide us with a
set of criteria that allow us to distinguish confirmation measures that capture
important evidential relationships from probabilistic measures that do not cap-
ture important evidential relationships. Hence, proponents of pluralism have
to provide criteria that exclude measures that are not required by an adequate
confirmation theory.

5 Summary

The present paper strengthens the problem of measure sensitivity by showing
that there are four valuable and important arguments within Bayesian confir-
mation theory that are not validated by any single measure of confirmation.
These considerations show that providing some form of solution to Fitelson’s
(1999) problem of measure sensitivity is central to Bayesian confirmation theory.

It is important to note that I do not take the strengthened version of the prob-
lem of measure sensitivity to provide any reason to abandon Bayesian confirma-
tion theory. What these considerations show, however, is that any solution to
the problem of measure sensitivity will require enormous modifications of the
standard Bayesian conception of confirmation, beyond anything anticipated in
Fitelson (1999): either one has to give up monism of confirmation measures or

7I am grateful to an anonymous referee for making me aware of this passage. In addition,
the referee correctly pointed out that Carnap’s distinction between absolute confirmation
and incremental confirmation is an historical instance of such a controversy. In particular,
Carnap was able to show that the triviality result achieved in Hempel’s Studies in the Logic of
Confirmation is based on the erroneous belief that there is just one explicandum designated
by the term “confirmation” instead of the two explicanda nowadays designated by the terms
absolute confirmation and incremental confirmation.

8An argument to the effect that an adequate theory of confirmation needs at least two
measures of confirmation is presented in Brössel (2012).

18



one has to give up the validity of one of the four important arguments consid-
ered in Section 3 of the paper.

More specifically, the first option is giving up monism of confirmation measures
– which I take to be part of the standard Bayesian conception of confirmation
– and embracing pluralism as suggested by Joyce and others. This I consider
to be a major (but nevertheless very attractive) deviation from the standard
Bayesian conception of confirmation. The challenge connected with this option
is to show that a complete Bayesian confirmation theory requires at least two
notions of confirmation to serve all the purposes of confirmation theory and to
present criteria that distinguish between confirmation measures that capture
important evidential relationships and probabilistic measures that do not cap-
ture important evidential relationships.

The second option is to uphold monism of confirmation measures and to give
up the validity of one of the four arguments considered in Section 3. The
incompatibility of Requirement 3.5 and Requirement 3.6 suggests that one has
to give up considering confirmation theory as an inductive logic, or one is left
with no argument that it leads one to accept the most informative of all true
theories. Although there is the possibility that there are different arguments for
similar conclusions with weaker or different requirements, according to the state
of the art in Bayesian confirmation theory, monism requires that at least one of
the two requirements is given up and that the arguments that presuppose these
requirements are therefore not sound. The challenge connected with this option
is to say which argument should be given up and to provide a justification for
why a confirmation measure that does not validate this argument is nevertheless
adequate. This includes in particular showing that the “one true” confirmation
measure can serve all the purposes associated with Bayesian confirmation theory,
even if it does not validate all the above arguments. Hence, proponents of
monism have to show that an adequate Bayesian theory of confirmation does
not require two measures of confirmation. This presents an enormous challenge
to those Bayesian confirmation theorists who want to uphold monism.

19



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