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 confirmation) have been proposed and defended in the philosophical literature. This plurality of measures of support is problematic, since it affects a great many arguments surrounding Bayesian confirmation theory (and inductive logic, generally)” (Reference Eells and Fitelson2002, 129).
To date, the most forceful display of the problem of measure sensitivity is one provided by Fitelson (Reference Fitelson1999). 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 current article adds to the results set out in Fitelson (Reference Fitelson1999), expanding on them in two principal respects. First, this article considers more confirmation measures; in Fitelson (Reference Fitelson1999) only a proper subclass of those confirmation measures considered here is taken into account. Second, this article shows that the problem of measure sensitivity actually runs deeper and is far more severe than is noted in Fitelson (Reference Fitelson1999).
The following section introduces the plurality of confirmation measures of Bayesian confirmation theory. Section 3 introduces the problem of measure sensitivity. In particular, this section considers four prominent arguments in Bayesian confirmation theory and shows that there is no confirmation measure that validates all of them. Subsections 3.1 and 3.2 introduce arguments already considered in Fitelson (Reference Fitelson1999), while subsections 3.3 and 3.4 consider two new arguments from Fitelson (Reference Fitelson2006) and Huber (Reference Huber2005), respectively. Section 4 discusses the ramifications that the strengthened problem of measure sensitivity has for Bayesian confirmation theory.
2. The Plurality of Bayesian Measures of Confirmation
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 reasoning. 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 the possible experimental outcomes would confirm the theory or hypothesis 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 on it by Bayesian philosophers of science and epistemologists cannot be investigated until a complete theory of confirmation has been provided.
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 measure 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 current article will not address the former task, that is, answering the question of how one should interpret the probabilities (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 confirmation 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 commonly agreed on. 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 
. Second, most Bayesian epistemologists lean explicitly or implicitly toward a monistic position regarding confirmation measures (Milne Reference Milne1996; Christensen Reference Christensen1999; Fitelson Reference Fitelson2001; Eells and Fitelson Reference Eells and Fitelson2002). Monism of confirmation measures is the position that there is only one adequate confirmation 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 quantitative 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. Earman (Reference Earman1992, chap. 5), Fitelson (Reference Fitelson1999, Reference Fitelson2001), and Crupi, Tentori, and Gonzalez (Reference Crupi, Tentori and Gonzalez2007) 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 
is a measure of (incremental) confirmation relative to probability function Pr, only if there is some (marker or neutral value) 
so that, for all hypotheses H and evidence E, if 
and 
, then
This first requirement on confirmation measures is taken from Fitelson (Reference Fitelson2001) and requires that they should be measures of incremental confirmation.Footnote 1 In particular, 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 probability 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 (Reference Joyce and Zalta2003a) calls the weak law of likelihood requirement, which “encapsulates a minimal form of Bayesianism to which all parties can agree” (sec. 3).
Requirement 2.2 (Weak Law of Likelihood). If 
and 
, then 
.
Joyce (Reference Joyce and Zalta2003a, sec. 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’ Theorem for questions of evidence: namely, that hypotheses are confirmed by data they predict” (see the appendix). This second requirement holds that when the evidence E is more probable in the light of hypothesis H 1 than in the light of H 2 and if the evidence E is also less probable under the assumption that H 1 is false than it is under the assumption that H 2 is false with respect to probability function Pr, then E confirms H 1 more than it confirms H 2.
What follows are among the most prominent Bayesian confirmation measures that satisfy both requirements.
(Carnap Reference Carnap1962):
if 
.
(Carnap Reference Carnap1962):
(Christensen Reference Christensen1999; Joyce Reference Joyce1999):
if 
.
(Crupi et al. Reference Crupi, Tentori and Gonzalez2007):
(Good Reference Good1960; Fitelson Reference Fitelson2001):
(Kemeny and Oppenheim Reference Kemeny and Oppenheim1952):
(Milne Reference Milne1996):
(Mortimer Reference Mortimer1988):
if 
.
(Nozick Reference Nozick1981):
if 
.
As already noted, this plurality of Bayesian measures of (incremental) confirmation 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: “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” (Reference Fitelson1999, S364). This section of the article 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 measure of confirmation is adopted. However, the results displayed here are in two respects more powerful than those achieved in Fitelson (Reference Fitelson1999). First, this article considers more confirmation measures. In Fitelson’s considerations, only a proper subclass of the confirmation measures introduced in section 2 is taken into account. In particular, Fitelson considers only the confirmation measures d, 
, r, and l. Second, this article shows that the problem of measure sensitivity actually runs deeper and is far more severe than is noted in Fitelson (Reference Fitelson1999). Fitelson draws only the following, relatively tentative, conclusion: “Many 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 contexts in question). I have also shown that there are compelling reasons to prefer d over the log-ratio measure r. Unfortunately, like Rosenkrantz (Reference Rosenkrantz1981), I have found no compelling reasons offered in the literature to prefer d over the log-likelihood ratio measure l” (S371).
According to this, it is possible to adopt the difference measures of confirmation, and no argument of confirmation theory considered in Fitelson (Reference Fitelson1999) 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 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 that validates all of them. This strengthens the results achieved in Fitelson (Reference Fitelson1999) since it shows that there cannot be a single confirmation measure that 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 confirmation should satisfy. Subsections 3.1 and 3.2 introduce arguments already considered in Fitelson (Reference Fitelson1999), while subsections 3.3 and 3.4 consider two new arguments from Fitelson (Reference Fitelson2006) and Huber (Reference Huber2005), respectively.
3.1. Earmann and the Problem of Irrelevant Conjunction
First, let us look at what is called the problem of irrelevant conjunction.Footnote 2 There are two (closely related) formulations of this problem. The first is the traditional formulation, and the second is Fitelson’s (Reference Fitelson2002) version. The traditional formulation of the problem is as follows:
The Problem of Irrelevant Conjunction 1. If H implies E, then E confirms 
for arbitrary I, provided 
.
Fitelson (Reference Fitelson2002) introduces a slightly different formulation of the problem. According to him, the problem of irrelevant conjunction is as follows:
The Problem of Irrelevant Conjunction 2. If E confirms H, and I is confirmationally irrelevant to H, E, and 
, then E also confirms 
.
Both formulations are statements of the problem of irrelevant conjunction because one can tack a confirmationally irrelevant conjunct I (the irrelevant conjunct) to the hypothesis H (the relevant conjunct) and still get the result that 
(the irrelevant conjunction) is confirmed.
This first formulation of the problem was first discussed in Hempel (Reference Hempel1945) in connection with the hypothetico-deductive method of confirmation. If one follows Hempel in adopting the special consequence condition, which requires that “if an observation report confirms a hypothesis H, then it also confirms every consequence of H” (31), 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 confirmation: 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 notion 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 literature one can find examples that seem to strongly support the claim that this result is counterintuitive. For example, Glymour (Reference Glymour1980) writes, “Nor are we greatly impressed by a theory of evidence which claims, for instance, that the Stark effect confirms the conjunction of quantum mechanics and the principle that the president of the Church of Jesus Christ of Latter Day Saints is infallible, all the while reassuring us that the Stark effect does not confirm the latter principle alone” (31).
However, Earman (Reference Earman1992) argues that examples like this are not too bad, as long as according to any reasonable confirmation measure the irrelevant conjunction 
is less confirmed than the relevant conjunct H. This is supposed to “soften the impact” (as Fitelson [Reference Fitelson2002] 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 (Reference Earman1992) requires that any reasonable confirmation measure satisfy the following condition:
Requirement 3.1 (Irrelevant Conjunction I). If 
is a reasonable confirmation measure, 
, and I is logically independent of H, E, and 
, then:
Hawthorne and Fitelson (Reference Hawthorne and Fitelson2004) suggest an alternative and, in their view, better-suited requirement to “cushion” the impact of the problem of irrelevant conjunction in its formulation by Fitelson (Reference Fitelson2002).
Requirement 3.2 (Irrelevant Conjunction II). If 
is a reasonable confirmation measure, 
, 
, and 
, then:
It is an interesting question which confirmation measures of section 2 satisfy these requirements. The following observation provides a list of which confirmation measures satisfy requirements 3.1 and 3.2 and which do not.
Observation 1. Confirmation measures d, 
, S, Z, l, F, and N satisfy requirements 3.1 and 3.2; r and M do not satisfy requirements 3.1 and 3.2.Footnote 3
One might be tempted to argue that observation 1 shows that (a) 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. This is because they “soften the impact” of the counterintuitive result insofar as they can be used to explain why philosophers have the preexplicatory intuition that irrelevant conjunctions are or should not be confirmed. The explanation is that they are confirmed to such a minimal degree that, preexplicatory, many philosophers and scientists do not have the intuition that they are confirmed. Thus, Bayesian confirmation theory can explain (away) these intuitions, given that they reject the ratio measure of confirmation.
However, inferring from observation 1 that all other Bayesian measures of confirmation 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 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 disconfirmation. 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 
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 (Reference Crupi and Tentori2010).
Requirement 3.3 (Irrelevant Conjunction III).Footnote 4 If 
is a reasonable confirmation measure with marker or neutral value r, 
, and 
and 
, then:
According to this requirement, “if 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 
but to a minor extent” (Crupi and Tentori Reference Crupi and Tentori2010, 4).
Observation 2. Confirmation measures d, 
, S, l, F, and N satisfy requirement 3.3; Z, r, and M do not satisfy requirement 3.3.
It is 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 and the Paradox of the Ravens
Another requirement on confirmation measures arises from arguments surrounding the well-known paradox of the ravens (Hempel Reference Hempel1945). This paradox is fueled by the following inconsistent intuitions:
Ravens Paradox 1. (i) A nonblack nonraven (e.g., a white shoe) does not confirm the hypothesis “All ravens are black.” (ii) A nonblack nonraven confirms the hypothesis “All nonblack things are nonravens.” (iii) Logically equivalent hypotheses are equally confirmed by a given piece of evidence (Equivalence Condition).
Now let H 1 be the hypothesis “All ravens are black” and let H 2 be the hypothesis “All nonblack things are nonravens.” Obviously, H 1 and H 2 are logically equivalent. According to i, nonblack nonravens do not confirm H 1. Hence, by the equivalence condition iii, nonblack nonravens do not confirm H 2. This contradicts intuition ii, according to which nonblack nonravens should confirm H 2 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 problem of irrelevant conjunction. In particular, the idea is that the observation of a nonblack nonraven 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 (Reference Horwich1982) 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 
is a reasonable confirmation measure and 
then:
Observation 3. Confirmation measures d, Z, l, F, and r satisfy requirement 3.4; 
, 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 preexplicatory intuition that a nonblack nonraven does not confirm “All ravens are black.” The explanation is that a nonblack nonraven confirms this hypothesis to such a minimal degree that philosophers and scientists do not have the preexplicatory intuition that it confirms it at all. However, the problem of irrelevant conjunction and the paradox of the ravens together do not show that Bayesians can exclude 
, 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, 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, Fitelson (Reference Fitelson2006) stresses that, since Carnap (Reference Carnap1962), the concept of confirmation has been closely linked to that of inductive logic. Carnap’s project was to provide a theory of confirmation or inductive logic that can be considered as an extension of deductive logic (Fitelson Reference Fitelson2006; Crupi et al. Reference Crupi, Tentori and Gonzalez2007). Fitelson (Reference Fitelson2006, 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 argument. Similarly, there is no worse argument than an argument with premises that imply the falsity of the conclusion. In accordance with this consideration, Fitelson suggests that any theory of confirmation that is supposed to be an account of inductive logic has to satisfy the requirement of logicality.Footnote 5
Requirement 3.5 (Logicality). If 
is an adequate confirmation measure, then (i) if E implies H, then 
, and (ii) if E implies 
, then 
.
Observation 4. Confirmation measures Z, l, and F satisfy requirement 3.5; d, 
, S, r, M, and N do not satisfy requirement 3.5.
The confirmation measures d, 
, S, r, M, and N fail to assign the maximum value if H is implied by E. Measures 
, 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 that is not valid according to these confirmation measures.
3.4. Huber and the Assessment of Theories
Huber (Reference Huber2007) focuses on another, more venerable question that arises in the context of Bayesian confirmation theory and inductive logic: Hume’s problem of induction. Huber writes: “Let us grant that Bayesian confirmation theory adequately explicates 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 justify 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?” (sec. 7). In an earlier article, 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 confirmation 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, incremental confirmation does not further this goal in general” (Reference Huber2005, 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 problem of measure sensitivity. Again, one interesting and valuable argument in favor of Bayesian confirmation theory depends on the choice of the confirmation measure. Skipping the details, Huber (Reference Huber2005) proposes a requirement dubbed “continuity” on confirmation measures so that they take one to “(the most) informative (among all) true theories.” Continuity requires that “any surplus in informativeness succeeds, if the difference in plausibility is small enough” (1156). Following philosophers such as Carnap and Bar-Hillel (Reference Carnap and Bar-Hillel1952), Popper (Reference Popper1954), and Hempel (Reference Hempel1960), Huber (Reference Huber2005, Reference Huber2008) suggests measuring the amount of information provided by a hypothesis with the help of the measure 
. The idea of this measure of information is that a hypothesis 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 Reference Huber2008, 91). With the help of this measure of the informativity of confirmation measures at hand, Huber (Reference Huber2005, Reference Huber2008) 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 
is an adequate confirmation measure defined in terms of Pr, then 
, so that if 
, 
, and 
, then 
.
With the help of the continuity requirement, Huber (Reference Huber2005) 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, not all incremental confirmation measures satisfy the continuity requirement.
Observation 5. Confirmation measures d, 
, S, r, M, and N satisfy requirement 3.6; Z, l, and F do not satisfy requirement 3.6.
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 Reference Hempel1960, 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 confirmation measures and the problem of the measure sensitivity of various arguments surrounding confirmation theory present 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 measures holds true, this strengthened problem of measure sensitivity is especially problematic. As already noted, monism of confirmation measures says that there is only one adequate confirmation measure (up to ordinal equivalence). Hence, monism requires providing plausible desiderata that single out the “one true measure of confirmation,” as Milne (Reference Milne1996) puts it.
Table 1 displays whether the different confirmation measures introduced in section 2 satisfy the requirements presupposed in the context of various arguments of Bayesian confirmation theory. However, one might object that these considerations do not show that there is no confirmation measure that validates all of them. The considerations 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 sec. 2) that validate all 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, that is, the confirmation measures l and F, and there is one confirmation measure that satisfies requirements 3.1–3.4 and requirement 3.6, that is, the confirmation measure d. However, there is no confirmation measure that satisfies requirements 3.5 and 3.6. The reason for this is that both requirements are inconsistent.
Observation 6. It is logically impossible to construct a confirmation measure 
that satisfies requirement 3.5 and requirement 3.6.
| Requirement | d |  |
S | Z | l | F | r | M | N |
|---|---|---|---|---|---|---|---|---|---|
| 3.1 | Yes | Yes | Yes | Yes | Yes | Yes | No | No | Yes |
| 3.2 | Yes | Yes | Yes | Yes | Yes | Yes | No | No | Yes |
| 3.3 | Yes | Yes | Yes | No | Yes | Yes | No | No | Yes |
| 3.4 | Yes | No | No | Yes | Yes | Yes | Yes | No | No |
| 3.5 | No | No | No | Yes | Yes | Yes | No | No | No |
| 3.6 | Yes | Yes | Yes | No | No | No | Yes | Yes | Yes |
| 3.1–3.6 | No | No | No | No | No | No | No | No | No |
Note “Yes” or “no” denotes whether a confirmation measure satisfies a requirement.
According to requirement 3.5 (Fitelson’s logicality requirement), two hypotheses H 1 and H 2 should be assigned the maximum degree of confirmation if they are implied by the evidence, independently of any surplus in informativeness between 
and 
(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 
and 
(i.e., 
and 
) succeeds if the difference in plausibility is small enough. Since by assumption both hypotheses are implied by the evidence, there is no difference in plausibility since 
. Hence, the surplus in informativeness should succeed, but this contradicts what is required by logicality. Accordingly, one can give up searching for confirmation measures that satisfy all six requirements.
In conclusion, the strengthened problem of measure sensitivity teaches two important lessons. First, one cannot use the above requirements as desiderata that an adequate measure of confirmation must satisfy to single out the one true confirmation measure. According to the arguments considered in Fitelson (Reference Fitelson1999), it would be possible to adopt the confirmation measure d and have 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 introduced 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 chestnut of confirmation theory.
Second, given monism of confirmation measures, one of the four arguments considered 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 [Reference Fitelson2006]), 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 satisfy all six requirements. Of course, there might be different arguments for similar conclusions with weaker or different requirements. However, 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. Assuming that the Bayesian monist aims at maximizing the number of arguments validated, two options present themselves: option 1 is to choose measure d, which softens the impact of the problem of irrelevant conjunction and the paradox of the ravens and which leads us to accept the most informative of all true theories. Option 2 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 (Reference Fitelson2006).
4. The Problem of Measure Sensitivity as an Argument 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 (Joyce Reference Joyce1999, Reference Joyce and Zalta2003a; Steel Reference Steel2007; Huber Reference Huber2008). 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 (Reference Zalta2003b) 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 central role in any adequate epistemology. … We need a theory of confirmation that is able to characterize a variety of evidential relationships 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” (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. 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 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” (Reference Carnap1962, 26).Footnote 6 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 Carnap’s words, that they are “equally fruitful scientific concepts.” In particular, they have to show that these (ordinally) different confirmation measures reflect 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.Footnote 7 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 capture 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 current article strengthens the problem of measure sensitivity by showing that there are four valuable and important arguments within Bayesian confirmation theory that are not validated by any single measure of confirmation. These considerations show that providing some form of solution to Fitelson’s (Reference Fitelson1999) 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 problem of measure sensitivity to provide any reason to abandon Bayesian confirmation 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 (Reference Fitelson1999): either one has to give up monism of confirmation measures or one has to give up the validity of one of the four important arguments considered in section 3 of the article.
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 capture 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.
Appendix In the light of Joyce’s strong claims about the weak law of likelihood, it is interesting to note that Crupi, Festa, and Buttasi (Reference Crupi, Festa, Buttasi, Suárez, Dorato and Rédei2010) 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 
and 
, or if 
and 
, then 
.
Joyce’s Weak Law of Likelihood requires that if the evidence E is strictly more probable in the light of hypothesis H 1 than in the light of H 2 and if it is strictly less probable in the light of 
than in the light of 
, then E confirms H 1 more than H 2. The strengthened weak law of likelihood slightly weakens the antecedent of Joyce’s requirement and requires that E confirms H 1 more than H 2, if the evidence E is strictly more probable in the light of hypothesis H 1 than in the light of H 2 and if E is strictly less or equally probable in the light of 
than 
or if the evidence E is strictly more or equally probable in the light of hypothesis H 1 than in the light of H 2 and if E is strictly less probable in the light of 
than 
. This slightly weakened antecedent seems to suffice to underscore the claim that H 1 predicts E better than H 2 and thus should be confirmed to a higher degree. Confirmation measures discussed in the literature that do not satisfy the strengthened weak law of likelihood are the Z measure by Crupi et al. (Reference Crupi, Tentori and Gonzalez2007), the r measure of confirmation defended in Milne (Reference Milne1996), and the M measure of confirmation by Mortimer (Reference Mortimer1988). These confirmation measures are introduced in detail section 2.
, i.e., if 
(without this caveat the requirement would be satisfied only by the measures d, 
, 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 the 
