1. Introduction
Explanation is a central notion in human reasoning and a traditional topic in the philosophy of science. In a recent article, Schupbach and Sprenger (Reference Schupbach and Sprenger2011) have put forward a probabilistic analysis of how successful a hypothesis or theory h is in explaining an event or state of affairs of interest e. In our view, the general philosophical assumptions of Schupbach and Sprenger's approach are sensible, and the theoretical endeavor that they pursue is highly valuable. Thus, in what follows, we will rely on how they define their framework of inquiry. However, we will subject their specific proposal, results, and interpretations to critical discussion.
We will use E to label a probabilistic measure of explanatory power and write 
, with e and h denoting contingent statements for the explanandum and candidate explanans, respectively, and P a relevant probability function, which is assumed to be regular (i.e., such that, for any contingent statement 
).Footnote 1 Notably, 
is meant as a measure of how successful h is in explaining e assuming that h qualifies as potentially relevant to e in explanatory terms. Following Schupbach and Sprenger (Reference Schupbach and Sprenger2011, 106–8), we will see the latter caveat as implied in the use of 
but not encoded in 
itself. Accordingly, 
will reflect the strength of an explanans-explanandum relation between h and e, if any is present. A systematic characterization of when such a relation per se obtains, although of course relevant and much deserving, is seen here as a distinct topic of investigation.
2. Reviewing Adequacy Conditions
Schupbach and Sprenger's favorite probabilistic measure of explanatory power is as follows:
In their main formal result, they prove that 
must hold (where f is a strictly increasing function), provided that four conditions CA1–CA4 are satisfied (see their theorem 1; Reference Schupbach and Sprenger2011, 111). This pinpoints a class of ordinally equivalent measures that are represented by some increasing function of the posterior ratio 
. (To appreciate this, consider the following rendition of 
.) We will now briefly comment on each of Schupbach and Sprenger's adequacy conditions in turn.
CA1. There exists an analytic function g such that, for any contingent 
and any regular 
. Values of 
range in 
.
Schupbach and Sprenger label this condition formal structure and describe it as “rather uncontentious” (Reference Schupbach and Sprenger2011, 109). Some cautionary notes are in order, though. As they show, all measures satisfying their CA1–CA4 are ordinally equivalent posterior ratio measures, as defined above. However, it is not the case that all posterior ratio measures satisfy CA1–CA4, and this is especially due to CA1, which goes beyond constraining the ordinal structure. For instance, there is no doubt that it is mathematically elegant to have 
normalized to range within 
. If that specific range is imposed, however, we have that posterior ratio measures such as 
(with 
) are ruled out just because the range becomes 
. Similar consequences arise from the analyticity requirement in CA1, for there exist posterior ratio measures that are not analytic, thus violating CA1.Footnote 2 By this restrictive character, CA1 prevents Schupbach and Sprenger's main result from being a proper representation theorem for posterior ratio measures. We will come back to this point soon.
CA2. Ceteris paribus, the greater the degree of statistical relevance between e and h, the greater the value of 
.
Schupbach and Sprenger's label for this condition is positive relevance. Our sole remark here is that CA2 is left rather unspecified, as statistical relevance can be measured in various ways. This is innocuous for their proof. Yet, to allow for a discussion of the content and plausibility of this assumption, a sharper rendition would be helpful.
CA3. If h 2 is probabilistically independent from e, h 1, and their conjunction (i.e., 
, 
, and 
), then 
.
Schupbach and Sprenger's line of argument for this condition, which we do not dispute, goes as follows. Statement h 2 is assumed to be irrelevant, so we have 
. But then, as adding h 2 leaves the degree to which e is expected unchanged, it does not alter the degree to which e is explained either. Their label for CA3 is irrelevant conjunctions. We suggest conjunction of irrelevant explanantes, though, for we will have to deal with the conjunction of irrelevant explananda as a separate issue later on.
CA4. If ¬h entails e, then the values of 
do not depend on the values of P(h). Formally, there exists a function f so that, if 
, then either 
or 
.
We struggle to see a clear connection between the informal and the formal clause in CA4 and, thus, to interpret this condition as conveying irrelevance of priors, as Schupbach and Sprenger label it.Footnote 3 In fact, we decidedly concur with them that intuition is hardly “strong enough to make a conclusive case for CA4” (Reference Schupbach and Sprenger2011, 111).
3. A Representation Theorem for Posterior Ratio Measures
The above remarks were meant to motivate a partly different approach to the derivation of posterior ratio measures of explanatory power to be pursued in the present section. To this purpose, let us introduce a propositional language L and the set Lc of the contingent formulas in L. Further, let P be the set of all regular probability functions that can be defined over L (so that for any 
and any 
, 
) and posit 
. Each 
can be seen as representing a possible state of belief concerning a domain described in L. As a first move, we propose the following explication of CA2 above:
E1. Positive relevance. For any 
and any 
, 
if and only if (iff) 
.
In essence, E1 states that 
is a strictly increasing function of 
when 
is kept fixed. Indeed, this assumption closely fits Schupbach and Sprenger's remark that (positive) explanatory power reflects “the strength” of the inequality 
(see Reference Schupbach and Sprenger2011, 110). An additional feature of E1 is that we immediately get an analogue of CA3 above, that is, an appropriate principle concerning the conjunction of irrelevant explanantes. Indeed, E1 implies that, for any 
and any 
, if h 2 is probabilistically independent from e, h 1, and their conjunction, then 
. As it is redundant on our E1, this condition does not need to be stated on its own.
Second, it will prove convenient to retain a different statement of CA4 that Schupbach and Sprenger had employed in earlier versions of their work.Footnote 4 The informal wording simply read “values of 
do not depend on the values of P(h)” (i.e., without the restrictive caveat about ¬h implying e that is involved in CA4 as spelled out above). Upon scrutiny, however, the proper formal rendition of the assumption that was intended by this statement turns out to yield a substantial constraint on the functional form of 
.Footnote 5 For lack of anything better, we will label this condition posteriors (its implications and plausibility will be discussed later on).
E2. Posteriors. There exists a function g such that, for any 
and any 
, 
.
It is then possible to prove the following theorem (see app. A):
Theorem 1. E1 and E2 hold iff there exists a strictly increasing function f such that, for any 
and any 
, 
.
As a bidirectional implication, theorem 1 shows that E1 and E2 are both sufficient and necessary to single out the set of posterior ratio measures of explanatory power, thus providing a representation theorem, that is, an axiomatic foundation carving out distinctive traits of these ordinally equivalent measures taken together. Notably, this has not been achieved by invoking a more demanding set of conditions than CA1–CA4, for the range of 
is left entirely unconstrained, and no analyticity, differentiability, or continuity requirement is involved.Footnote 6 We now feel in a better position to precisely discuss how appealing posterior ratio measures of explanatory power are in philosophical terms.
4. Explanatory Justice
A clear and general axiomatization of posterior ratio measures fosters insight into their distinctive properties and thus a focused discussion of their implications. As a starting point for such a discussion, we will consider a quote from Schupbach and Sprenger (Reference Schupbach and Sprenger2011, 115) in which they criticize an important competing measure of explanatory power—that is, 
—originally adopted by Good (Reference Good1960) and more recently by McGrew (Reference McGrew2003):
Consider the following example: let e be a general description of the Brownian motion observed in some particles suspended in a particular liquid, and let h be Einstein's atomic explanation of this motion. Of course, h constitutes a lovely explanation of e. … However, take any irrelevant new statement e* and conjoin it to e; for example, let e* be the proposition that the mating season for an American green tree frog takes place from mid-April to mid-August. In this case, measure 
judges that Einstein hypothesis explains Brownian motion to the same extent that it explains Brownian motion and this fact about tree frogs. Needless to say, this result is deeply unsettling.
On the basis of this remark, Schupbach and Sprenger put forward the requirement that 
in such circumstances (i.e., when e is an explanatory success of h and e* is probabilistically independent from e, h, and their conjunction), which is in fact implied by any posterior ratio measure. Presumably, the intuitive motivation here reflects a sense of “explanatory justice,” to be informally understood as follows: a certain amount of explanatory success of h concerning e, no matter how large, cannot be extended “for free.” That is, it would not make sense to claim the same amount of explanatory success concerning any logically stronger statement 
—unless of course some positive explanatory import of h on e* independently exists.Footnote 7 Along this line of argument, we urge, careful and separate consideration should be given to the case of explanatory failure.
Let us illustrate with a mundane example. Assume there is a fair coin that can only be of one of two kinds: either it is a normal coin or it has tails on both sides. Label h the former hypothesis (normal coin) and ¬h the latter (no heads, two tails). Let e be a streak of tosses of, say, 10 tails. Clearly, ¬h explains e perfectly well, while h fares poorly on this account. But now let e* be involved once again (i.e., the American green tree frogs’ mating season). Should we say that the explanatory failure of h is at all mitigated if e* is conjoined to e? We think not. Again it seems a matter of “explanatory justice.” For, should it be the case that 
in these circumstances, then one would be allowed to indefinitely relieve a lack of explanatory power, no matter how large, by adding more and more irrelevant explananda, simply at will. A strongly unpalatable outcome—yet this is precisely what any posterior ratio measure of explanatory power implies, including of course 
. Schupbach and Sprenger are aware of this property (see Reference Schupbach and Sprenger2011, 115), but they do not offer a discussion dispelling what we see as its highly counterintuitive character. On the basis of the foregoing, we put forward the following principle:
E2*. Explanatory justice (upward and downward).
i). For any
and any , if and e* is probabilistically independent from e, h, and their conjunction, then .ii). For any
and any , if and e* is probabilistically independent from e, h, and their conjunction, then .
and any 
, if 
and e* is probabilistically independent from e, h, and their conjunction, then 
.
and any 
, if 
and e* is probabilistically independent from e, h, and their conjunction, then 
.
Recall that Schupbach and Sprenger criticized Good's measure of explanatory power 
for violating E2*i, which 
satisfies (along with all posterior ratio measures). For the sake of brevity and vividness, we will say that posterior ratio measures are just upward, while Good's measure is not. Notably, the pattern reverses with E2*ii, which is implied by Good's measure 
but violated by 
(and all posterior ratio measures). Good's measure is thus just downward, while posterior ratio measures are not. The question naturally arises, then, whether E2*i and E2*ii can ever be satisfied at once, that is, whether there exist measures displaying overall explanatory justice.Footnote 8 This is the topic of the next section.
5. A Further Representation Theorem
Let us go back to the properties of posterior ratio measures and pinpoint the culprit for their lack of downward explanatory justice. Based on our theorem 1, the problem is easily located in E2. To show this, we will now drop that assumption and replace it in such a way as to obtain a new and different representation theorem by which both clauses in E2* are fulfilled.
First, to dispense with E2, we will rely on another technical assumption about 
, that is,
E0. Formality. There exists a function g such that, for any 
and any 
, 
.
For a purely probabilistic theory of explanatory power, assumption E0 is almost as undemanding as it can be. It simply states that 
depends on the probability distribution over the algebra generated by h and e, which is entirely determined by 
, 
, and 
.Footnote 9
Now we only need to include what follows:
E3. Symmetry. For any 
and any 
, 
iff 
.
E3 postulates an inverse (ordinal) correlation between the explanatory success and explanatory failure of a hypothesis with regards to complementary events. Notably, it is purely ordinal in nature and, thus, a weaker version of a homonymous condition that Schupbach and Sprenger themselves advocate and motivate with the remark that “the less surprising (more expected) the truth of e is in light of a hypothesis, the more surprising (less expected) is e's falsity” (Reference Schupbach and Sprenger2011, 113).
The announced result is thus as follows (see app. B for a proof):
Theorem 2. E0, E1, E2*, and E3 hold iff there exists a strictly increasing function f such that, for any 
and any 
, 
, where
The answer to our question above concerning overall explanatory justice is then in the positive. There exists a class of ordinally equivalent measures of explanatory power that—unlike either Schupbach and Sprenger's 
or Good's 
—displays justice both upward and downward, that is, for both explanatory success and failure. The crucial step is to assume E2* straight away as an axiom and to get rid of E2.Footnote 10
Even beyond E0, E1, and E3, measures 
and 
share further desirable properties spelled out by Schupbach and Sprenger. For instance, both of them take maximum (minimum) value iff h implies e (¬e). Yet given the conflicting consequences of E2 and E2*, our theorems 1 and 2 imply that 
and 
are nonequivalent, not even in purely ordinal terms. For instance, ε may rank e 1 rather than e 2 as a stronger explanatory success of a given h, while ∊* implies the opposite; that is, there exist cases in which 
, while 
. So 
and 
represent two genuinely alternative ways to explicate the notion of explanatory power. As a historical note, it should be pointed out that 
is not an entirely new idea, as its positive branch already appears in Niiniluoto and Tuomela (Reference Niiniluoto and Tuomela1973, 66, 89; also see Pietarinen Reference Pietarinen, Hintikka and Suppes1970). The next section will discuss some further properties of this measure.
6. Explanatory Power, Confirmation, and Miracles
There exists a rather effective rule of thumb to generate a decent candidate measure of explanatory power; that is, take a plausible probabilistic measure of (incremental) confirmation and invert the positions of e and h. Indeed, Schupbach and Sprenger readily notice that their measure 
entertains this relationship with a very interesting measure of confirmation or inductive support originally defined by Kemeny and Oppenheim (Reference Kemeny and Oppenheim1952) and more recently revived by Fitelson (Reference Fitelson, Sarkar and Pfeifer2005); that is,
(The label l is employed here because this quantity is a strictly increasing function of the likelihood ratio 
.)
As shown below, the same kind of structural analogy holds between 
and yet another measure of confirmation, 
, which is based on the notion of the “relative distance” between the initial and final probability of a hypothesis (see Crupi, Tentori, and Gonzalez Reference Crupi and Tentori2007; Crupi, Festa, and Buttasi Reference Crupi and Tentori2010; Crupi and Tentori Reference Crupi, Festa, Buttasi, Suárez, Dorato and Rédei2010, Reference Crupi, Tentori and Gonzalezforthcoming):
In considering pairs of analogous expressions for explanatory power and inductive confirmation, such as those above, one should always keep in mind a crucial theoretical difference. In the measurement of explanatory power, the hypothesis h at issue is meant to be in some explanatory relation (to be separately defined) with evidence e—a caveat with no apparent counterpart in probabilistic confirmation theory. That said, 
is itself a measure of relative distance, the latter notion involving the background probability of the explanandum, 
, and the associated likelihood of the candidate explanans, 
. In particular, when h enjoys some amount of explanatory success relative to e (so that 
), 
assesses that explanatory success by the proportion of the background surprisingness of e (i.e., 
) that is removed by assuming h (i.e., by covering the positive difference between 
and 
). However, in case e is at odds with h (so that 
), 
rates explanatory failure by a negative value that is lower and lower, the higher the proportion of the background expectedness of e (i.e., 
) that is removed by assuming h (i.e., by covering the negative difference between 
and 
).Footnote 11
In view of these structural analogies, investigating the connections between probabilistic explanatory power 
and confirmation 
appears appropriate (see Crupi Reference Crupi2012). Interestingly, if relative distance measures of confirmation and explanatory power are concurrently adopted—that is, if one posits both 
and 
, where f and g are strictly increasing functions—then one can easily derive what follows (proof omitted):
M. No miracle (retail version). For any 
and any 
, if 
and 
, then 
iff 
.
M postulates an inverse (ordinal) correlation between the degree of positive confirmation that a successful explanatory hypothesis h receives from the occurrence of explanandum e and the degree to which e fails to be explained by ¬h. So explanatory hypothesis h is confirmed by e to the extent that e appears inexplicable (a miracle, as it were), assuming the falsity of h. Notably, if 
and 
are chosen among the corresponding classes of ordinal equivalence, a straightforward quantitative counterpart of M obtains; that is 
just in case 
.
While admittedly only a first step in a formal analysis of the subtle relations between confirmation and explanatory power, this result strikes us nonetheless as a telling implication of relative distance measures. In fact, the label for M is freely adapted after Worrall (Reference Tentori, Crupi and Osherson2006), who contrasts “retail” instances of a so-called no-miracle argument with its “wholesale” variant. In Worrall's terms, retail instances of the argument involve specific (scientific) hypotheses, while the wholesale argument (which he criticizes) would concern the global philosophical stance of a realist view of “all” science. Despite the differences between Worrall's and our theoretical framework and objectives, we believe that the meaning of M vindicates our terminological suggestion.Footnote 12
7. Conclusion
In our discussion, we have retained much of Schupbach and Sprenger's original approach in the inquiry on the notion of explanatory power. We have proved a representation theorem for posterior ratio measures, which Schupbach and Sprenger favor, along with a different representation theorem for an alternative set of measures relying on the notion of relative distance in probability. We end up advocating the latter and 
as a suitable exemplar of its class of ordinal equivalence. We have motivated this preference through several remarks. First, 
shares the properties set out by Schupbach and Sprenger that are genuinely appealing, while overcoming an undesirable trait of posterior ratio measures, that is, departure from downward justice upon the conjunction of irrelevant explananda. Second, 
conveys in probabilistic terms a view of explanatory power that is conceptually sound, as it transparently involves how the background surprisingness/expectedness of explanandum e is reduced by assuming candidate explanans h. Finally, relative distance measures yield a telling preliminary result concerning the relation between formal representations of explanatory power and inductive confirmation, thus bridging discussions in this area with a so-called no-miracle argument.
Theorem 1 E1 and E2 hold iff there exists a strictly increasing function f such that, for any 
and any 
, 
.
Proof Right-to-Left Implication
E1. Let O denote odds; that is, 
. By the probability calculus (Bayes's theorem), 
. Keeping in mind that f is an increasing function and 
is an increasing function of the posterior ratio 
, we have 
iff 
iff 
iff 
.
E2. Fulfillment of E2 immediately follows from 
.
Left-to-Right Implication
As a trivial consequence of E2, there exists a function j such that, for any 
and any 
, 
. With no loss of generality, we will convey probabilistic coherence and regularity by constraining the domain of j to include pairs of values (x, y) such that (i) 
, and (ii) 
. We thus posit 
and denote the domain of j as Dj.
Lemma 1.1. For any x, y 1, y 2 such that 
, 
, and 
, there exist 
and 
such that 
, 
, and 
.
Proof. Let 
be given. The equalities in lemma 1.1 arise from the following scheme of probability assignments:
Suppose there exist 
such that 
. Then, by lemma 1.1 and the definition of Dj, there exist 
and 
such that 
, 
, and 
. By the probability calculus, if the latter equalities hold, then 
. Thus, there exist 
and 
such that 
, even if 
, contradicting E1. Conversely, E1 implies that, for any 
, 
, 
. So, for E1 to hold, there must exist f such that, for any 
and any 
, 
, and 
. We thus posit 
and denote the domain of f as Df.
Lemma 1.2. For any 
, there exist 
and 
such that 
and 
.
Proof. Let 
and 
be given so that 
(as the latter quantity must be positive, y 1 exists) and 
(as the latter quantity must be positive, y 2 exists). The equalities in lemma 1.2 arise from the following scheme of probability assignments:
Suppose there exist 
such that 
and 
. Then, by lemma 1.2 and the definition of Df, there exist 
and 
such that 
and 
. By the probability calculus, if the latter equalities hold, then 
. Thus, there exist 
and 
such that 
, even if 
, contradicting E1. Conversely, E1 implies that, for any 
, if 
then 
. By a similar argument, E1 also implies that, for any 
, if 
then 
. So, for E1 to hold, it must be that, for any 
and any 
, 
, and f is a strictly increasing function. This completes the proof of theorem 1.
Theorem 2 E0, E1, E2*, and E3 hold iff there exists a strictly increasing function f such that, for any 
and any 
, 
, where
Proof Right-to-Left Implication
E0. If there exists a strictly increasing function f such that 
, then E0 is trivially satisfied.
E1. Let 
be given. Three classes of cases can obtain. (i) Let 
be such that 
and 
. It is easy to verify that, for any 
, 
iff 
. So we have that, for any 
, 
iff 
and 
iff 
. It follows that, for any 
, 
iff 
. (ii) Let 
be such that 
and 
. Then we have that, for any 
, 
iff 
iff 
iff 
iff 
. (iii) Finally, let 
be such that 
and 
. Then we have that, for any 
, 
iff 
iff 
iff 
. As i–iii are exhaustive, for any 
and any 
, 
iff 
. By ordinal equivalence, if there exists a strictly increasing function f such that 
, then E1 follows.
E2*. Let 
be given. (i) Let 
be given so that 
. Then, for any 
, if e* is probabilistically independent from e, h, and their conjunction, 
iff 
, by which 
, which obtains iff 
iff 
, by which 
, which obtains iff 
. By ordinal equivalence, if there exists a strictly increasing function f such that 
, then E2*i follows. (ii) Let 
be given so that 
. Then, for any 
, if e* is probabilistically independent from e, h, and their conjunction, 
iff 
iff 
. By ordinal equivalence, if there exists a strictly increasing function f such that 
, then E2*ii follows.
E3. Let 
and 
be given. Three classes of cases can obtain. (i) Let 
be such that 
and 
. It is easy to verify that, for any 
, 
iff 
iff 
iff 
. So we have that, for any 
, 
iff 
iff 
iff 
and 
iff 
iff 
. It follows that, for any 
, 
iff 
. (ii) Let 
be such that 
and 
. Then we have that, for any 
, 
iff 
iff 
iff 
iff 
. (iii) Finally, let 
be such that 
and 
. Then we have that, for any 
, 
iff 
iff 
iff 
iff 
. As i–iii are exhaustive, for any 
and any 
, 
iff 
. By ordinal equivalence, if there exists a strictly increasing function f such that 
, then E3 follows.
Left-to-Right Implication The Case of Explanatory Failure 
Note that 
. As a consequence, by E0, there exists a function j such that, for any 
and any 
, 
. With no loss of generality, we will convey probabilistic coherence, regularity, and explanatory failure by constraining the domain of j to include triplets of values (x, y, w) such that the following conditions are jointly satisfied:
•
;•
, by which , so that , and thus ;•
(conveying explanatory failure; that is, ), by which , so that , and thus , and , so that , and thus ;•
, by which , and thus .
;
, by which 
, so that 
, and thus 
;
(conveying explanatory failure; that is, 
), by which 
, so that 
, and thus 
, and 
, so that 
, and thus 
;
, by which 
, and thus 
.We thus posit 
and denote the domain of j as Dj.
Lemma 2.1. For any 
such that 
, 
, and 
, there exist 
and 
such that 
, 
, 
, and 
.
Proof. The equalities in lemma 2.1 arise from the following scheme of probability assignments:
Suppose there exist 
such that 
. Then, by lemma 2.1 and the definition of Dj, there exist 
and 
such that 
, 
, 
, and 
. Clearly, if the latter equalities hold, then 
. Thus, there exist 
and 
such that 
, even if 
, contradicting E1. Conversely, E1 implies that, for any 
, 
. So, for E1 to hold, there must exist k such that, for any 
and any 
, if 
, then 
, and 
. We thus posit 
and denote the domain of k as Dk.
Lemma 2.2. For any 
such that 
, 
, and 
, there exist 
and 
such that 
, 
, 
, 
, 
, and 
.
Proof. Let 
be given so that 
(as the latter quantity must be positive, w exists). The equalities in lemma 2.2 arise from the following scheme of probability assignments:
Suppose there exist 
such that 
. Assume 
with no loss of generality. Then, by lemma 2.2 and the definition of Dk, there exist 
and 
such that 
, 
, 
, 
, 
, and 
. If the latter equalities hold, then 
and e* is probabilistically independent from e, h, and their conjunction. Thus, there exist 
and 
such that 
, even if 
and e* is probabilistically independent from e, h, and their conjunction, contradicting E2*ii. Conversely, E2* implies that, for any 
, 
. So, for E2* to hold, there must exist m such that, for any 
and any 
, if 
, then 
, and 
. We thus posit 
and denote the domain of m as Dm.
Lemma 2.3. For any 
, there exist 
and 
such that 
and 
.
Proof. Let 
be given so that 
(as the latter quantity must be positive, w 1 exists) and 
(as the latter quantity must be positive, w 2 exists). The equalities in lemma 2.3 arise from the following scheme of probability assignments:
Suppose there exist 
such that 
and 
. Then, by lemma 2.3 and the definition of Dm, there exist 
and 
such that 
and 
. Clearly, if the latter equalities hold, then 
. Thus, there exist 
and 
such that 
, even if 
, contradicting E1. Conversely, E1 implies that, for any 
, if 
, then 
. By a similar argument, E1 also implies that, for any 
, if 
, then 
. So, for E1 to hold, it must be that, for any 
and any 
, if 
, then 
, and m is a strictly increasing function.
The Case of Explanatory Success
Notice that 
and 
. As a consequence, by E0, there exists a function r such that, for any 
and any 
, 
. With no loss of generality, we will convey probabilistic coherence, regularity, and explanatory success by constraining the domain of r to include triplets of values 
such that the following conditions are jointly satisfied:
•
;•
, by which , so that , and thus ;•
(conveying explanatory success; that is, ), by which , so that , and thus , and , so that , and thus ;•
, by which , and thus .
;
, by which 
, so that 
, and thus 
;
(conveying explanatory success; that is, 
), by which 
, so that 
, and thus 
, and 
, so that 
, and thus 
;
, by which 
, and thus 
.We thus posit 
and denote the domain of r as Dr.
Lemma 2.4. For any 
such that 
, 
, and 
, there exist 
and 
such that 
, 
, 
, and 
.
Proof. The equalities in lemma 2.4 arise from the following scheme of probability assignments:
Suppose there exist 
such that 
. Then, by lemma 2.4 and the definition of Dr, there exist 
and 
such that 
, 
, 
, and 
. By the probability calculus, if the latter equalities hold, then 
. Thus, there exist 
and 
such that 
, even if 
, contradicting E1. Conversely, E1 implies that, for any 
, 
. So, for E1 to hold, there must exist s such that, for any 
and any 
, if 
, then 
, and 
. We thus posit 
and denote the domain of s as Ds.
Lemma 2.5. For any 
such that 
and 
, there exist 
and 
such that 
, 
, and 
.
Proof. Let 
be given so that 
(as the latter quantity must be positive, w exists) and 
(as the latter quantity must be positive, w exists). The equalities in lemma 2.5 arise from the following scheme of probability assignments:
Suppose there exist 
such that 
. Then, by lemma 2.5 and the definition of Ds, there exist 
and 
such that 
, 
, and 
. If the latter equalities hold, then 
. Thus, there exist 
and 
such that 
, even if 
, contradicting E3. So, for E3 to hold, there must exist t such that, for any 
and any 
, if 
, then 
, and 
. We thus posit 
and denote the domain of t as Dt.
Lemma 2.6. For any 
, there exist 
and 
such that 
, and 
.
Proof. Let 
be given so that 
(as the latter quantity must be positive, w 1 exists) and 
(as the latter quantity must be positive, w 2 exists). The equalities in lemma 2.6 arise from the following scheme of probability assignments:
Suppose there exist 
such that 
and 
. Then, by lemma 2.6 and the definition of Dt, there exist 
and 
such that 
and 
. By the probability calculus, if the latter equalities hold, then 
. Thus, there exist 
and 
such that 
, even if 
, contradicting E1. Conversely, E1 implies that, for any 
, if 
, then 
. By a similar argument, E1 also implies that, for any 
, if 
, then 
. So, for E1 to hold, it must be that, for any 
and any 
, if 
, then 
, and t is a strictly decreasing function.
Summing up, if E0, E1, E2*, and E3 hold, then for any 
and any 
, (i) if 
, then 
and m is a strictly increasing function, thus 
is a strictly increasing function of 
, and (ii) if 
, then 
and t is a strictly decreasing function, thus 
is a strictly increasing function of 
. As i and ii are exhaustive, for E0, E1, E2*, and E3 to hold, it must be that, for any 
and any 
, 
, and f is a strictly increasing function. This completes the proof of theorem 2.
. Such a term will be omitted from our notation for simple reasons of convenience, as it is inconsequential for our discussion.
if 
; 
if 
.
is a function of either just 
or just 
in the target class of cases. Yet none of these quantities can be said to make the prior 
“irrelevant,” lacking additional details. In fact, it all depends on what is or is not being concurrently kept fixed. For instance, if we happen to keep 
and 
fixed, then of course 
and 
vary together. If 
and 
are kept fixed instead, then it is 
and 
that vary together. Surely, it could also be the case that, say, 
varies “independently” from 
(i.e., with the latter possibly remaining fixed), provided that room for variation is allowed for somewhere else. But condition CA4 is silent on all this. For this reason, we think that the connection with the “irrelevance of priors” is not well specified. (We thank Jan Sprenger for prompting this clarification.)
in these cases would become quite inconsequential. We find this worry legitimate, although by no means conclusive. Space limitations force us to postpone more thorough discussion to another occasion, but a relevant remark will arise in n. 10 below.
, and so that h and ¬e are inconsistent. If and only if this holds, for each of Schupbach and Sprenger's posterior ratio measures and each of our relative distance measures alike (see below), 
gets a fixed maximum value and 
a fixed minimum value, which seems rather appealing (also see Schupbach and Sprenger 
(for the conjunctive explanandum 
, unlike e, is not implied by h) along with 
(for the conjunctive explanandum 
is still inconsistent with h, just as ¬e was). While the strict inequality in E2*i neatly generalizes the former fact, the strict equality in E2*ii does so with the latter. (We thank an anonymous reviewer for raising this issue.)
and any 
, if e* is probabilistically independent from e, h, and their conjunction, then 
. Note, however, that 
does not imply W, as it gives 
whenever 
(also see n. 8). Thus, posterior ratio measures do not convey the idea of watering down in a fully coherent fashion. Second, by allowing for watering down in (nonextreme) cases of explanatory failure, 
also allows for e being a worse explanatory failure than 
, despite e* being itself a (mild) explanatory failure for a given h, and even if h screens off e* from e (proof omitted). So, e.g., let h be the hypothesis that Lee Harvey Oswald acted alone in the assassination of JFK. There exist sensible subjective probability estimates by which, according to a posterior ratio measure, h would fail to explain the sequence of shots in Dallas (e) but fail to a lesser extent with the conjunctive explanandum consisting in both that sequence of shots and the subsequent killing of Oswald (
). We find it hard to see how rankings of this kind could be rationalized and highlight as a virtue of explanatory justice (E2*) that it prevents them from arising.
can be written in yet another way that is possibly instructive, as follows: 
if 
; 
if 
.
