1. Explanation and Explanatory Power
Since the publication of Hempel and Oppenheim's (Reference Hempel and Oppenheim1948) classic investigation into “the logic of explanation,” philosophers of science have earnestly been seeking an analysis of the nature of explanation. Necessity (Glymour Reference Glymour1980), statistical relevance (Salmon Reference Salmon and Salmon1971), inference and reason (Hempel and Oppenheim Reference Hempel and Oppenheim1948; Hempel Reference Hempel1965), familiarity (Friedman Reference Friedman1974), unification (Friedman Reference Friedman1974; Kitcher Reference Kitcher, Kitcher and Salmon1989), causation (Salmon Reference Salmon1984; Woodward Reference Woodward2003), and mechanism (Machamer, Darden, and Craver Reference Machamer, Darden and Craver2000) are only some of the most popular concepts that such philosophers draw on in the attempt to describe necessary and sufficient conditions under which a theory explains some proposition.Footnote 1 A related project that is, on the other hand, much less often pursued by philosophers today is the attempt to analyze the strength of an explanation—that is, the degree of explanatory power that a particular explanans has over its explanandum. Such an analysis would clarify the conditions under which hypotheses are judged to provide strong versus weak explanations of some proposition, and it would also clarify the meaning of comparative explanatory judgments such as “hypothesis A provides a better explanation of this fact than does hypothesis B.”
Given the nature of these two projects, the fact that the first receives so much more philosophical attention than the second can hardly be explained by appeal to any substantial difference in their relative philosophical imports. Certainly, the first project has great philosophical significance; after all, humans on the individual and social levels are constantly seeking and formulating explanations. Given the ubiquity of explanation in human cognition and action, it is both surprising that this concept turns out to be so analytically impenetrable and critical that philosophers continue to strive for an understanding of this notion.Footnote 2 The second project is, however, also immensely philosophically important. Humans regularly make judgments of explanatory power and then use these judgments to develop preferences for hypotheses or even to infer outright to the truth of certain hypotheses. Much of human reasoning—again, on individual and social levels—makes use of judgments of explanatory power. Ultimately then, in order to understand and evaluate human reasoning generally, philosophers need to come to a better understanding of explanatory power.
The relative imbalance in the amount of philosophical attention that these two projects receive is more likely due to the prima facie plausible but ultimately unfounded assumption that one must have an analysis of explanation before seeking an analysis of explanatory power. This assumption is made compelling by the fact that in order to analyze the strength of something, one must have some clarity about what that thing is. However, it is shown to be much less tenable in light of the fact that humans do generally have some fairly clear intuitions concerning explanation. The fact that there is no consensus among philosophers today over the precise, necessary, and sufficient conditions for explanation does not imply that humans do not generally have a firm semantic grasp on the concept of explanation. Just how firm a semantic grasp on this concept humans actually have is an interesting question. One claim of this article will be that our grasp on the notion of explanation is at least sufficiently strong to ground a precise formal analysis of explanatory power, even if it is not strong enough to determine a general account of the nature of explanation.
This article attempts to bring more attention to the second project above by formulating a Bayesian analysis of explanatory power. Moreover, the account given here does this without committing to any particular theory of the nature of explanation. Instead of assuming the correctness of a theory of explanation and then attempting to build a measure of explanatory power derivatively from this theory, we begin by laying out several more primitive adequacy conditions that, we argue, an analysis of explanatory power should satisfy. We then show that these intuitive adequacy conditions are sufficient to define for us a unique probabilistic analysis and measure of explanatory power.
Before proceeding, two more important clarifications are necessary. First, we take no position on whether our analysis captures the notion of explanatory power generally; it is consistent with our account that there be other concepts that go by this name but which do not fit our measure.Footnote 3 What we do claim, however, is that our account captures at least one familiar and epistemically compelling sense of explanatory power that is common to human reasoning.
Second, because our explicandum is the strength or power of an explanation, we restrict ourselves in presenting our conditions of adequacy to speaking of theories that do in fact provide explanations of the explanandum in question.Footnote 4 This account thus is not intended to reveal the conditions under which a theory is explanatory of some proposition (that is, after all, the aim of an account of explanation rather than an account of explanatory power); rather, its goal is to reveal, for any theory already known to provide such an explanation, just how strong that explanation is. Ultimately then, this article offers a probabilistic logic of explanation that tells us the explanatory power of a theory (explanans) relative to some proposition (explanandum), given that that theory constitutes an explanation of that proposition. In this way, this article forestalls the objection that two statements may stand in the probabilistic relation described while not simultaneously constituting an explanation.
2. The Measure of Explanatory Power 
The sense of explanatory power that this article seeks to analyze has to do with a hypothesis's ability to decrease the degree to which we find the explanandum surprising (i.e., its ability to increase the degree to which we expect the explanandum). More specifically, a hypothesis offers a powerful explanation of a proposition, in this sense, to the extent that it makes that proposition less surprising. This sense of explanatory power dominates statistical reasoning in which scientists are “explaining away” surprise in the data by means of assuming a specific statistical model (e.g., in the omnipresent linear regression procedures). But the explaining hypotheses need not be probabilistic; for example, a geologist will accept a prehistoric earthquake as explanatory of certain observed deformations in layers of bedrock to the extent that deformations of that particular character, in that particular layer of bedrock, and so on, would be less surprising given the occurrence of such an earthquake.
This notion finds precedence in many classic discussions of explanation. Perhaps its clearest historical expression occurs when Peirce (Reference Peirce, Hartshorne and Weiss1931–35, 5.189) identifies the explanatoriness of a hypothesis with its ability to render an otherwise “surprising fact” as “a matter of course.”Footnote 5 This sense of explanatory power may also be seen as underlying many of the most popular accounts of explanation. Most obviously, Deductive-Nomological and Inductive-Statistical accounts (Hempel Reference Hempel1965) and necessity accounts (Glymour Reference Glymour1980) explicitly analyze explanation in such a way that a theory that is judged to be explanatory of some explanandum will necessarily increase the degree to which we expect that explanandum.
Our formal analysis of this concept proceeds in two stages: in the first stage, a parsimonious set of adequacy conditions is used to determine a measure of explanatory power up to ordinal equivalence. In other words, we show that, for all pairs of functions f and f′ that satisfy these adequacy conditions, 
if and only if 
; all such measures thus impose the same ordinal relations on judgments of explanatory power. This is already a substantial achievement. In the second stage, we introduce more adequacy conditions in order to determine a unique measure of explanatory power (from among the class of ordinally equivalent measures).
In the remainder of the article, we make the assumption that the probability distribution is regular (i.e., only tautologies and contradictions are awarded rational degrees of belief of 1 and 0). This is not strictly required to derive the results below, but it makes the calculations and motivations much more elegant.
2.1. Uniqueness Up to Ordinal Equivalence
The first adequacy condition is, we suggest, rather uncontentious. It is a purely formal condition intended to specify the probabilistic nature and limits of our explication (which we denote 
):
CA1 (Formal Structure): For any probability space and regular probability measure 
, 
is a measurable function from two propositions 
to a real number 
. This function is defined on all pairs of contingent propositions (i.e., cases such as 
, etc., are not in the domain of 
).Footnote 6 This implies by Bayes's Theorem that we can represent 
as a function of 
, 
, and 
, and we demand that any such function be analytic.Footnote 7
The next adequacy condition specifies, in probabilistic terms, the general notion of explanatory power that we are interested in analyzing. As mentioned, a hypothesis offers a powerful explanation of a proposition, in the sense that we have in mind, to the extent that it makes that proposition less surprising. In order to state this probabilistically, the key interpretive move is to formalize a decrease in surprise (or increase in expectedness) as an increase in probability. This move may seem dubious, depending on one's interpretation of probability. Given a physical interpretation (e.g., a relative frequency or propensity interpretation), it would be difficult indeed to saddle such a psychological concept as surprise with a probabilistic account. However, when probabilities are themselves given a more psychological interpretation (whether in terms of simple degrees of belief or the more normative degrees of rational belief), this move makes sense. In this case, probabilities map neatly onto degrees of expectedness.Footnote 8 Accordingly, insofar as surprise is inversely related to expectedness (the more surprising a proposition, the less one expects it to be true), it is straightforwardly related to probabilities. Thus, if h decreases the degree to which e is surprising, we represent this with the inequality 
. The strength of this inequality corresponds to the degree of statistical relevance between e and h, giving us:
CA2 (Positive Relevance): Ceteris paribus, the greater the degree of statistical relevance between e and h, the greater the value of 
.
The third condition of adequacy defines a point at which explanatory power is unaffected. If 
does nothing to increase or decrease the degree to which e, 
, or any logical combination of e and 
are surprising, then 
will not make e any more or less surprising than 
by itself already does. In this case, tacking 
on to our hypothesis has no effect on the degree to which that hypothesis alleviates our surprise over e. Given that explanatory power has to do with a hypothesis's ability to render its explanandum less surprising, we can state this in other words: if 
has no explanatory power whatever relative to e, 
, or any logical combination of e and 
, then explanandum e will be explained no better or worse by conjoining 
to our explanans 
. Making use of the above probabilistic interpretation of a decrease in surprise, this can be stated more formally as follows:
CA3 (Irrelevant Conjunction): If 
and 
and 
, then 
.
The fourth adequacy condition postulates that the measure of explanatory power should, if the negation of the hypothesis entails the explanandum, not depend on the prior plausibility of the explanans. This is because the extent to which an explanatory hypothesis alleviates the surprising nature of some explanandum does not depend on considerations of how likely that hypothesis is in and of itself, independent of its relation to the evidence. More precisely, it would strike us as odd if we could, given that 
entails e, rewrite 
as a function of 
plus either 
or 
. In that case, the plausibility of the hypothesis in itself would affect the degree of explanatory power that h lends to e. Since we find such a feature unattractive, we require what follows:
CA4 (Irrelevance of Priors): If 
entails e, then the values of 
do not depend on the values of 
.Footnote 9
We acknowledge that intuitions might not be strong enough to make a conclusive case for CA4. However, affirming that condition is certainly more plausible than denying it. Moreover, it only applies to a very specific case (
), while making no restrictions on the behavior of the explanatory power measure in more general circumstances.
These four conditions allow us to derive the following theorem (proof in app. A):
Theorem 1. All measures of explanatory power satisfying CA1–CA4 are monotonically increasing functions of the posterior ratio 
.
From this theorem, two important corollaries follow. First, we can derive a result specifying the conditions under which 
takes its maximal and minimal values (proof in app. A):
Corollary 1. Measure 
takes maximal value if and only if h entails e and minimal value if and only if h implies 
.
Note that this result fits well with the concept of explanatory power that we are analyzing, according to which a hypothesis explains some proposition to the extent that it renders that proposition less surprising (more expected). Given this notion, any h ought to be maximally explanatorily powerful regarding some e when it renders e maximally unsurprising (expected), and this occurs whenever h guarantees the truth of e (
). Similarly, h should be minimally explanatory of e if e is maximally surprising in the light of h, and this occurs whenever h implies the falsity of e (
).
The second corollary constitutes our desired ordinal equivalence result:
Corollary 2. All measures of explanatory power satisfying CA1–CA4 are ordinally equivalent.
To see why the corollary follows from the theorem, let r be the posterior ratio of the pair 
, and let r′ be the posterior ratio of the pair 
. Without loss of generality, assume 
. Then, for any functions f and f′ that satisfy CA1–CA4, we obtain the following inequalities:
where the inequalities are immediate consequences of theorem 1. So any f and f′ satisfying CA1–CA4 always impose the same ordinal judgments, completing the first stage of our analysis.
2.2. Uniqueness of
This section pursues the second task of choosing a specific and suitably normalized measure of explanatory power out of the class of ordinally equivalent measures determined by CA1–CA4. To begin, we introduce an additional, purely formal requirement of our measure:
CA5 (Normality and Form): Measure 
is the ratio of two functions of 
, 
, 
, and 
, each of which are homogeneous in their arguments to the least possible degree 
.Footnote 10
Representing 
as the ratio of two functions serves the purpose of normalization. The terms 
, 
, 
and 
fully determine the probability distribution over the truth-functional compounds of e and h, so it is appropriate to represent 
as a function of them. Additionally, the requirement that our two functions be “homogenous in their arguments to the least possible degree 
” reflects a minimal and well-defined simplicity assumption akin to those advocated by Carnap (Reference Carnap1950) and Kemeny and Oppenheim (Reference Kemeny and Oppenheim1952, 315). This assumption effectively limits our search for a unique measure of explanatory power to those that are the most cognitively accessible and useful.
Of course, larger values of 
indicate greater explanatory power of h with respect to e. Measure 
's maximal value, 
, indicates the point at which explanans h fully explains its explanandum e, and 
(
's minimal value) indicates the minimal explanatory power for h relative to e (where h provides a full explanation for e being false). The neutral point at which h lacks any explanatory power whatever relative to e is represented by 
.
While we have provided an informal description of the point at which 
should take on its neutral value 0 (when h lacks any explanatory power whatever relative to e), it is still left to us to define this point formally. Given our notion of explanatory power, a complete lack of explanatory power is straightforwardly identified with the scenario in which h does nothing to increase or decrease the degree to which e is surprising. Probabilistically, in such cases, h and e are statistically irrelevant to (independent of) one another:
CA6 (Neutrality): For explanatory hypothesis h, 
if and only if 
.
The final adequacy condition requires that the more h explains e, the less it explains its negation. This requirement is appropriate given 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. Corollary 1 and neutrality provide a further rationale for this condition. Corollary 1 tells us that 
should be maximal only if 
. Importantly, in such a case, 
, and this value corresponds to the point at which this same corollary demands 
to be minimal. In other words, given corollary 1, we see that 
takes its maximal value precisely when 
takes its minimal value and vice versa. Also, we know that 
and 
should always equal zero at the same point given that 
if and only if 
. The formal condition of adequacy that most naturally sums up all of these points is as follows.
CA7 (Symmetry): 
.
These three conditions of adequacy, when added to CA1–CA4, conjointly determine a unique measure of explanatory power as stated in the following theorem (proof in app. B).Footnote 11
Theorem 2. The only measure that satisfies CA1–CA7 is
Remark. Since, for 
,
it is easy to see that 
is indeed an increasing function of the posterior ratio.
Thus, these conditions provide us with an intuitively grounded, unique measure of explanatory power.Footnote 12
3. Theorems of
We have proposed the above seven conditions of adequacy as intuitively plausible constraints on a measure of explanatory power. Accordingly, the fact that these conditions are sufficient to determine 
already constitutes a strong argument in this measure's favor. Nonetheless, we proceed in this section to strengthen the case for 
by highlighting some important theorems that follow from adopting this measure. Ultimately, the point of this section is to defend further our assertion that 
is well behaved in the sense that it gives results that match our clear intuitions about the concept of explanatory power, even in one case where other proposed measures fail to do so.Footnote 13
3.1. Addition of Irrelevant Evidence
Good (Reference Good1960) and, more recently, McGrew (Reference McGrew2003) both explicate h's degree of explanatory power relative to e in terms of the amount of information concerning h provided by e. This results in the following intuitive and simple measure of explanatory power:Footnote 14
According to this measure, the explanatory power of explanans h must remain constant whenever we add an irrelevant proposition e′ to explanandum e (where proposition e′ is irrelevant in the sense that it is statistically independent of h in the light of e):
This is, however, a very counterintuitive result. To see this, 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, and this fact is reflected nicely by measure I:
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 I judges that Einstein's 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.
Instead, it seems that, as the evidence becomes less statistically relevant to some explanatory hypothesis h (with the addition of irrelevant propositions), it ought to be the case that the explanatory power of h relative to that evidence approaches the value at which it is judged to be explanatorily irrelevant to the evidence (
). Thus, if 
, then this value should decrease with the addition of e′ to our evidence: 
. Similarly, if 
, then this value should increase with the addition of e′: 
. And finally, if 
, then this value should remain constant at 
. Measure 
gives these general results as shown in the following theorem (proof in app. C):
Theorem 3. If 
—or equivalently, 
—and 
, then:
if
, then ,if
, then , andif
, then .
, then 
,
, then 
, and
, then 
.3.2. Addition of Relevant Evidence
Next, we explore whether 
is well behaved in those circumstances in which we strengthen our explanandum by adding to it relevant evidence. Consider the case in which h has some explanatory power relative to e so that 
(i.e., h has any degree of explanatory power relative to e greater than the minimal degree). What should happen to this degree of explanatory power if we gather some new information e′ that, in the light of e, we know is explained by h to the worst possible degree?
To take a simple example, imagine that police investigators hypothesize that Jones murdered Smith (h) in light of the facts that Jones's fingerprints were found near the dead body and Jones recently had discovered that his wife and Smith were having an affair (e). Now suppose that the investigators discover video footage that proves that Jones was not at the scene of the murder on the day and time that it took place (e′). Clearly, h is no longer such a good explanation of our evidence once e′ is added; in fact, h now seems to be a maximally poor explanation of 
precisely because of the addition of e′ (h cannot possibly explain 
because e′ rules h out entirely). Thus, in such cases, the explanatory power of h relative to the new collection of evidence 
should be less than that relative to the original evidence e; in fact, it should be minimal with the addition of e′. This holds true in terms of 
, as shown in the following theorem (proof in app. D):
Theorem 4. If 
and 
(in which case, it also must be true that 
), then 
.
Alternatively, we may ask what intuitively should happen in the same circumstance (adding the condition that h does not have the maximal degree of explanatory power relative to e—i.e., 
) but where the new information we gain e′ is fully explained by h in the light of our evidence e. Let h and e be the same as in the above example, and now imagine that investigators discover video footage that proves that Jones was at the scene of the murder on the day and time that it took place (e′). In this case, h becomes an even better explanation of the evidence precisely because of the addition of e′ to the evidence. Thus, in such cases, we would expect the explanatory power of h relative to the new evidence 
to be greater than that relative to e alone. Again, 
agrees with our intuition here (proof in app. D):
Theorem 5. If 
and h does not already fully explain e or its negation (
) and 
, then 
.
While these last two theorems are highly intuitive, they are also quite limited in their applicability. Both theorems require in their antecedent conditions that one's evidence be strengthened with the addition of some e′ that is itself either maximally or minimally explained by h in the light of e. However, our intuitions reach to another class of related examples in which the additional evidence need not be maximally or minimally explained in this way. In situations in which h explains e to some positive degree, it is intuitive to think that the addition of any new piece of evidence that is negatively explained by (made more surprising by) h in the light of e will decrease h's overall degree of explanatory power. Similarly, whenever h has some negative degree of explanatory power relative to e, it is plausible to think that the addition of any new piece of evidence that is positively explained by (made less surprising by) h in the light of e will increase h's overall degree of explanatory power. These intuitions are captured in the following theorem of 
(proof in app. D):
Theorem 6. If 
, then if 
, then 
. Alternatively, if 
, then if 
, then 
.
4. Conclusions
Above, we have shown the following: first, 
is a member of the specific family of ordinally equivalent measures that satisfy our first four adequacy conditions. Moreover, among the measures included in this class, 
itself uniquely satisfies the additional conditions CA5–CA7. The theorems presented in the last section strengthen the case for 
by showing that this measure does indeed seem to correspond well and quite generally to many of our clear explanatory intuitions. In light of all of this, we argue that 
is manifestly an intuitively appealing formal account of explanatory power.
The acceptance of 
opens the door to a wide variety of potentially fruitful, intriguing further questions for research. Here, we limit ourselves to describing very briefly two of these projects that seem to us to be particularly fascinating and manageable with 
in hand.
First, measure 
makes questions pertaining to the normativity of explanatory considerations much more tractable, at least from a Bayesian perspective. Given this probabilistic rendering of the concept of explanatory power, one has a new ability to ask and attempt to answer questions such as, “Does the ability of a hypothesis to explain some known fact itself constitute reason in that hypothesis's favor in any sense?” or, relatedly, “Is there any necessary sense in which explanatory power is tied to the overall probability of an hypothesis?” Such questions call out for more formal work in terms of 
attempting to show whether, and how closely, 
might be related to 
. This further work would have important bearing on debates over the general normativity of explanatory power; it would also potentially lend much insight into discussions of Inference to the Best Explanation and its vices or virtues.
Second, we have presented and defended 
here as an accurate normative account of explanatory power in the following sense: in the wide space of cases in which our conditions of adequacy are rationally compelling and intuitively applicable, one ought to think of explanatory power in accord with the results of 
. However, one may wonder whether people actually have explanatory intuitions that accord with this normative measure. With 
in hand, this question becomes quite susceptible to further study. In effect, the question is whether 
is, in addition to being an accurate normative account of explanatory power, a good predictor of people's actual judgments of the same. This question is, of course, an empirical one and thus requires an empirical study into the degree of fit between human judgments and theoretical results provided by 
. Such a study could provide important insights both for the psychology of human reasoning and for the philosophy of explanation.Footnote 15
Appendix A Proof of Theorem 1 and Corollary 1
Theorem 1. All measures of explanatory power satisfying CA1–CA4 are monotonically increasing functions of the posterior ratio 
.
Proof. 
, 
, and 
jointly determine the probability distribution of the pair 
, so we can represent 
as a function of these values: there is a 
such that 
.
First, note that whenever the assumptions of CA3 are satisfied (i.e., whenever 
is independent of all e, 
, and 
), the following equalities hold:
Now, for all values of 
, we can choose propositions e, 
, and 
and probability distributions over these such that the independence assumptions of CA3 are satisfied, and 
, 
, 
, and 
. Due to CA1, we can always find such propositions and distributions so long as 
is applicable. The above equations then imply that 
and 
. Applying CA3 (
) yields the general fact that
Consider now the case that 
entails e; that is, 
. Assume that 
could be written as a function of 
alone. Accordingly, there would be a function 
such that
If we choose 
, it follows from equations (A2) and (A3) that
In other words, g (and 
) would then be constant on the triangle 
for any fixed 
. Now, since g is an analytic function (due to CA1), its restriction 
(for fixed x) must be analytic as well. This entails in particular that if 
is constant on some nonempty open set 
, then it is constant everywhere:
1. All derivatives of a locally constant function vanish in that environment (Theorem of Calculus).
2. We write, by CA1,
as a Taylor series expanded around a fixed point :Since all derivatives of in the set are zero, all terms of the Taylor series, except the first one () vanish.
as a Taylor series expanded around a fixed point 
:
in the set 
are zero, all terms of the Taylor series, except the first one (
) vanish.Thus, 
must be constant everywhere. But this would violate the statistical relevance condition CA2 since g (and 
) would then depend on 
alone and not be sensitive to any form of statistical relevance between e and h.
Thus, whenever 
entails e, 
either depends on its second argument alone or on both arguments. The latter case implies that there must be pairs 
and 
with 
such that
Note that if 
, we obtain
and so we can write 
as a function of 
and 
.
Combining (A5) and (A6), and keeping in mind that g cannot depend on 
alone, we obtain that there are pairs 
and 
such that
This can only be the case if the prior probability (
and 
, respectively) has an impact on the value of g (and thus on 
), in contradiction with CA4. Thus, equality in (A5) holds whenever 
. Hence, 
cannot depend on both arguments, and it can be written as a function of its second argument alone.
Thus, for any 
, there must be a 
such that
This establishes that 
is a function of the posterior ratio if h and e are negatively relevant to each other. By applying analyticity of 
once more, we see that 
is a function of the posterior ratio 
in its entire domain (i.e., also if e and h are positively relevant to each other or independent).
Finally, CA2 implies that this function must be monotonically increasing since, otherwise, explanatory power would not increase with statistical relevance (of which the posterior probability is a measure). Evidently, any such function satisfies CA1–CA4. QED
Corollary 1. Measure 
takes maximal value if and only if h entails e and minimal value if and only if h implies 
.
Proof. Since 
is an increasing function of the posterior ratio 
, 
is maximal if and only if 
. Due to the regularity of 
, this is the case if and only if 
entails 
, in other words, if and only if h entails e. The case of minimality is proven analogously. QED
Appendix B Proof of Theorem 2 (Uniqueness of )
)
Theorem 2. The only measure that satisfies CA1–CA7 is
Let 
, 
, 
, and 
with 
. Write 
(by CA5).
Lemma 1. There is no normalized function 
of degree 1 that satisfies our desiderata CA1–CA7.
Proof. If there were such a function, the numerator would have the form 
. If e and h are independent, the numerator must vanish, by means of CA6. In other words, for those values of 
, we demand 
. Below, we list four different realizations of 
that make e and h independent, namely, (1/2, 1/4, 1/6, 1/12), (1/2, 1/3, 1/10, 1/15), (1/2, 3/8, 1/14, 3/56), and (1/4, 1/4, 1/4, 1/4). Since these vectors are linearly independent (i.e., their span has dimension 4), it must be the case that 
. Hence, there is no such function of degree 1. QED
Lemma 2. CA3 entails that for any value of 
,
Proof. For any 
, we choose e, 
such that
Moreover, we choose a 
such that the antecedent conditions of CA3 are satisfied, and we let 
. Applying the independencies between 
, e, and 
and recalling (A1), we obtain
and similarly
Making use of these equations, we see directly that CA3—that is, 
—implies equation (B1). QED
Proof of Theorem 2 (Uniqueness of 
). Lemma 1 shows that there is no normalized function 
of degree 1 that satisfies our desiderata. Our proof is constructive: we show that there is exactly one such function of degree 2, and then we are done, due to the formal requirements set out in CA5. By CA5, we look for a function of the form
We begin by investigating the numerator.Footnote 16 CA6 tells us that it has to be zero if 
, in other words, if
Making use of 
, we conclude that this is the case if and only if 
:
The only way to satisfy the constraint (B3) is to set 
and to set all other coefficients in the numerator to zero. All other choices of coefficients do not work since the dependencies are nonlinear. Hence, f becomes
Now, we make use of corollary 1 and CA7 in order to tackle the coefficients in the denominator. Corollary 1 (maximality) entails that 
if 
, and CA7 (symmetry) is equivalent to
First, applying corollary 1 yields 
, and by a comparison of coefficients, we get 
and 
. Similarly, we obtain 
and 
from 
, combining corollary 1 with CA7 (i.e., eq. [B4]).
Now, f has the form
Assume now that 
. Let 
. We know by corollary 1 that in this case, 
. Since the numerator vanishes, the denominator must vanish too, but by 
it stays bounded away from zero, leading to a contradiction (
). Hence, 
. In a similar vein, we can argue for 
by letting 
and for 
by letting 
(making use of [B4] again: 
).
Thus, f can be written as
by letting 
.
It remains to make use of CA3 in order to fix the value of 
. Set 
in (B1) and make use of 
(lemma 2) and the restrictions on f captured in (B5). By making use of (B1), we obtain the general constraint
For (B6) to be true in general, we have to demand that 
, which implies that 
. Hence,
implying
which is the unique function satisfying all our desiderata. QED
Appendix C Proof of Theorem 3
Theorem 3. If 
—or equivalently, 
—and 
, then:
if
, then ,if
, then , andif
, then .
, then 
,
, then 
, and
, then 
. Proof. Since 
and the posterior ratio 
are ordinally equivalent, we can focus our analysis on that quantity:
This quantity is greater than one if and only if the numerator exceeds the denominator, that is, if and only if
which is satisfied if and only if 
and not satisfied otherwise. Thus, 
(and 
) if and only if 
. The other two cases follow directly from (C2).
It remains to show that 
and 
always have the same sign. This follows from the fact that
Thus, h is positively relevant to e if and only if it is positively relevant to 
. By CA2 and CA6, this implies that 
if and only if 
and vice versa for negative relevance. QED
Appendix D Proofs of Theorems 4–6
Theorem 4. If 
and 
(in which case, it also must be true that 
), then 
.
Proof. Under the assumptions of theorem 1, by application of Bayes's Theorem,
Thus,
QED
Theorem 5. If 
and h does not already fully explain e or its negation (
) and 
, then 
.
Proof. Note first that
Analogous to theorem 3, we prove this theorem by comparing the posterior ratios 
and 
and applying equation (D1):
since, by assumption, 
. This implies that 
. QED
Theorem 6. If 
, then if 
, then 
. Alternatively, if 
, then if 
, then 
.
Proof. First, we note that if 
, then also 
. Then we apply the same approach as in the previous proofs:
This is exactly the term in the last line of (C1). We have already shown in the proof of theorem 3 that this quantity is greater than 1 if and only if 
(i.e., if 
). This suffices to prove the first half of theorem 6. The reverse case is proved in exactly the same way. QED
” in Bayesian formalizations. Nonetheless, here and in the remainder of this article, we choose for the sake of transparency and simplicity in exposition to leave k implicit in all formalizations.
so that, if 
implies e, either 
or 
. Note that in this case, 
, so 
. If a function f with the above properties did not exist, we could not speak of a way in which 
would be independent of considerations of prior plausibility of h.
. It can be proven that 
is also the only measure that satisfies CA3, CA5, CA6, CA7, and corollary 1, although we do not include this separate proof in this article.
is closely related to Kemeny and Oppenheim's (
is 
with h and e reversed (h is replaced by e, and e is replaced by h).
. It should be noted that Popper's measure is ordinally equivalent to Good's in the same sense that 
is ordinally equivalent to the posterior ratio 
. Thus, the problem we present here for Good's measure is also a problem for Popper's.
(and other candidate measures of explanatory power), see Schupbach (
by means of symmetry (CA7) alone. (Take the counterexample 
, which even satisfies corollary 1.) Hence, our proof is truly original.
