Purely Probabilistic Measures of Explanatory Power: A Critique

Abstract

All extant purely probabilistic measures of explanatory power satisfy the following technical condition: if Pr(E | H1) > Pr(E | H2) and Pr(E | ∼H1) < Pr(E | ∼H2), then H1’s explanatory power with respect to E is greater than H2’s explanatory power with respect to E. We argue that any measure satisfying this condition faces three serious problems—the Problem of Temporal Shallowness, the Problem of Negative Causal Interactions, and the Problem of Nonexplanations. We further argue that many such measures face a fourth problem—the Problem of Explanatory Irrelevance.

1. Introduction

There are clear cases in which two hypotheses each explain an observation, but one of them does so with greater explanatory power (or strength). Here’s one example:

Suppose that a randomized control trial is performed to see how the incidence of heart disease is affected by eliminating fried foods from one’s diet, and that the result is that the reduction is 10%. Now consider two causal models that might explain why this is the outcome. The first says that the reduction should have been between 9% and 11% while the second says that the reduction should have been between 2% and 20%. Suppose that both models are theoretically well-grounded, and that each is explanatorily relevant to the observed reduction. We think that there is a clear sense in which the first model has greater explanatory power than the second with respect to the observed reduction. Each model explains the observed reduction to some extent, but the explanation provided by the first is stronger than the one provided by the second.

Here’s another:

Suppose Jones smoked cigarettes and inhaled asbestos, and that both of these causally contributed to her getting lung cancer, where a person’s risk of lung cancer is greater if she inhales both carcinogens than if she inhales just one. Jones’s getting lung cancer is explained by her smoking cigarettes, by her inhaling asbestos, and by her doing both, but the smoking + asbestos explanation is more powerful than the smoking explanation is and it’s also more powerful than the asbestos explanation is.

We’ll consider other examples in what follows, but these two suffice to set the stage for our main question: Is there a plausible mathematical measure of the power with which a hypothesis H explains an explanadum E?

That the answer is affirmative has prima facie plausibility. Each of the previous examples can be fleshed out so that the different explanans statements have different probabilities and confer different probabilities on their explananda. Perhaps these probabilistic differences ground differences in explanatory power, and perhaps more generally explanatory power can be measured in terms of probabilities.

The idea of a mathematical measure of explanatory power carries baggage that more modest ideas about explanatory power do not shoulder. Here is a less demanding question:

For any H1, H2, and O, if H1 and H2 each explains O, is there a plausible criterion that determines whether H1’s explanation of O is more powerful than H2’s?

This is the question of whether there is a criterion for comparative explanatory power, where both hypotheses address the same observation. No numbers are involved. A measure of explanatory power does much more. For example, it would describe whether the power of H1’s explanation of O1 exceeds the power of H2’s explanation of O2, and by how much. That may sound like a paradigmatic example of apples and oranges, but that’s what a mathematical measure delivers. If you think that the power of an explanation has multiple orthogonal dimensions, with the result that H1 may be a more powerful explanation of O than H2 is on some dimensions, while the reverse is true on others (as do Ylikoski and Kuorikoski 2010), you may think that the quest for such a measure is quixotic.

That said, several able philosophers have dared to plunge ahead. Are they tilting at windmills? One such attempt is Schupbach and Sprenger’s (2011). They set out several conditions of adequacy on measures of explanatory power, which they use to derive the following measure:

$$\[{\rm{EXP}}{{\rm{O}}_{{\rm{SS}}}}\left( {{\rm{E}},\;{\rm{H}}} \right) = \frac{{\Pr ({\rm{H}}\;{\rm{|E}}) - \Pr ({\rm{H}}\;{\rm{|}} \sim {\rm{E}})}}{{\Pr ({\rm{H}}\;{\rm{|E}}) + \Pr ({\rm{H}}\;{\rm{|}} \sim {\rm{E}})}}\]$$

EXPO_SS is a purely probabilistic measure of explanatory power (a PPM, for short) in that the value of EXPO_SS(E, H) is fully determined by probabilities.^{1, 2}

Schupbach and Sprenger’s measure is not the only game in town. There are many—indeed, very many—purely probabilistic measures of explanatory power in logical space in addition to EXPO_SS. Here are four:

$${\rm{EXP}}{{\rm{O}}_{{\rm{CT}}}}\left( {{\rm{E}},\;{\rm{H}}} \right) = \!\left\{ {\matrix{ {{{\Pr ({\rm{E}}\;{\rm{|H}}) - {\rm{Pr}}\left( {\rm{E}} \right)} \over {1 - {\rm{Pr}}\left( {\rm{E}} \right)}}\;{\rm{if}}\Pr ({\rm{E}}\;{\rm{|H}})\; \ge {\rm{Pr}}\left( {\rm{E}} \right)} \cr {{{\Pr ({\rm{E}}\;{\rm{|H}}) - {\rm{Pr}}\left( {\rm{E}} \right)} \over {{\rm{Pr}}\left( {\rm{E}} \right)}}{\rm{\ if}}\Pr ({\rm{E}}\;{\rm{|H}}) \lt Pr\left( E \right)} \cr } } \right.$$

$${\rm{EXP}}{{\rm{O}}_{\rm{G}}}\left( {{\rm{E}},{\rm{\;H}}} \right) = {\rm{log}}\left[ {{{\Pr {\rm{(E\;|H}})} \over {{\rm{Pr}}\left( {\rm{E}} \right)}}} \right]$$

$$\rm EXP{O_C}\left( {E,{\rm{ }}H} \right){\rm{ }} = {\rm{ }}Pr\left( {E{\rm{ }}|{\rm{ }}H} \right){\rm{ }}-{\rm{ }}Pr\left( {E{\rm{ }}|{\rm{ }}\sim H} \right)$$

$$\rm EXP{O_D}\left( {E,{\rm{ }}H} \right){\rm{ }} = {\rm{ }}Pr\left( {E{\rm{ }}|{\rm{ }}H} \right){\rm{ }}-{\rm{ }}Pr\left( E \right)$$

EXPO_CT is defended in Crupi and Tentori (2012). EXPO_G is defended in Good (1960).³ EXPO_C and EXPO_D are mentioned but not defended in Schupbach (2011). In what follows, we’ll use “The Five” to refer to these five measures.⁴

No two of The Five are ordinally equivalent, but they all satisfy the following two conditions, each of which involves two likelihood inequalities:

TLI-1. If (a) Pr(E | H1) > Pr(E | H2) and (b) Pr(E | ∼H1) < Pr(E | ∼H2), then EXPO(E, H1) > EXPO(E, H2).

TLI-2. If (a) Pr(E | H1) ≥ Pr(E | H2) and (b) Pr(E | ∼H1) ≤ Pr(E | ∼H2), then EXPO(E, H1) ≥ EXPO(E, H2).

EXPO_SS, EXPO_CT, EXPO_G, and EXPO_D meet these conditions because they meet the following stronger conditions, each of which involves one likelihood inequality:

OLI-1. If Pr(E | H1) > Pr(E | H2), then EXPO(E, H1) > EXPO(E, H2).

OLI-2. If Pr(E | H1) ≥ Pr(E | H2), then EXPO(E, H1) ≥ EXPO(E, H2).

Neither OLI-1 nor OLI-2 is satisfied by EXPO_C because this measure says that H’s explanatory power with respect to E depends not just on Pr(E | H) but also on Pr(E | ∼H).⁵

Our focus in what follows will be on “TLI” measures, defined as purely probabilistic measures of explanatory power that meet both TLI-1 and TLI-2. These include each of The Five along with many additional measures. Consider, for example, the following:

$${\rm{EXP}}{{\rm{O}}_{{\rm{LR}}}}\left( {{\rm{E}},{\rm{\;H}}} \right) = {\rm{log}}\left[ {{{\Pr {\rm{(E\;|H}})} \over {{\rm{Pr(E\;|}}\sim{\rm{H}})}}} \right]$$

This measure and measures ordinally equivalent to it have been discussed in the context of confirmation (as a measure of the degree to which E incrementally confirms H),⁶ but not, as far as we are aware, in the context of explanatory power. But, as EXPO_LR meets both TLI-1 and TLI-2, it is included in our target class of purely probabilistic measures of explanatory power. The question is whether any TLI measure is adequate.

We want to emphasize that the issue here is not about inference. For example, in the cancer case described at the outset, we aren’t claiming that the smoking + asbestos explanation is more probable than the smoking explanation. It isn’t. The claim, rather, is that the smoking + asbestos explanation is stronger than the smoking explanation in that it sheds more light on why Jones has lung cancer. Both explanations shed light on why Jones has lung cancer, but the first does this to a greater extent.

The remainder of this article is organized as follows. In section 2, we present the Problem of Temporal Shallowness. In section 3, we introduce the Problem of Negative Causal Interactions. In section 4, we consider a reply that might be made to the objections we presented in the previous two sections. In section 5, we describe the Problem of Nonexplanations. In section 6, we consider a possible reply to what we’ve said about that problem. In section 7, we discuss the Problem of Explanatory Irrelevance. In section 8, we conclude.

2. The problem of temporal shallowness

Consider a causal chain from a distal cause C_d, to a proximate cause C_p, and then to an effect E. For example, you throw a baseball, which causes the baseball to hit a window, which causes the window to break. Someone then asks: Why did the window break? Consider two answers:

1. (2.1) A baseball hit the window and was going 65 mph.

2. (2.2) You threw a baseball, which caused the baseball to hit the window at 65 mph.

Notice that proposition (2.2) entails proposition (2.1) but not conversely. Explanation (2.1) is temporally shallow, in that it describes only the more proximate cause. Explanation (2.2) has more temporal depth, in that it traces the effect back to a proximate cause, and traces that proximate cause back to your dastardly deed. This is a case, we suggest, in which greater temporal depth makes for greater explanatory power: Explanation (2.2) is stronger than explanation (2.1).

The baseball example is mundane, but the point of the example applies to scientific examples as well. Evolutionary biologists and philosophers of biology routinely distinguish “proximate” from “ultimate” explanations. Ernst Mayr (1961) used this terminology to distinguish two ways of explaining the phenotypes of organisms. For example, why do sunflowers turn toward the sun? A proximate explanation of that behavior would describe the mechanisms inside individual sunflowers that cause phototropism. An ultimate explanation would describe the evolutionary processes, including natural selection, that caused that trait to increase in frequency and remain at high frequency in the lineage leading to present day sunflowers. As Mayr (1961, 1503) says, “[t]here is always a proximate set of causes and an ultimate set of causes; both have to be explained and interpreted for a complete understanding of the given phenomenon.” Mayr opposes temporal shallowness, but he also rejects the idea that only evolutionary considerations should be mentioned in explaining the present phenotypes of organisms.⁷

To put the sunflower example into the baseball format, consider these two explanations of why sunflowers now turn toward the sun:

1. (2.3) Sunflowers now have mechanism M inside their cells (where M has features F1, F2, …, Fn).

2. (2.4) The ancestors of present day sunflowers were subject to selection for phototropism with the result that genes (G1, G2, …, Gm) that cause sunflowers to develop mechanism M (where M has features F1, F2, …, Fn) increased in frequency. These genes were passed down the generations, and present-day sunflowers inherited them, with the result that they have mechanism M.

What goes for the broken window also goes for the sunflowers. Propositions (2.3) and (2.4) are both explanatory, but we suggest that (2.4) has more explanatory power than (2.3).

Some readers may balk at this last, absolute, pronouncement. Perhaps they believe that what counts as the more powerful explanation depends on your interests. Sometimes all you want is the temporally shallow cause; at other times you want more. Neither is better than the other unconditionally. We sympathize with that pluralistic claim, but we do feel the tug of the idea that temporally more inclusive explanations are often more explanatorily powerful than explanations that are temporally less inclusive (Sober 1999).

How do TLI measures address the question of temporal shallowness? Are some or all of them pluralistic? Consider two types of cases in which there is a causal chain from X to Y to Z (or, strictly speaking, from the events described by X to the events described by Y to the events described by Z):

Type 1 (i) X causes Y, (ii) Y causes Z, (iii) X and Y are logically independent of each other, (iv) Y and Z are logically independent of each other, (v) X increases the probability of Y, (vi) Y increases the probability of Z, and (vii) X is screened-off from Z both by Y and by ∼Y in that Pr(Z | Y&X) = Pr(Z | Y) and Pr(Z | ∼Y&X) = Pr(Z |∼Y).

Type 2 (i) X causes Y, (ii) Y causes Z, (iii) X entails and increases the probability of Y, (iv) Y increases the probability of Z, and (v) X is screened-off from Z by Y in that Pr(Z | Y&X) = Pr(Z | Y).

We show in Appendix 1 that TLI measures are heavily biased in favor of temporally shallow explanations in the following sense:

Theorem 1 In cases of Type 1, TLI measures entail that EXPO(Z, Y) > EXPO(Z, X) and EXPO(Z, Y) ≥ EXPO(Z, X&Y).

Theorem 2 In cases of Type 2, TLI measures entail that EXPO(Z, Y) ≥ EXPO(Z, X) and EXPO(Z, Y) ≥ EXPO(Z, X&Y).⁸

Surely, however, an adequate measure of explanatory power should allow EXPO(Z, Y) to be less than EXPO(Z, X) or less than EXPO(Z, X&Y) in causal chains of either kind.⁹ In such causal chains, TLI measures say that explanatory power is never increased by adding information about distal causes. Ernst Mayr is not the only scientist who would scoff at that suggestion.

It might be objected that our argument here depends on a wholesale commitment to the thesis that a more complete explanation of E is always more powerful than a less complete explanation of E. Philosophers often argue against this thesis by pointing out that more complete explanations are sometimes harder to understand and work with, and that less complete explanations sometimes apply to more explananda (see, e.g., Ylikoski and Kuorikoski [2010, 216–17]). We agree with both these points, but they aren’t relevant to the thesis we want to endorse. We’re interested in whether more complete explanations ever have greater explanatory power than less complete explanations of the same explanandum when both are easy to grasp. In addition, L is not shown to be a better explanation of E1 than M is by the fact that L explains E1, E2, …, En whereas M explains only E1.¹⁰ In any event, the examples we have discussed in this section do not fall in the cracks that Ylikoski and Kuorikoski’s observations reveal. Neither of the explanations of window breaking that we described strains our cognitive capacities to the breaking point, and the same point holds of the two explanations we described of sunflower phototropism.

3. The problem of negative causal interactions

At the start of this article, we considered an example in which asbestos exposure and smoking cigarettes both causally promote lung cancer. We assumed that lung cancer goes up in probability if an individual smokes, if an individual inhales asbestos, and if an individual does both, with lung cancer being more probable if both causes are present than it would be if there were just one.

Here we tweak the example, as shown in Table 1. Suppose that c, a, and i are positive, and that i > c and i > a. Now there is a causal interaction between smoking and asbestos; the model is not additive (which is the condition that obtains precisely when the interaction term i=0). In particular, suppose the interaction is negative. Although smoking cigarettes increases the probability of lung cancer if Jones doesn’t inhale asbestos, and inhaling asbestos increases the probability of lung cancer if Jones doesn’t smoke, smoking cigarettes decreases the probability of lung cancer if Jones inhales asbestos, and inhaling asbestos decreases the probability of lung cancer if Jones smokes. Somehow if Jones inhales cigarette smoke and asbestos particles, each inhibits the other’s carcinogenic propensities.¹¹

Table 1. The probability of Jones’s having lung cancer conditional on four conjunctions

Pr(Jones has lung cancer | –)
	Jones inhales asbestos	Jones does not inhale asbestos
Jones smokes cigarettes	x + c + a – i	x + a
Jones does not smoke cigarettes	x + c	x

Now suppose Jones has lung cancer (this is the explanandum E) and each of the following three hypotheses is true:

H1: Jones smoked cigarettes.

H2: Jones inhaled asbestos.

H1&H2: Jones smoked cigarettes and Jones inhaled asbestos.

Jones’s cancer is explained by each of these hypotheses, but H1 and H2 provide censored pictures of the causal facts whereas H1&H2’s explanation is more causally complete. H1 and H2 each resemble the one-sided story presented by the prosecutor in a trial who neglects to mention details that would cast doubt on the defendant’s guilt. Doing so may be rhetorically powerful, but it is evidentially inferior to a fuller accounting that mentions both the bits of evidence that favor guilt and the bits that favor innocence. So it is with explanation—powerful explanations should describe both the causal promoters and the causal inhibitors that impacted the explanandum event. Removing all the latter reduces explanatory power.¹²

How do TLI measures handle cases like that of Jones? Consider, more generally, cases of the following type:

Type 3 (i) Y causes X, (ii) Z causes X, and (iii) there’s a negative causal interaction in that (a) Pr(X | Y) is greater than Pr(X), (b) Pr(X | Z) is greater than Pr(X), (c) Pr(X | ∼Y&∼Z) is less than Pr(X | Y&Z), and (d) the latter is less than each of Pr(X | Y&∼Z) and Pr(X | ∼Y&Z).

We show in Appendix 2 that TLI measures are heavily biased in favor of censored explanations in that:

Theorem 3 In cases of Type 3, TLI measures entail that EXPO(X, Y) > EXPO(X, Y&Z) and EXPO(X, Z) > EXPO(X, Y&Z).

The upshot is that each of The Five, and all other TLI measures, entail an erroneous interpretation of negative causal interactions of the kind we’ve described. They all say that a hypothesis that describes both positive and negative causal factors has less explanatory power than a hypothesis that describes the positive and ignores the negative.^{13, 14}

We note that this point about negative causal interactions involves no commitment to the general thesis that more complete explanations are always more powerful. What we said at the end of the previous section about window-breaking and phototropism also applies to the present example of cigarettes and asbestos.

4. A possible reply to the temporal shallowness and the negative causal interactions problems

The arguments offered in the previous two sections concerning temporal shallowness and negative causal interactions could be defused if there were an important sense of explanatory power that demands the verdicts that TLI measures deliver on those topics. A possible defusion of this type can be found in the motivation that Schupbach and Sprenger (2011) offer for their proposed measure of explanatory power. They begin with the idea that there’s a sense of explanatory power on which explanatory power is a matter of surprise reduction. They further hold that surprise reduction is psychological and so should be understood in terms of actual or rational degrees of belief. They defend this idea by arguing that their interpretation of explanatory power “finds precedence in many classical discussions of explanation”:

Perhaps its clearest historical expression occurs when Peirce (1931–35, 5.189) identifies the explanatoriness of a hypothesis with its ability to render an otherwise “surprising fact” as “a matter of course.” 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 1965) and necessity accounts (Glymour 1980) 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. (Schupbach and Sprenger 2011, 108)

We find this dubious, especially when it comes to Hempel’s D-N and I-S models. First, it’s clear from passages such as the following that Hempel sees his D-N and I-S models in terms of a nonpsychological, or objective, conception of explanation:

Scientific research seeks to give an account—both descriptive and explanatory—of empirical phenomena which is objective in the sense that its implications and its evidential support do not depend essentially on the individuals who happen to apply or to test them. This ideal suggests the problem of constructing a nonpragmatic conception of scientific explanation—a conception that requires reference to puzzled individuals no more than does the concept of mathematical proof. And it is this nonpragmatic conception of explanation with which the two covering-law models are concerned. (Hempel 2001, 82)

Second, in Hempel’s D-N model of explanation, what matters (roughly) is that the explanans propositions entail the explanandum proposition. Your degrees of belief (normative or actual) play no role, much less the idea that learning the explanans must change your degrees of belief in the explanandum. Even if it’s true that you increase your degree of belief in E when you use Hempel’s D-N model and conclude that H explains E, it doesn’t follow that Hempel’s model asserts that explanatory power is a matter of increase in degree of belief. Compare: Even if it’s true that you become excited when you use Hempel’s D-N model and conclude that H explains E, it doesn’t follow that Hempel’s model asserts that explanatory power is a matter of becoming excited.^{15, 16}

What about Peirce (1931–35, 1958)? It’s true that the passage from Peirce that Schupbach and Sprenger cite includes the expressions “surprising fact” and “a matter of course.” However, the main subject of that passage is abduction, not the question of what an explanation is. Peirce thinks that when H is abductively inferred (or conjectured) from E, H explains E and reduces E’s surprisingness, but it doesn’t follow that he takes this to be essential to explanation. In fact, there are many passages where Peirce characterizes explanation with no appeal to surprise reduction or to an agent’s degrees of belief. One such passage is cited by Schupbach and Sprenger (2011, 108, n. 5). They quote Peirce as saying that “to explain a fact is to show that it is a necessary or, at least, a probable result from another fact, known or supposed.” This sounds like Hempel but without all the formal niceties.¹⁷

In addition to appealing to the precedents of Hempel and Peirce, Schupbach and Sprenger cite Glymour (1980) as a precursor of the idea of explanatory power as surprise reduction. Glymour’s basic idea is that when an explanans is deduced from necessary truths involving property identities, the resulting explanans shows that the explanandum is necessary rather than contingent. This is a contingency-reducing explanation, which Glymour doesn’t think of psychologistically.

Schupbach and Sprenger (2011, 108) also appeal to statistical practice, saying that the surprise-reduction concept 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).” Our response is that regression procedures are part of the frequentist toolkit, and frequentists have no truck with degrees of belief.

In summary, the historical precedents that Schupbach and Sprenger (2011) note do not lend credence to the idea that there is an important sense of “explanatory power” according to which explanatory power should be understood in terms of surprise reduction where the latter concept is understood in terms of the changes in credence that an agent makes or ought to make. This leaves it open that their starting idea can be motivated in some other way. We now turn to our third argument against TLI measures of explanatory power.

5. The problem of nonexplanations

Take some case of Type 1 (as described in section 2)—that is, an example in which X causes and increases the probability of Y, Y causes and increases the probability of Z, X and Y are logically independent of each other, Y and Z are logically independent of each other, and X is screened-off from Z both by Y and by ∼Y. Is X’s explanatory power with respect to Y greater than Z’s explanatory power with respect to Y? The answer is clearly yes. Because X is a cause of Y whereas Z is an effect of Y, X explains Y at least to some extent, whereas Z explains Y not at all.

This simple example in which an explanation is contrasted with a nonexplanation suggests a condition of adequacy on measures of explanatory power:

Nonexplanations Worse. If (i) X is an explanation of E in that X explains E at least to some extent and (ii) Y is a nonexplanation with respect to E in that Y explains E not at all, then EXPO(E, X) > EXPO(E, Y).¹⁸

Nonexplanations have no explanatory power. Explanations have some explanatory power. Hence nonexplanations have less explanatory power than explanations.

TLI measures fail to satisfy Nonexplanations Worse. It’s straightforward to construct cases of Type 1, for example, in which:

(5.1) $${\rm{Pr(Y}}\,{\rm{|}}\,{\rm{Z)}} \gt {\rm{Pr(Y|}}\,\,{\rm{X)}}$$

(5.2) $${\rm{Pr(Y}}|\sim{\rm{Z}}) \lt {\rm{Pr}}({\rm{Y}}|\sim{\rm{X}})$$

We describe an example of this sort in Appendix 3. Because X is an explanation of Y whereas Z is not an explanation of Y, Nonexplanations Worse implies that EXPO(Y, X) > EXPO(Y, Z). However, given (5.1) and (5.2), TLI measures deliver the result that EXPO(Y, X) < EXPO(Y, Z).¹⁹

Consider, for comparison, the following condition of adequacy on probabilistic measures of confirmation in the sense of increase in probability:

Nonconfirmers Worse. If (i) E confirms X in that E increases the probability of X at least to some extent and (ii) E is a nonconfirmer with respect to Y in that E decreases or has no impact on the probability of Y, then CONF(X, E) > CONF(Y, E).

This condition, qua condition of adequacy, is uncontroversial in Bayesian confirmation theory. The situation with respect to explanatory power seems similar: just as confirmers confirm to a greater degree than nonconfirmers, explanations explain more powerfully than nonexplanations.

6. A possible reply to the problem of nonexplanations

We have been understanding PPMs as unrestricted in the sense that the EXPO(E, H) is well defined even if H is not an explanation of E. The reason why is straightforward: Some of our beliefs about explanatory power concern nonexplanations, and we want to know how well PPMs do in terms of all our considered judgments about explanatory power. In contrast, Schupbach and Sprenger (2011) restrict EXPO_SS, along with rival PPMs, as follows:

Potential-Explanation Restriction. EXPO(E, H) is defined precisely when H is a potential explanation of E in that H would explain E if H were true.²⁰

Given this way of understanding PPMs, TLI measures do not deliver the result that EXPO(Y, X) < EXPO(Y, Z) in our example of a causal chain from X to Y to Z because Z isn’t a potential explanation of Y, and so EXPO(Y, Z) isn’t defined.²¹ The restriction allows TLI measures to avoid generating false claims about explanatory power. But these theories still fail to satisfy Nonexplanations Worse because they fail to deliver some manifestly true propositions about explanatory power.²² The result is that the measures are incomplete, not that they entail falsehoods.

We argued in the previous section that it is a condition of adequacy on a quantitative measure of explanatory power that it judge nonexplanations as having less explanatory power than genuine explanations. This condition of adequacy of course obviously clashes with the restriction that Schupbach and Sprenger embrace. We hope the following comments will help adjudicate.

It is one thing for a mathematical criterion to fail to generate a result, as when the ratio x/y fails to be defined when y=0. It is another thing for a mathematical criterion to entail a result, which friends of the criterion wish to disown. Schupbach and Sprenger will understandably dislike what their mathematical criterion says about the causal chain from X to Y to Z discussed in section 5, but to stipulate that their criterion does not apply to such cases is not a legitimate solution. Where do such stipulations stop? Will friends of TLI measures of explanatory power urge that their theories do not apply to cases involving temporal shallowness or to cases involving negative causal factors? We are not saying that they are inclined to do so. Our question concerns the ground rules for using restrictive stipulations to avoid apparent counterexamples.

An anonymous referee for this journal suggests that friends of TLI measures of explanatory power could set aside the restriction strategy and instead simply stipulate that if H is not an explanation of E, then EXPO(E, H) has the lowest possible value. Consider, once more, the example of Schupbach and Sprenger’s proposed measure. EXPO_SS(E, H) might be understood in terms of a function from inputs to outputs such that:

1. (i) If the input consists of (a) explanandum E and (b) hypothesis H where H is not an explanation of E, then the output is –1.

2. (ii) If the input consists of (a) explanandum E, (b) hypothesis H where H is an explanation of E, and (c) probability function Pr(–), then the output is a number equal to ${{\Pr ({\rm{H}}\;{\rm{|E}}) - \Pr ({\rm{H}}\;{\rm{|}}\sim{\rm{E}})} \over {\Pr ({\rm{H}}\;{\rm{|E}}) + \Pr ({\rm{H}}\;{\rm{|}}\sim{\rm{E}})}}$ .

The same referee further notes that this suggestion is similar to the fact that the likelihood ratio measure of confirmation—CONF(H, E) = ${{\Pr ({\rm{E}}\;{\rm{|H}})} \over {\Pr ({\rm{E}}\;{\rm{|}}\sim{\rm{H}})}}$ —is standardly understood so that CONF(H, E) is maximal at ∞ when Pr(E | H)>0 and Pr(E | ∼H)=0, even though, strictly speaking, division by 0 is undefined. There is an important difference here, however. Take some value for Pr(E | H) greater than Pr(E | ∼H). The limit of ${{\Pr ({\rm{E}}\;{\rm{|H}})} \over {\Pr ({\rm{E}}\;{\rm{|}}\sim{\rm{H}})}}$ approaches positive infinity if Pr(E | H) is held fixed and Pr(E | ∼H) is made smaller and smaller. This is the rationale behind the standard way of understanding the likelihood ratio measure. However, nothing about surprise reduction or ${{\Pr ({\rm{H}}\;{\rm{|E}}) - \Pr ({\rm{H}}\;{\rm{|}}\sim{\rm{E}})} \over {\Pr ({\rm{H}}\;{\rm{|E}}) + \Pr ({\rm{H}}\;{\rm{|}}\sim{\rm{E}})}}$ justifies the stipulation put forward in (i). Rather, what motivates (i) is the idea that nonexplanations have no explanatory power. We of course agree with that idea, but we think it’s reasonable to expect that a mathematical measure should capture that idea without recourse to a special stipulation.

We think our criticism of the referee’s suggested stipulation can be further motivated by the following “continuity argument.” Consider a sequence of hypotheses H1, H2, etc. where each entry has less explanatory power than the one before, relative to a fixed explanandum proposition E. According to Schupbach and Sprenger’s measure, these hypotheses can be ranked for their explanatory power just by looking at the value of Pr(E|Hi)—the bigger this is, the more explanatory it is, with the result that Pr(E|BestPossible)=1, and Pr(E|worst possible) = 0. The point of relevance here is that numerous nonexplanations are nowhere “close” to Pr(E|WorstPossible) = 0. Consider, for instance, some hypothesis H about the present that deductively retrodicts E. Although H fails to explain E (assuming that the past isn’t explained by the present), the fact remains that Pr(E | H)=1, which is also the value of Pr(E|BestPossible). Here the reduction in E’s surprisingness is maximal, and yet the referee’s suggested stipulation puts H in the same camp as hypotheses that entail ∼E.

In summary, Schupbach and Sprenger (2011) stipulate that their measure applies only to potential explanations. The referee mentioned suggests that if this stipulation fails to satisfy, proponents of TLI measures can stipulate instead that a nonexplanation has the minimum possible degree of explanatory power. These are different stipulations, but there is a common flaw, or so we have suggested. The use of stipulations in either of these two ways reminds us of Russell’s (1919, 71) famous quip that “[t]he method of ‘postulating’ what we want has many advantages; they are the same as the advantages of theft over honest toil.”

7. The problem of explanatory irrelevance

In the previous two sections we discussed the dichotomy between an explanation of E and a nonexplanation of E. We now want to consider something that is a matter of degree. If H is a genuine explanation of E, and you conjoin explanatorily irrelevant proposition I to H, does the resulting conjunction, H&I, provide a less powerful explanation of E than H provides?

A related question has a distinguished history in philosophical discussions of explanation. Consider two influential examples:

This sample of table salt dissolves in water, for it had a dissolving spell cast upon it, and all samples of table salt that have had dissolving spells cast on them dissolve in water. (Kyburg 1965, 147)

John Jones avoided becoming pregnant during the past year, for he has taken his wife’s birth control pills regularly, and every man who regularly takes birth control pills avoids pregnancy. (Salmon 1971, 34)

Kyburg and Salmon present these examples to illustrate how a genuine explanation can be turned into a nonexplanation by adding information that is explanatorily irrelevant. They seem to be committed to the general thesis that if H contains even one detail that is explanatorily irrelevant to E, then H is not an explanation of E. We think this may go too far.

Shifting from the question of whether a hypothesis explains to the question of how powerfully it does so, our modest thesis about irrelevant additions is this:

Adding Irrelevance Sometimes Hurts. EXPO(E, H) is sometimes greater than EXPO(E, H&I) when (a) I is explanatorily irrelevant to E and (b) I is probabilistically independent of E, H, and E&H.

Salmon and Kyburg appear to be purists; for them adding even a teaspoon of irrelevancy to a sea of explanatorily relevant information is enough to turn the sea into a nonexplanation. We aren’t purists about such additions when the question is degree of explanatory power. That’s why we say sometimes. Here we have in mind examples in which a brief, telling H is conjoined with a giant, complicated, and doubly irrelevant proposition I.

How do TLI measures fair in terms of the Adding Irrelevance Sometimes Hurts condition? The situation is mixed, in that some TLI measures meet Adding Irrelevance Sometimes Hurts whereas others do not. We show in Appendix 4 that Adding Irrelevance Sometimes Hurts is met by EXPO_C and EXPO_LR but not by EXPO_SS, EXPO_CT, EXPO_G, or EXPO_D. ²³

TLI measures would satisfy the Adding Irrelevance Sometimes Hurts condition if they were understood so that H&I fails to explain E when I is explanatorily irrelevant to E, and if nonexplanations are stipulated to have the minimum possible degree of explanatory power. However, this way of understanding those measures leads back to our earlier discussion of the role of stipulation in the formulation of PPMs of explanatory power.

8. Concluding comments

We have focused on TLI measures of explanatory power, which include all extant purely probabilistic measures of explanatory power. All TLI measures face three problems—the Problem of Temporal Shallowness, the Problem of Negative Causal Interactions, and the Problem of Nonexplanations. Some TLI measures face a fourth problem—the Problem of Explanatory Irrelevance. These problems together raise the suspicion that an adequate measure of explanatory power can’t be purely probabilistic. However, the door has not slammed shut absolutely. It is left open that an adequate PPM of explanatory power can be found; the place to search, we suggest, is in measures that violate the TLI constraints.