2.5 The Measure of Explanatory Power E
2.5.2 Uniqueness, Version 2
As mentioned above, one could take issue with this first uniqueness theorem because of its reliance on the simplicity requirement expressed in CA 1; that is, one might object that the above theorem and corresponding proof do not show that there is only one intuitively- satisfying measure of explanatory power, but rather that there is only one simplest such measure. Insofar as someone is skeptical that the notion of simplicity required by CA 1 has any epistemic merit then, that person will not likely be persuaded that E is uniquely satisfactory by the above result. (Note, however, that this concern would not raise any
two measures are structurally equivalent; however, regarding the interpretation of the measure, E (e, h) is F (h, e) with h and e reversed (h is replaced by e, and e is replaced by h).
real challenge to E ’s status as a uniquely satisfactory Carnapian explication, given that the simplicity requirement is built in to this method.)
To alleviate this worry, this section introduces an alternative result showing that, even if we set aside our simplicity requirement, E is still the uniquely best measure of explanatory power, up to ordinal equivalence – where any two proposed measures of explanatory power f and f0 are ordinally equivalent if and only if it is true that, f (e, h) > (=, <)f (e0, h0) if and only if f0(e, h) > (=, <)f0(e0, h0). In other words, the main result of this section states that all functions that satisfy a set of clear adequacy conditions (probabilistic versions of a subset of Conditions 1-6) will agree on all ordinal judgments. This implies that all such functions are strictly monotonic functions of one another; one can say that they are merely rescaled versions of one another.17
This is quite a substantial achievement. This result shows that, even without making a simplicity assumption, we can derive a unique probabilistic account of explanatory power. The fact that this account comes in the way of a class of ordinally equivalent functions might worry some; however, with only one relatively minor exception, all of the applications of this account of explanatory power in this dissertation will not depend upon one’s choice of measure from among this set.18 The upshot is that, for those who are wary of requiring that
the intended explication be simple, there is an alternative theorem that singles out a class of ordinally equivalent measures of explanatory power; and thankfully, accepting this class of measures is sufficient for deriving all of the key results that will follow in this dissertation’s study of the epistemology of explanation.
To present this second uniqueness result, it is necessary first to introduce and motivate some more adequacy conditions. The first adequacy condition is again one that sets out the purely formal requirements of our measure. Like CA 1, its main purpose is to specify the probabilistic nature of our explicatum. Unlike CA 1, however, this condition does not require that our measure be simple.
17The remaining content of this chapter is based upon my joint work with Jan Sprenger, as published in (Schupbach and Sprenger 2011).
18The one exception is the work accomplished in Chapter 3; in that chapter, one’s choice of measure will influence the degree of fitness between the theoretical results derived from a measure and experimental participants’ explanatory judgments. Accordingly, in that chapter, I will have to make use of the first uniqueness result given above – and thus also of the simplicity requirement made in CA1– in order to single out one measure to test from among the class of ordinally equivalent measures.
CA 6. For any probability space and regular probability measure (Ω, A, P r(·)), E is a mea- surable function from two propositions e, h ∈ A to a real number E (e, h) ∈ [−1, 1]. More precisely, given Bayes’s Theorem, E is represented as a function of P r(e), P r(h|e) and P r(h|¬e), and we demand that any such function be analytic.19
The next adequacy condition specifies, in probabilistic terms, the general notion of ex- planatory power that we are interested in explicating. As mentioned in Section 2.4.2, an explanans has explanatory power over some explanandum, in the sense that we have in mind, to the extent that it makes that explanandum less surprising. More specifically Condition 1 tells us that a hypothesis has positive explanatory power over a proposition to the extent that it decreases the degree to which that proposition is surprising (i.e., increases the degree to which we expect that proposition to be true), while Condition 2 states that a hypothesis has negative explanatory power over a proposition to the extent that it increases the degree to which that proposition is surprising. If h decreases (increases) the degree to which e is surprising, we represent this with the inequality P r(e) < (>)P r(e|h). The strength of this inequality corresponds to the degree of statistical relevance between e and h, and so we can capture all of this probabilistically by requiring the following:
CA 7. (Positive Relevance). Ceteris paribus, the greater the degree of statistical relevance between e and h, the higher E (e, h).
The following adequacy condition observes that explanatory power, in our sense, does not depend upon 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. Rather, to decide the effect of a hypothesis upon the surprisingness (expectedness) of some explanandum, one
19A real-valued function f is analytic if we can represent it as the Taylor expansion around a point in its domain. This requirement is, first of all, quite weak insofar as it does not rule out any normal mathematical function. Furthermore, and more importantly, this requirement is needed in order to ensure that our measure cannot be composed in an arbitrary or ad-hoc way.
Since E is represented via certain conditional probabilities, one might worry about logically extreme cases where, e.g., P r(e) = 0. I suggest that this worry can easily be avoided by remembering that h is assumed to provide a potential explanation of e. Cases of zero probability will not present a problem simply because self-contradictory propositions (those propositions that have zero probability) cannot act as explanans or explanandum in a potential explanation. In effect then, E is defined on all pairs of contingent propositions; i.e., cases such as P r(e) = 0 etc. are not in the domain of E .
compares how surprising (expected) the explanandum is apart from considerations of the hypothesis to how surprising (expected) it would be granting the truth of the hypothesis. In making this specific comparison, it is simply not necessary (and not helpful) to know how plausible the explanatory hypothesis is on its own. With this sense of explanatory power in mind then, it is perfectly sensible to talk about two hypotheses that are vastly unequal in their respective plausibilities having the same amount of explanatory power over an explanandum. For example, dehydration and cyanide poisoning may be (approximately) equally powerful explanations of symptoms of dizziness and confusion insofar as they both make such symptoms less surprising to the (approximately) same degree. And this is true despite the fact that dehydration is typically by far the more plausible explanans. In light of these considerations, we require the following
CA 8. (Irrelevance of Priors). Values of E (e, h) do not depend upon the values of P r(h).20
Retaining CA 5 from Section 2.5.1 and now also requiring CA 6, CA 7, and CA 8, one can derive the following theorem (proof in Appendix B):
Theorem 2. All measures of explanatory power satisfying CA 5 - CA 8 are monotonically increasing functions of the posterior ratio P r(h|e)/P r(h|¬e).
Note that the specific measure of explanatory power introduced and defended in Section 2.5.1,
E(e, h) = P r(h|e) − P r(h|¬e) P r(h|e) + P r(h|¬e),
is one such measure. We have already seen, via Theorem 1, that this measure satisfies CA 5; moreover it is easy to see that it does satisfy CA6. E can be shown to satisfy both CA7 and CA 8 simultaneously by proving that E is purely – no ceteris paribus clause required – an increasing function of the degree of statistical relevance between h and e, and so that it can be represented purely as a function of P r(e|h) and P r(e). This is shown in the proof of the following representation theorem (Appendix C):
20The following weaker version of CA8actually suffices in the proof of Theorem2: When either h or ¬h implies e, values of E (e, h) and E (e, ¬h) do not depend upon the values of P r(h) and P r(¬h). Nonetheless, the notion of explanatory power analyzed here motivates the condition that explanatory power does not depend upon P r(h) generally – not merely when h or ¬h implies e. Accordingly, I include this stronger condition here.
Theorem 3. E can be represented as a function only of P r(e) and P r(e|h). Moreover, E is a decreasing function – at constant P r(e|h) – of P r(e) and an increasing function – at constant P r(e) – of P r(e|h).
Given that E thus satisfies CA 5 - CA 8, Theorem 2 implies that E is a monotonically increasing function of the posterior ratio P r(h|e)/P r(h|¬e). This result is proved more directly in Lemma 3 of AppendixC.
From Theorem2, two important corollaries follow. First, we can derive a result specifying the conditions under which E takes its maximal and minimal values. In other words, we can derive CA 3(Maximality) and the corresponding Minimality condition from CA5 - CA8 (proof in Appendix B):
Corollary 1. E (e, h) takes maximal value if and only if h entails e, and minimal value if and only if h implies ¬e.
The second corollary constitutes our desired ordinal equivalence result:
Corollary 2. All measures of explanatory power satisfying CA5- CA8are ordinally equiv- alent.
To see why this corollary follows from Theorem2, let r be the posterior ratio of the pair (e, h), and let r0 be the posterior ratio of the pair (e0, h0). Without loss of generality, assume r > r0. Then, for any functions f and f0 that satisfy CA 5 - CA8, we obtain the following inequalities:
f (e, h) = g(r) > g(r0) = f (e0, h0) f0(e, h) = g0(r) > g0(r0) = f0(e0, h0),
where the inequalities are immediate consequences of Theorem2. So any f and f0 satisfying CA 5- CA 8always impose the same ordinal judgments.