Uniqueness, Version 1

Before interpreting Conditions 1-6 probabilistically, the following formal condition of ade- quacy is first needed in order to specify that the measure of explanatory power that explanans h has over explanandum e (denoted E (e, h)), which we seek as our explicatum, must be prob- abilistic in nature and simple in a well-defined sense – in accordance with Carnap’s precision and simplicity desiderata.15

CA 1. 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, E is the ratio of two functions of P r(e∧h), P r(¬e∧h), P r(e∧¬h) and P r(¬e∧¬h), each of which are homogeneous in their arguments to the least possible degree k ≥ 1.

Representing E as the ratio of two functions serves the purpose of normalization. P r(e ∧ h), P r(¬e ∧ h), P r(e ∧ ¬h) and P r(¬e ∧ ¬h) fully determine the probability distribution over the truth-functional compounds of e and h, so it is appropriate to represent E as a function of them. The requirement that the two functions be “homogeneous in their arguments” ensures that the functional form of E itself does not determine which of the terms (P r(e ∧ h), P r(¬e ∧ h), P r(e ∧ ¬h), P r(¬e ∧ ¬h)) should have more weight.

The requirement that E be the ratio of two functions, each having “the least possible degree k ≥ 1” reflects a minimal and well-defined Carnapian simplicity condition akin to the version advocated by Kemeny and Oppenheim (1952, p. 315). Below, in Section 2.5.2, I show that this simplicity requirement is not needed to determine E as the unique measure of explanatory power up to ordinal equivalence. Nonetheless, there are several reasons one might want to retain this requirement. First, such a simplicity requirement is part and parcel of Carnap’s notion of explication. Accordingly, we build Carnap’s simplicity requirement into our conditions of adequacy. Second, this requirement effectively limits the search for a unique measure to those that are the most cognitively accessible and applicable. Some

15_{Up until this point in the dissertation, I have been able to avoid the topic of just what sort of thing the} explanandum is. At this point, however, I should clarify the following: For the sake of the following account, all explananda are ultimately categorized as propositions. At times, it is natural to talk instead about the explaining of evidence or events. In either case, the proper explanandum actually may be thought of as the proposition describing the relevant evidence or event. Such a proposition may of course be as complex as is necessary to describe the corresponding evidence or event (or conglomerate of events) accurately.

such restraint is appropriate insofar as we want to ensure that our resulting measure is not complex to the point of being hopelessly opaque and unusable. Third, in addition to this pragmatic virtue, whether or not simplicity is of epistemic virtue is an open question, and many philosophers and scientists endorse the idea that there are good epistemic reasons to prefer simpler theories. The result that there is a unique, simplest measure of explanatory power will be of great interest to any such thinker.

Of course, larger values of E (e, h) indicate greater explanatory power of h with respect to e. Accordingly, E (e, h) = 1 (E ’s maximal value) is the value at which h is interpreted as a maximally powerful potential explanation of e; similarly, E (e, h) = −1 indicates the minimal degree of explanatory power for h relative to e, where h is interpreted as providing a maximally powerful potential explanation for e being false. E (e, h) = 0 is the “neutral point” at which h lacks any explanatory power relative to e (i.e., where h is explanatorily irrelevant to e).

While CA 1 gives us an informal idea of when E should take on certain values, it is still left to us to define these points formally. Here is where Conditions 1-6 become especially pertinent. According to the above, E (e, h) should take the value 0 precisely when h lacks any explanatory power relative to e. Condition 3 specifies that such irrelevance occurs if and only if e is neither more nor less surprising in light of h. Given the inverse relation between surprise and probability, the way to formalize this probabilistically is to say that, in such cases, h and e are statistically irrelevant to (independent of) one another – in which case, P r(e|h) = P r(e), or equivalently, P r(h ∧ e) = P r(h) × P r(e):

CA 2. (Neutrality). For explanatory hypothesis h, E (e, h) = 0 if and only if P r(h ∧ e) = P r(h) × P r(e).

CA1also demands that E (e, h) takes a maximum value of 1 if and only if h is a maximally powerful explanation of e. Condition 4 clarifies that such will be the case precisely when h leads us to expect with certainty that e is true. Such a notion is straightforwardly formalized with the equality P r(e|h) = 1, resulting in the following condition:

CA 3. (Maximality). For explanatory hypothesis h, E (e, h) = 1 if and only if P r(e|h) = 1.

that of h relative to ¬e decreases. In other words, the more h explains the truth of e, the less it explains its falsity. CA2and CA3provide us with further rationale for this condition. CA 3 tells us that E (e, h) should be maximal only if P r(e|h) = 1. Importantly, in such a case, P r(¬e|h) = 0, and this value intuitively corresponds to the point at which we should expect E(¬e, h) to be minimal (see Condition 5 above). In other words, given CA 3, we see that E(e, h) takes its maximal value 1 precisely when E(¬e, h) takes its minimal value −1 and vice versa. Also, we know from CA 2 that E (e, h) and E (¬e, h) should always equal zero at the same point given that P r(h∧e) = P r(h)×P r(e) if and only if P r(h∧¬e) = P r(h)×P r(¬e). These considerations lead to the following requirement:

CA 4. (Symmetry). E (e, h) = −E (¬e, h).

The final condition of adequacy appeals to a scenario in which degree of explanatory power is unaffected. If a hypothesis h2 is explanatorily irrelevant to another hypothesis h1,

to some proposition e, and to any logical combination of h1 and e, then Condition 3 tells

us that it does nothing to increase or decrease the degree to which these are surprising. In such a case, conjoining h2 to h1 will do nothing to increase or decrease the degree to which

e is surprising in light of the hypothesis. Given CA2(Neutrality), we can state this in other words: if h2 has no explanatory power whatever relative to e, h1, or any logical combination

of e and h1, then its presence will not affect the overall explanatory power of h1 relative to

e. This gives us the following condition:

CA 5. (Irrelevant Conjunction). If P r(e ∧ h2) = P r(e) × P r(h2) and P r(h1∧ h2) = P r(h1) ×

P r(h2) and P r(e ∧ h1∧ h2) = P r(e ∧ h1) × P r(h2), then E (e, h1∧ h2) = E (e, h1).

These five adequacy conditions conjointly determine a unique measure of explanatory power as stated in the following theorem (Proof in Appendix A).

Theorem 1. The only measure that satisfies CA 1- CA 5 is E(e, h) = P r(h|e) − P r(h|¬e)

P r(h|e) + P r(h|¬e).

Thus, as desired, the function E provides a measure of the strength of a potential explanation; the higher the value of E (e, h), the more powerful the potential explanation that h provides of e.16

Note that this measure also satisfies the conditions from Section 2.4.2 that were not needed in order to prove Theorem 1. Conditions 1 and 2 require that explanatory power increases (decreases) as the degree to which e is surprising decreases (increases) in light of h. Put more formally, these conditions require that E (e, h) > 0 to the extent that P r(e) < P r(e|h). These conditions are satisfied by E given that

E(e, h) = P r(h|e) − P r(h|¬e) P r(h|e) + P r(h|¬e) > 0

to the extent that P r(h|e) > P r(h|¬e). And this inequality holds to the extent that P r(e|h) > P r(e) – this is easy to see in light of the fact that

P r(h|e) P r(h|¬e) = P r(e|h) P r(e) × 1 − P r(e) 1 − P r(e|h).

Additionally, Condition 5 requires that explanatory power is minimal if and only if e is certainly false in light of h. This fact also follows necessarily from E given that E (e, h) = −1 if and only if

E(e, h) = −P r(h|¬e) P r(h|¬e).

But this equality follows only if P r(h) 6= 0 and P r(h|e) = 0, which implies that P r(e|h) = 0. Hence, E (e, h) = −1 if and only if P r(e|h) = 0. Thus, E is the unique measure of explanatory power that is able to satisfy the intuitive requirements described in Conditions 1-6.