1. Introduction

Is unification the key to understanding what a scientific explanation is? This idea has been developed creatively by Michael Friedman (Reference Friedman1974) and Philip Kitcher (Reference Kitcher1981, Reference Kitcher, Kitcher and Salmon1989), both of whom take their inspiration from Kant’s idea that scientific understanding is achieved via the unification of diverse phenomena. In this article, we examine Friedman’s development of the idea that law X explains laws Y ₁, Y ₂, …, Y_n if and only if X unifies the Y’s. The special spin he gives to this idea is his use of the concept of one sentence’s being acceptable independently of another.Footnote ¹

Kitcher (Reference Kitcher1976) develops an objection to Friedman’s theory that is widely taken to be devastating, so it may be asked why new objections are worth examining. One reason is that Friedman’s theory can be modified to evade Kitcher’s objection (as we explain in sec. 3). Another is that philosophers sometimes think that Friedman’s basic idea concerning explanation, unification, and independent acceptability is on the right track even though his technical implementation of that basic idea is flawed. This is true, for example, of Psillos, who writes: “There may be a way to defend the spirit of Friedman’s model of unification. … I want to motivate my suggestion by noting that the basic idea behind Friedman’s program is right: explanation does amount to reduction of the number of independently acceptable laws. The crux of this idea is that all apparently independently acceptable regularities are consequences of more basic laws” (Reference Psillos2002, 271–72). Friedman (Reference Healey1983, 261–62) himself took that position in the aftermath of Kitcher’s criticism. The concept of independent acceptability is therefore worth examining. Friedman depends on this concept but offers little by way of elucidation. Kitcher’s criticism of Friedman does not focus on that concept at all.Footnote ²

Before setting to work, we want to mention two reasons for being skeptical about unificationist approaches to explanation. First, it seems implausible that an explanation of two or more events must unify them. True, the motion of the moon and the motion of an apple falling to earth are explained by Newton’s unifying theory, but that is just one favorable example. Consider, in contrast, the fact that this rose is red and so is that fire hydrant. The right explanation here is disunifying. The shared color is a coincidence, but that does not mean that it is inexplicable. The explanation is in terms of separate independent causes, not common causes (Sober Reference Stadler, de regt, Hartmann and Okasha2012). Notice that the same point would apply if the explananda were generalizations (“all roses are red” and “all fire hydrants are red”) rather than two token events.Footnote ³

The second skeptical idea we want to float is that equating explanation with unification derives its appeal, at least in part, from conflating an epistemological question with the question of what an explanation is. We agree that hypotheses that unify two or more bodies of phenomena are often epistemically “better” than hypotheses that disunify them. Sometimes unifying explanations have higher likelihoods, and sometimes they can be expected to be more predictively accurate (Sober Reference de regt, Hartmann and Okasha2015). However, these epistemic points do not address the problem that Hempel (Reference Hempel1965) crystallized and that subsequent philosophers, including Friedman and Kitcher, have addressed. Hempel’s problem was this: given a set of propositions (or sentences), which of them explains which others? Unification can be a guide to belief without being the right solution to Hempel’s problem (Sober Reference Sober and Stadler2003).

Having noted these two skeptical thoughts, we freely admit that unification might be the key to understanding how laws explain other laws, even if it is misguided when the task is to understand what it is to explain events and accidental generalizations. Since Friedman’s theory concerns the explanation of laws, the possibility remains that his approach is correct.

2. Friedman’s Theory of Explanation

Friedman motivates the idea that explanation and unification are tightly linked by describing an example:

Friedman develops this idea by stating some assumptions:

The expression “independent” in these two passages is crucial. K is deductively closed, so it is infinite in size, and thus no law-like sentence can reduce the number of sentences in K. However, it does not follow that no law-like sentence can reduce the number of independent sentences in K. Friedman’s view, we take it, is that the number of independent sentences in K is finite; this makes it possible for a law-like sentence to reduce the number of independent sentences in K.

When does a given law-like sentence reduce the number of independent law-like sentences in K? Friedman considers and rejects the idea that this reduction can be achieved just by replacing two or more independent sentences in K with a single sentence S. Suppose that K’s membership is restricted to the Boyle-Charles law (B), Galileo’s law (G), and their consequences.Footnote ⁴ Let M be the laws of mechanics. It might seem that B and G can be replaced by M and that this replacement reduces the number of independent sentences in K from two to one.Footnote ⁵ Consider, though, the conjunction B&G: B and G can be replaced by B&G, but, clearly, B&G does not reduce the number of independent sentences in K. It might be countered that the conjunction B&G is really two independent sentences, because it is logically equivalent to the two-membered set {B, G}. This, however, will not do, for, as Friedman notes, any sentence is logically equivalent to a set of n sentences for any finite and positive number n. Law M, for instance, is logically equivalent to the one-membered set {M}, the two-membered set {$B,B\supset M$}, the three-membered set {$B,B\supset M,G$}, the four-membered set {$B,B\supset M,G,G\supset M$}, and so on.

Friedman then points out that the situation is different when the issue concerns independently acceptable sentences as opposed to just (logically) independent sentences: “although every sentence is equivalent to a set of n independent sentences, it is not the case that every sentence is equivalent to a set of n independently acceptable sentences—e.g., the members of the set {$P,P\supset S$} may not be acceptable independently of S; for our only grounds for accepting $P\supset S$, say, might be that it is a consequence of S” (Reference Friedman1974, 16). But how is the notion of independent acceptability to be understood? Friedman begins with a modest remark: “I don’t have anything very illuminating to say about what it is for one sentence to be acceptable independently of another. Presumably, it means something like: there are sufficient grounds for accepting one which are not also sufficient grounds for accepting the other” (16). He then advances the following two theses:

That is all that Friedman has to say on the subject.

FIA1 and FIA2, we take it, can be reformulated as follows:

Here we assume that by “grounds” Friedman means evidence.

When is evidence sufficient for acceptance? A natural answer is this:

Three clarifications are in order here. First, the right-hand side of HP should be understood as short for “either E is the total evidence available to an agent (or a scientific community) at a time and $p\left(S|E\right)>c$ or E is a subset of the total evidence E* available to an agent (or a scientific community) at a time and both $p\left(S|E\right)>c$ and $p\left(S|E^{*}\right)>c$.” Second, c is a “caution threshold”; it says how probable S must be, given the evidence, for accepting S. Third, we assume that $0.5\le c<1$, but we do not assume any particular value for c.Footnote ⁶ If, as HP says, acceptability is a matter of high probability, then FIA1 and FIA2 are provably correct.Footnote ⁷ We assume in what follows that FIA1 and FIA2 are to be understood in terms of HP (although in sec. 6 we consider a strengthened definition of what acceptability is). Friedman never says otherwise, and his principles FIA1 and FIA2 are true when thus understood.Footnote ⁸

In addition to using the concept of one sentence’s being acceptable independently of another, Friedman also uses the concept of reduction. Although he says only a little about independent acceptability, Friedman provides more details about reduction. He writes:

So S reduces Δ if and only if the smallest K-partition of $\Delta \cup \left\{S\right\}$ (i.e., the smallest set of K-atomic sentences that is logically equivalent to $\Delta \cup \left\{S\right\}$) is smaller than the smallest K-partition of Δ (i.e., the smallest set of K-atomic sentences that is logically equivalent to Δ).

We now are in a position to state Friedman’s theory of explanation. Let “con_K(X)” be the set of independently acceptable consequences of X, where each member of con_K(X) is acceptable independently of each of the others and of X as well. Here’s the theory:

We note that X is not a member of con_K(X). This follows from the fact that independent acceptability is irreflexive. Take some sentence X. Since X entails itself, it follows, by FIA1, that X is not acceptable independently of itself.

To help make Friedman’s theory comprehensible, we turn to a toy example. Initially, at time t ₁, K’s membership is restricted to L ₁, L ₂, and their consequences, where L ₁ and L ₂ are acceptable independently of each other. Suppose that the total evidence available at t ₁ is restricted to E ₁ and E ₂, where E ₁ makes L ₁ acceptable but does not make L ₂ acceptable, and E ₂ makes L ₂ acceptable but not L ₁. In other words, $p\left(L_1|E_1\right)>c$ but $p\left(L_2|E_1\right)\le c$, and $p\left(L_2|E_2\right)>c$ but $p\left(L_1|E_2\right)\le c$.Footnote ¹⁰ Law L ₃ entails each of L ₁ and L ₂, but $p\left(L_3|E_1\right)\le c$, $p\left(L_3|E_2\right)\le c$, and $p\left(L_3|E_1\text{\&}{E}_2\right)\le c$. Then later, at time t ₂, evidence E ₃ is obtained and $p\left(L_3|E_3\right)>c$, so, at t ₂, K’s membership is restricted to L ₁, L ₂, L ₃, and their consequences. Does L ₃, at t ₂, explain L ₁?

Law L ₃ entails L ₁, and since $p\left(L_1|E_1\right)>c$ but $p\left(L_3|E_1\right)\le c$, L ₁ is acceptable independently of L ₃.Footnote ¹¹ Thus, conditions i and ii in FE are satisfied. Now consider condition iii. This condition is satisfied if and only if the smallest K-partition of ${\mathrm{con}}_K\left(L_3\right)\cup \left\{L_3\right\}$ (i.e., the smallest set of K-atomic sentences that is logically equivalent to ${\mathrm{con}}_K\left(L_3\right)\cup \left\{L_3\right\}$) is smaller than the smallest K-partition of con_K(L ₃) (i.e., the smallest set of K-atomic sentences that is logically equivalent to con_K(L ₃)). Suppose that each of L ₁, L ₂, and L ₃ is K-atomic. Then, since L ₃ is a member of ${\mathrm{con}}_K\left(L_3\right)\cup \left\{L_3\right\}$ but is not a member of con_K(L ₃), the smallest K-partition of ${\mathrm{con}}_K\left(L_3\right)\cup \left\{L_3\right\}$ is {L ₃}, whereas the smallest K-partition of con_K(L ₃) is {L ₁, L ₂}. Given this, and given that {L ₃} is smaller than {L ₁, L ₂}, it follows that L ₃ reduces con_K(L ₃). So, by FE, L ₃ explains L ₁. This is just as it should be, according to Friedman.

Now consider a different question: Does L ₁&L ₂ explain L ₁? The conjunction L ₁&L ₂ entails L ₁, and since L ₁ is acceptable independently of L ₂, and since L ₁&L ₂ entails L ₂, it follows, by FIA2, that L ₁ is acceptable independently of L ₁&L ₂. So conditions i and ii in FE are satisfied. However, condition iii is not satisfied. It might seem otherwise, since it might seem that the smallest K-partition of ${\mathrm{con}}_K\left(L_1\text{\&}{L}_2\right)\cup \left\{L_1\text{\&}{L}_2\right\}$ is {L ₁&L ₂}, whereas the smallest K-partition of con_K(L ₁&L ₂) is {L ₁, L ₂}. However, L ₁&L ₂ is not K-atomic, since L ₁&L ₂ is logically equivalent to {L ₁, L ₂} each member of which is acceptable independently of L ₁&L ₂. Thus, {L ₁&L ₂} is not a K-partition of ${\mathrm{con}}_K\left(L_1\text{\&}{L}_2\right)\cup \left\{L_1\text{\&}{L}_2\right\}$. The smallest K-partition of ${\mathrm{con}}_K\left(L_1\text{\&}{L}_2\right)\cup \left\{L_1\text{\&}{L}_2\right\}$ is simply {L ₁, L ₂}. Thus, the smallest K-partition of ${\mathrm{con}}_K\left(L_1\text{\&}{L}_2\right)\cup \left\{L_1\text{\&}{L}_2\right\}$ and the smallest K-partition of con_K(L ₁&L ₂) are one and the same set. So L ₁&L ₂ does not reduce con_K(L ₁&L ₂), and thus, by FE, L ₁&L ₂ does not explain L ₁. This too is just as it should be, as Friedman (Reference Friedman1974, 18) notes: conjunctions never explain their conjuncts.

Our toy example is just an example, but our main points about it generalize. First, if X is K-atomic and has at least two independently acceptable consequences, then the smallest K-partition of ${\mathrm{con}}_K\left(X\right)\cup \left\{X\right\}$ is {X}, whereas the smallest K-partition of con_K(X) has at least two members, and so X reduces con_K(X). Second, if X is an n-fold conjunction and each conjunct is acceptable independently of the others, then the smallest K-partition of ${\mathrm{con}}_K\left(X\right)\cup \left\{X\right\}$ and the smallest K-partition of con_K(X) are one and the same set—the set of X’s conjuncts—and thus X does not reduce con_K(X).

The first of these points can be strengthened: X is K-atomic and has at least two independently acceptable consequences if and only if the smallest K-partition of ${\mathrm{con}}_K\left(X\right)\cup \left\{X\right\}$ is {X}, whereas the smallest K-partition of con_K(X) has at least two members, and so X reduces con_K(X).Footnote ¹² This means that the only way for X to reduce con_K(X) is for X to be K-atomic and have at least two independently acceptable consequences. It follows that FE can be reformulated as:

We hope that FE* makes it clear, in our toy example, why Friedman’s theory delivers the verdict that L ₃ explains L ₁ but L ₁&L ₂ does not. The problem with L ₁&L ₂ is that it is not K-atomic.

We mentioned in section 1 that Friedman’s theory is superior to Kitcher’s in at least one important respect. The point we had in mind involves a good objection that Woodward (Reference Woodward2003, Reference Woodward and Zalta2014) makes to Kitcher’s theory: the theory entails that explanation is “winner-takes-all.” That is, according to Kitcher’s theory, if X explains Y, then there is no X* distinct from X such that X* also explains Y. Friedman’s theory is not subject to this objection. This can be seen by adding to our toy example. Suppose that L ₆, which is K-atomic, comes to be accepted, where L ₆ entails each of L ₃, L ₄, and L ₅ and where L ₃, L ₄, and L ₅ are each acceptable independently of L ₆. It is still the case that L ₃ entails L ₁, L ₁ is acceptable independently of L ₃, L ₃ is K-atomic, and L ₃ has at least two independently acceptable consequences. Hence, by Friedman’s theory, L ₃ still explains L ₁ after L ₆ makes its appearance. It is also true, according to Friedman’s theory, that L ₆ explains L ₁. This is because L ₆ entails L ₁ (given that entailment is transitive), L ₁ is acceptable independently of L ₆ (given FIA2), L ₆ is K-atomic (by hypothesis), and L ₆ has at least two independently acceptable consequences (given that L ₁, L ₂, L ₃, L ₄, and L ₅ are each acceptable independently of L ₆). So, contra the idea that explanation is winner-takes-all, L ₁ is explained both by L ₃ and by L ₆.

We turn now to Kitcher’s critique of FE. The fact that Friedman’s theory is superior to Kitcher’s theory in at least one important respect makes it all the more important to ask whether Kitcher’s critique of FE is merely technical or, instead, is a stake through the heart of Friedman’s whole approach.

3. Kitcher’s Critique of FE

Kitcher’s critique of FE begins with a point that is clearly correct: FE, which is equivalent to FE*, implies that only K-atomic sentences can be explanatory. Kitcher argues that this consequence is a problem for Friedman’s theory, since some conjunctions of independently acceptable sentences have explanatory power. He gives as an example the conjunction of the Boyle-Charles law (B) and the first law of thermodynamics (T). Laws B and T are acceptable independently of each other. Given this, and given that B&T entails each of B and T, it follows by FIA2 that each of B and T is acceptable independently of B&T. This means that {B, T} is a partition of B&T, and thus B&T is not K-atomic. And yet B&T has explanatory power; for example, B&T explains the law of adiabatic expansion of an ideal gas.

Kitcher’s critique of FE is decisive; FE is false. However, it does not follow that Friedman’s core idea—that explanation is a matter of unification which in turn is a matter of reducing the number of independently acceptable sentences in K—is false. FE is just one way of fleshing out Friedman’s core idea. Perhaps there is an alternative development of this core idea that allows conjunctions of independently acceptable sentences to have explanatory power. But not just any such alternative formulation will do. What is needed is an alternative formulation according to which conjunctions of independently acceptable sentences can be explanatory, but conjunctions—whether of independently acceptable sentences or not—cannot explain their conjuncts.

Below we give an alternative formulation of exactly that sort. But first we need to state a stipulative definition, one in which we define what we mean by “relevant entailment” (r-entailment) in terms of classical entailment (c-entailment):

For example, let $X=\left[\left(F\supset G\right)\text{\&}F\right]$ and $Y=G$. Condition i holds since X c-entails Y. If the first occurrence of “F” in X is replaced by “H,” then the result is $X^{*}=\left[\left(H\supset G\right)\text{\&}F\right]$, which does not c-entail Y. The same point holds for the occurrence of “G” in X and also for the second occurrence of “F” in X. So condition ii holds. Thus, X r-entails Y. For a second example, let $X=\left[F\text{\&}G\right]$ and $Y=F$. Here too condition i holds. But the occurrence of “G” in X is irrelevant in that, for any H, $X^{*}=\left[F\text{\&}H\right]$ c-entails Y. So condition ii does not hold, and thus X fails to r-entail Y. With r-entailments, as opposed to mere c-entailments, there are no logically irrelevant, or idle, occurrences of nonlogical terms.Footnote ¹³

With “r-entailment” thus defined, we can reformulate Friedman’s theory of explanation (FE):

Here con*_K(X) is the set of independently acceptable sentences r-entailed by X.

No conjunction r-entails its conjuncts. So FES0031824800009788 resembles FE in that both imply that no conjunction explains its conjuncts. But FE** S0031824800009788, unlike FE, implies that conjunctions of independently acceptable sentences can be explanatory. Suppose, modifying our toy example a bit, that L ₄, L ₅, and L ₆ are each independently acceptable sentences r-entailed by L ₁&L ₂. Then the smallest K-partition of ${{\mathrm{con}}^{*}}_K\left(L_1\text{\&}{L}_2\right)\cup \left\{L_1\text{\&}{L}_2\right\}$ is {L ₁, L ₂}, whereas the smallest K-partition of con*_K(L ₁&L ₂) includes at least L ₄, L ₅, and L ₆. It follows that L ₁&L ₂ reduces con*_K(L ₁&L ₂) and thus explains each of L ₄, L ₅, and L ₆. The same seems to be true of Kitcher’s example concerning the Boyle-Charles law (B), the first law of thermodynamics (T), and the law of adiabatic expansion of an ideal gas (A). Conjunction B&T r-entails A, which is acceptable independently of B&T, and, assuming that con*_K(B&T) has at least two members in addition to A, B&T reduces con*_K(B&T). Hence, B&T explains A along with each of the other members of con*_K(B&T).

This shows that there is a way to flesh out Friedman’s core idea so that a conjunction of independently acceptable sentences can explain and so that, at the same time, no conjunction can explain its conjuncts. The trick is to use r-entailment instead of c-entailment.

It should be noted that there is independent reason to use r-entailment when developing a deductivist theory of explanation. If X c-entails Y but does not r-entail Y, then X involves a kind of irrelevancy. That X can classically entail Y without explaining Y (owing to the fact that X includes explanatorily irrelevant information) is a familiar point in the literature on explanation. Think, for example, of Kyburg’s (Reference Kyburg1965) discussion of a hexed piece of salt that dissolves in water, Salmon’s (Reference Salmon and Salmon1971, 34) discussion of a man who takes birth control pills and never becomes pregnant, and Putnam’s (Reference Putnam1975, 295–97) discussion of a peg that fits through one hole in a board but not through another.Footnote ¹⁴

Of course, it is possible that FE** has a technical flaw of some sort. However, if that turns out to be so, we see no reason to assume that there can be no technical fix in the offing. Regardless, we now want to return to FE and focus on cases in which X is not a conjunction. If FE is problematic in the way it applies to those cases, this not only tells against the specifics of FE but also, and more importantly, against the underlying idea that explanation is a matter of unification which in turn is a matter of reducing the number of independently acceptable sentences in K.Footnote ¹⁵

4. A New Objection to Friedman’s Theory

Consider two situations in which you might find yourself. In both, K’s membership is restricted to L ₁, L ₂, L ₃, and their consequences, where L ₁, L ₂, and L ₃ are each K-atomic. Law L ₃ entails each of L ₁ and L ₂ (but not vice versa). The situations are different in the following respect:

FE implies that whether L ₃ explains L ₁ depends on whether you are in situation 1 or situation 2. In situation 1, L ₁ is acceptable independently of L ₃, and L ₃ has two independently acceptable consequences, so L ₃ explains L ₁. In situation 2, L ₁ is not acceptable independently of L ₃, and L ₃ has no independently acceptable consequences, so L ₃ does not explain L ₁. According to FE, whether X explains Y depends on the evidence you have. Of course, whether you are justified in believing that L ₃ explains L ₁ depends on your evidence. However, our contention is that whether L ₃ in fact explains L ₁ should not depend on the evidence you happen to have.Footnote ¹⁶ There is a related point. Suppose you are in situation 1, whereas your twin is in situation 2. FE says that L ₃ explains L ₁ for you but not for your twin. This too is problematic for FE. Whether L ₃ explains L ₁ should not be agent relative in this sense.

Here we concur with a good point that Friedman himself makes—that an adequate theory of explanation should have the consequence that whether X explains Y should be an objective matter of fact: “It should be objective—what counts as an explanation should not depend on the idiosyncracies and changing tastes of scientists and historical periods. It should not depend on such nonrational factors as which phenomena one happens to find somehow more natural, intelligible, or self-explanatory than others. … If there is some objective and rational sense in which scientific theories explain, a philosophical theory of explanation should tell us what it is” (Reference Friedman1974, 14). Friedman is right, but we think he should have gone farther and insisted that whether X explains Y depends not at all on what sort of evidence an agent happens to have for X and Y.Footnote ¹⁷

We have faulted Friedman’s theory for making the question of whether law X explains laws Y and Z depend on the kind of evidence an agent has for X, Y, and Z. The sort of agent independence we endorse is a consequence of most theories of explanation, starting with Hempel’s, but is there an argument for thinking that this usual assumption is right? In the case of causal explanations of token events, the case for agent independence seems pretty clear. Whether one token event caused another depends not at all on whether anyone has evidence that these events occurred or were causally related. Causal facts existed aplenty long before scientists discovered them, and there are causal facts that scientists have yet to discover and some that scientists probably never will. And yet, in all these cases, it is true that one token event explains another, whether or not anyone knows that the events are so related. We feel that this fact about causal explanation provides a prima facie argument for the thesis that whether one law explains two or more other laws does not depend on anyone’s having evidence that the laws are true.

This argument can be supplemented with another. Consider any case in which it is uncontroversial that a given law explains one or more other laws. When you look at the explanation, you will usually see that it takes the form of a derivation. The derivation need not mention the evidence that supports the explaining law and the evidence that supports the laws that are being explained. In the derivation you see that the one law entails the others and that the character of the derivation is what allows the one law to explain the others. This does not mean that every derivation is an explanation. However, some are, and the fact that explanatory derivations do not need to cite evidence for the laws that figure in premises and conclusions provides a strong reason for thinking that the explanation relation does not depend on anyone’s having evidence for the laws’ being true.

It may be objected that this criticism of Friedman’s theory misses the mark, since Friedman talks about the law-like sentences that are accepted “by the scientific community,” not by an individual scientist. This objection fails, however, since our point can be frameshifted. Whether one proposition explains another should not depend on how a given scientific community obtained its evidence for those propositions. In addition, there need be no uniform pattern by which the members of a scientific community obtain their evidence; when some scientists in the community are in situation 1 while others are in situation 2, it may be an unacceptable simplification to talk about “the” situation in which the community finds itself.

5. A New Interpretation of Independent Acceptability

So far we have understood Friedman’s concept of independent acceptability as follows:

In the light of our criticism of IA1 in the previous section, we want to consider an alternative characterization of independent acceptability:

IA2 is like IA1 in that both are based on the high probability criterion for rational acceptability (HP). The key difference is that the right-hand side of IA1 is stronger than the right-hand side of IA2.Footnote ¹⁹

IA2 has the consequence that independent acceptability is not relative to an agent, a time, and the agent’s probability function. To see why, let us return to the case of situation 1 and situation 2. In situation 1, there is evidence E ₁ that makes L ₁ acceptable but does not have this impact on L ₃, and there is evidence E ₂ that makes L ₂ acceptable but does not have that impact on L ₃. It therefore follows by IA2 that in both situations each of L ₁ and L ₂ is acceptable independently of L ₃. This means that if Friedman’s theory were understood in terms of IA2, the theory would imply that in both situations L₃ explains L₁ and thus would evade the problem of subjectivity that plagues Friedman’s theory when understood in terms of IA1. But is this new formulation of Friedman’s theory otherwise adequate?

If FIA1 and FIA2—Friedman’s two theses about independent acceptability—are understood in terms of IA2, they say this:

FIA1_IA2 and FIA2_IA2 are both true, given IA2 and the high probability criterion of acceptance (HP).Footnote ²⁰ This is good news for Friedman’s theory. There is, however, some bad news. It is straightforward to show that:

(See app. A for proof.) FIA1*_IA2 is similar to but stronger than FIA1_IA2; FIA1*_IA2 says that a fact about entailment is necessary and sufficient for a failure of independent acceptability, whereas FIA1_IA2 says only that the entailment is sufficient.

FIA1*_IA2 poses a problem for Friedman’s theory. Let $S=\left(x\right)\left(\mathrm{Fx}\supset \mathrm{Gx}\right)$ and $\Gamma =\left\{\left(x\right)\left(\left[\mathrm{Fx}\text{\&}\mathrm{Hx}\right]\supset \mathrm{Gx}\right),\left(x\right)\left(\left[\mathrm{Fx}\text{\&}~\mathrm{Hx}\right]\supset \mathrm{Gx}\right)\right\}$. Neither member of Γ entails S, so by FIA1*_IA2 it follows that each member of Γ is acceptable independently of S. But then, since S is equivalent to Γ, S is not K-atomic, and so Friedman’s theory (FE) entails that S explains nothing. This means that FE, understood in terms of IA2, implies that no universal generalization is K-atomic, and thus no universal generalization explains anything.

This problem generalizes. Any law-like sentence L is equivalent to a set $\Gamma =\left\{L\vee M_1,L\vee M_2,\dots ,L\vee M_n\right\}$, where {M ₁, M ₂, …, M _n} is a partition (of law-like sentences), such that no member of Γ entails L. Thus, by FIA1*_IA2, each member of Γ is acceptable independently of L. It follows that no law-like sentence is K-atomic, and so FE, understood in terms of IA2, implies that no law-like sentence explains anything.Footnote ²¹

Recall Friedman’s claim that “although every sentence is equivalent to a set of n independent sentences, it is not the case that every sentence is equivalent to a set of n independently acceptable sentences—e.g., the members of the set $\left\{P,P\supset S\right\}$ may not be acceptable independently of S; for our only grounds for accepting $P\supset S$, say, might be that it is a consequence of S” (Reference Friedman1974, 16). This claim is right when understood in terms of IA1, but it is wrong when understood in terms of IA2. For, when IA2 is in play, S is acceptable independently of Q if and only if S is independent of Q in that S does not entail Q. This means that FE, understood in terms of IA2, is tantamount to the proposal that Friedman initially considers and immediately rejects that explanation is a matter of reducing the number of independent sentences in K.

Recall the problem with Friedman’s initial proposal. It might seem that reduction can be understood in terms of replacing two or more independent sentences with a single sentence. This does not work, though, since B and G can be replaced by B&G, and yet, clearly, B&G does not reduce the number of independent sentences in K. A natural reply is that B&G is really two independent sentences, because it is logically equivalent to the two-membered set {B, G}. But, as Friedman notes, any sentence is logically equivalent to an n-membered set of independent sentences. The problem, then, is that if reduction is understood in terms of replacing two or more independent sentences with a single sentence, and if a sentence is really n sentences provided that it is logically equivalent to an n-membered set of independent sentences, then no sentence reduces the number of independent sentences in K, and thus there is no explanation. The objection that Friedman makes to the first proposal he considers is really the same as our objection to FE when it is understood in terms of IA2: if FE is so understood, then no sentence reduces the set of its independently acceptable consequences, and thus there is no explanation.Footnote ²²

Would it help Friedman’s theory if HP were replaced by an alternative Bayesian conception of acceptability and thus if IA2 were replaced by an alternative Bayesian conception of independent acceptability? We are skeptical. Consider, for example, the following alternative to HP:

This leads to the following alternative to IA2:

It is straightforward to show that:

(See app. B for proof.) The points above concerning the case of $S=\left(x\right)\left(\mathrm{Fx}\supset \mathrm{Gx}\right)$ and $\Gamma =\left\{\left(x\right)\left(\left[\mathrm{Fx}\text{\&}\mathrm{Hx}\right]\supset \mathrm{Gx}\right),\left(x\right)\left(\left[\mathrm{Fx}\text{\&}~\mathrm{Hx}\right]\supset \mathrm{Gx}\right)\right\}$ carry over straightforwardly to FE understood in terms of IA2*. Furthermore, FIA1 and FIA2 are false when interpreted in terms of IA2*. (See app. C for proof.)

6. Concluding Comment

After showing how Friedman’s theory of explanation as unification can be reformulated so as to evade Kitcher’s (Reference Kitcher1976) objection, we explored three Bayesian interpretations of the concept of independent acceptability. Each of these interpretations is problematic for Friedman’s theory. When Friedman’s theory is coupled with IA1, the theory has the implausible consequence that whether one proposition explains another depends on the kind of evidence an agent (or a scientific community) has for the two propositions. IA2 and IA2* avoid this pitfall, but they saddle Friedman’s theory with a separate implausibility since Friedman’s theory, conjoined with IA2 or IA2*, implies that universal generalizations never explain anything. We acknowledge that there may be other Bayesian conceptions of independent acceptability and that non-Bayesian epistemologies may have suggestions that are worth considering as well. So our conclusion is conditional: if independent acceptability is interpreted in any of the ways we have examined, Friedman’s theory is in trouble.