1. McGrew and Myrvold on Unification

In three recent papers, Wayne Myrvold (Reference Myrvold1996, Reference Myrvold2003) and Timothy McGrew (Reference McGrew2003) have developed Bayesian accounts of the virtue of unification. In his account, Myrvold offers a specific measure of unification that can be applied to individual hypotheses. He argues that one must account for this measure in order to calculate correctly the degree of confirmation that a hypothesis receives from its evidence. On the other hand, McGrew demonstrates that, ceteris paribus, a hypothesis that unifies its evidence will have a higher posterior probability than a hypothesis that does not. In this paper, I prove that the two accounts of unification require the same underlying inequality, revealing the fact that McGrew and Myrvold have accounted for unification in fundamentally identical probabilistic terms. I then evaluate five putative counterexamples to this account and show that these examples, far from disqualifying it, serve to clarify our notion of unification by disentangling it from a host of other concepts.

McGrew describes a hypothesis's level of “theoretical consilience” (what we are here calling ‘unification’) as “the extent to which [otherwise independent] pieces of evidence can be seen to be positively relevant to each other in the light of that hypothesis” (Reference McGrew2003, 562). A unifying hypothesis, in this sense, is one that increases the positive relevance of its bits of evidence. This definition in place, McGrew shows that, ceteris paribus, a unifying hypothesis will achieve a greater posterior probability than a nonunifying hypothesis:

The fact that H ₁ is a unifying hypothesis and H ₂ is not can be symbolized:Footnote ¹

Given the ceteris paribus conditions, we know that the individual likelihoods of each bit of evidence are equal for the hypotheses, and so we are able to infer:

Utilizing our ceteris paribus conditions once more and Bayes’ theorem, we may validly conclude:

Myrvold suggests that unification is a hypothesis's ability “to render what, on prior grounds, appear to be independent phenomena informationally relevant to each other” (Reference Myrvold2003, 400). In order to develop a measure of this virtue, Myrvold first defines the “informational relevance” of evidence set $e_{1},\,\ldots ,\,e_{n}$:

He then defines the measure of unification in terms of informational relevance:

If we place Myrvold's unification measure under the ceteris paribus conditions that McGrew has set up and use this measure comparatively between two hypotheses, then Myrvold's account reduces to McGrew's. First, we compare the degrees of unification for two different hypotheses, where H ₁ unifies the evidence but H ₂ does not:

Because H ₂ does not unify the evidence set, the evidence remains independent and the numerator and denominator of the second term are equal. In this case the logarithm is equal to 0:

But this inequality is exactly that used by McGrew to account for the virtue of unification. Thus, the two accounts are placing the virtue of unification in fundamentally identical Bayesian terms.

A parallel proof of the equality of these two accounts is available in the case that both hypotheses unify the evidence, but H ₁ to a greater degree than H ₂. In this case, the positive relevance that H ₁ bestows upon the evidence set is greater than that bestowed upon the evidence set by H ₂. McGrew's account changes as follows:

Both relations are now inequalities; we cannot tell by these inequalities that the first is stronger than the second. One easy way to show this relation is by dividing the left-hand terms by the right-hand terms and comparing these new quantities:

Given Myrvold's account, we have only to show that $U^{(n) }(e_{1},\,\ldots ,\,e_{n}\mathrm{;}\,\,H_{1}) $ is greater than $U^{(n) }(e_{1},\,\ldots ,\,e_{n}\mathrm{;}\,\,H_{2}) $. But, as presented above, this reduces to showing

Again, the fundamental inequality necessary for detecting a difference in unification is the same in both accounts. Thus, regardless of whether H ₂ leaves the evidence set statistically independent or less positively relevant than H ₁, both accounts ultimately require the same underlying inequalities.

2. Lange's Counterexamples

Marc Lange (Reference Lange2004) argues that a significant measure of unification is both too easy and too difficult for theories to achieve given Myrvold's (and, as it turns out, McGrew's) account of unification (MM). I will argue that Lange's counterexamples fail. Showing why this is so in the case of each counterexample sharpens the sense of unification that MM is accounting for by disentangling it from other distinct virtues. This section represents an attempt to carry out McGrew's practical advice that we “distinguish diverse evidential virtues that have been conflated in our existing terminology” (Reference McGrew2003, 557). As he writes:

Example 1. Lange argues that a positive value for the measure of unification U is too easy for a hypothesis to achieve—i.e, the fact that a hypothesis achieves a positive U-value is not a sufficient condition for the assertion that the hypothesis has the virtue of unification—through the use of five potential counterexamples. In the first example, H is the hypothesis that “two decay products annihilate as particle and antiparticle (so they must be electron and positron).” There are two bits of evidence in support of H: e ₁ = “atom #1 [i.e., decay product #1] emits an electron” and e ₂ = “atom #2 emits a positron.” According to Myrvold's measure of unification,Footnote ²

Thus, according to MM, H unifies e ₁ and e ₂. But, Lange argues, H can only unify its evidence if it possesses some sort of explanatory or ontological priority to e ₁ and e ₂. He concludes, “[T]his statistical relevance relation is too crude to indicate whether there obtain the ontological and explanatory relations involved in genuine unification” (Reference Lange2004, 207).

Lange's argument can be summarized as follows:

By two applications of modus tollens, C2 follows from P1, P2, and P3; thus, Lange's argument is valid. P1 is unquestionably true, given that H does have a positive U-value. P3 seems correct as well, but P2 is questionable for several reasons. First, unification is often—but not always—a virtue that an explanatory hypothesis possesses, and this is perhaps the main reason why P2 intuitively seems true. But why should one think that if H is a unifying hypothesis, it must also be a good explanation? Unification is only one explanatory virtue among many, and a unifying hypothesis might not be an explanatory hypothesis if it is sufficiently lacking in other explanatory virtues. Lange's example depicts precisely such a situation—H unifies, but does not explain, the evidence e ₁ and e ₂. Thus, H need not have explanatory priority to its evidence in order to unify the evidence.

What about ‘ontological priority’? This term is vague, and Lange does little to clarify its precise intended meaning. Within the context of Lange's first example, it seems most likely that the term refers to a temporal priority of existence that a certain state of affairs or fact has over another. Given this meaning, it is true that the state of affairs referred to in H is not ontologically prior to the state of affairs referred to in e ₁ or e ₂—i.e., it is not the case that the two decay products annihilate as matter and antimatter until one atom emits an electron and the other emits a positron. However, this category of ontological priority is not a necessary feature of a unifying hypothesis. Indeed, the state of affairs of there being fresh-baked chocolate chip cookies at home (H) may unify the state of affairs of my hurrying home (e ₁) with the state of affairs of my brother's coming over to visit this afternoon (e ₂), even if the two latter states of affairs are actualized before the cookies are baked. Thus, upon the most obvious interpretation of ‘ontological priority’, and given the discussion of explanatory priority above, P2 is false—it is entirely possible for H to be a unifying hypothesis (according to the MM account) that does not have explanatory or ontological priority to its evidence. Consequently, neither C1 nor C2 can be soundly inferred from this argument.

Lange's first example seems initially plausible because it appeals to our sense that H has no explanatory power regarding e ₁ and e ₂. It seems that H must not have the explanatory virtue of unification because it is an overall explanatorily worthless hypothesis. However, granting that H is explanatorily vacuous, it is not necessarily true that it thus has no explanatory virtue regarding e ₁ and e ₂. The MM account easily responds to Lange's first example by claiming that H does have the explanatory virtue of unification but is not thus explanatorily virtuous; one explanatory virtue does not a good explanation make.

Example 2. In his second attempted counterexample, Lange shows that one can get a positive U-value for $H\thinspace{:}\thinspace e_{1}\supset e_{2}$, where e ₁ and e ₂ represent any two arbitrary propositions (e.g., Lange uses e ₁: “Queen Elizabeth I was crowned on a Tuesday,” and e ₂: “There are basalt outcroppings on the far side of the moon”):

(given that e ₂ entails H). Because e ₂ logically entails H and not vice versa, $P(e_{2}) \leq P(H) $. But this means that $U\geq 0$. Lange claims that this result is counterintuitive: “But genuinely to unify [e ₁] and [e ₂], a theory must reveal them to have some deep common explanatory basis. It should not suffice for a theory merely to staple [e ₁] and [e ₂] together with a ⊃” (Reference Lange2004, 208).

Like his first attempted counterexample, the efficacy of Lange's second example hinges upon his assertion that a unifying theory must be able to explain the evidence. But again, unification is only one explanatory virtue among many, and a unifying hypothesis might not be an explanatory hypothesis if it is sufficiently lacking in other explanatory virtues. A theory can possess the virtue of unification while at the same time not explaining its evidence; thus, there is little reason to accept Lange's bold claim that a unifying theory “must reveal [e ₁ and e ₂] to have some deep common explanatory basis.” The ability of H to unify is not a sufficient condition for the ability of H to explain, and Lange's first two examples represent this insufficiency relation nicely. The fact that H ‘staples’ e ₁ and e ₂ together by a conditional relation does suffice to unify e ₁ and e ₂; however, it does not suffice to explain e ₁ and e ₂.

One reason why this example seems to work is that we are intuitively inclined to reject H as a reasonable hypothesis. However, this intuition does not necessarily mean that the hypothesis does not render the evidence positively relevant—i.e., that the hypothesis does not unify the evidence in MM's sense. Instead, our intuitive rejection of H seems to be caused by the fact that H is contrived to link together two arbitrary propositions and is thus ad hoc. It should come as no surprise that H unifies e ₁ and e ₂ according to the MM account, because H is a proposition gerrymandered for this sole purpose. It succeeds in doing so simply by the fact that H rules out one of the possible ways for e ₁ and e ₂ to be negatively correlated—in the case that e ₁ is true and e ₂ is false. However, H does not explain e ₁ and e ₂. This is obvious because H makes a weaker claim than the conjunction of e ₁ and e ₂. H is entailed by the conjunction of e ₁ and e ₂. But any such hypothesis that does not go beyond its evidence set, but rather merely states a logically entailed consequence of this set, will not be explanatory. Any valuable explanation must go outside of its evidence set in order to describe why that evidence happens to be the way that it is.

This example reveals one further important point. The conjunction (e ₁ & e ₂) is only true in the case that both conjuncts are true, whereas H is true by material implication anytime that e ₂ is true—regardless of the truth value of e ₁—or in the case that both e ₁ and e ₂ are false. In this case, H's prior probability P(H) may turn out to be very high. P(H) can be calculated as the sum of H's truth conditions:

If $P(e_{1}) $ is sufficiently low, H will have a high prior probability. Plugging this high value into Bayes’ theorem, H may also achieve a significant posterior probability. However, one would still not want to explain e ₁ and e ₂ by appealing to the gerrymandered hypothesis H. This example shows that explanatory power is not equivalent to a high prior probability or to a high posterior probability. In this example, H may be a highly plausible claim, but it is not a highly explanatory hypothesis—despite the fact that it does unify e ₁ and e ₂.

Example 3. Lange's third example compares the U-value of two different hypotheses:

As Lange notes, as a result of Einstein's strong belief that all phenomena involving light's generation, conversion, and interaction with matter are as if light is quantized, he was able to derive—among other things—Stokes's rule of photoluminescence and an equation governing the Volta effect. For Lange's example, e ₁ and e ₂ may be set as these two results derived through the use of the light-quantum hypothesis. For both hypotheses, we can calculate the U-value:

As Lange notes, this value will be positive, revealing the fact the H ₁ unifies the evidence set. More important, Lange asserts, is the fact that the results will be identical for H ₂. He objects to this: “This replacement should make a dramatic difference to the degree of unification. … Myrvold's notion of unification is insensitive to the great difference in unifying power between [H ₁] and [H ₂]” (Reference Lange2004, 210).

Lange is again confusing the virtue of unification as explicated in the MM model with some other notion. Both H ₁ and H ₂ would render the evidence set equally positively relevant, and this makes sense. Whether a hypothesis possesses the virtue of unification, in MM's sense, is solely a question of whether e ₁ bears an increased informational relevance to e ₂ in light of H. Both of these hypotheses are identical in this respect and only differ in their explanations of why they have this unifying ability. For Lange to suggest that this constitutes a successful counterexample to the account of unification is mistaken. Any account of this theoretical virtue would be inaccurate if it followed Lange's advice and distributed different values for these two hypotheses.

Lange's example has, prima facie, a ring of plausibility to it because there are differences between H ₁ and H ₂ that render H ₁ significantly less theoretically virtuous. However, these differences are not found in the relative abilities of the hypotheses to unify the evidence set. First, unlike H ₂, it is far from obvious that H ₁ is even sufficiently clear to have a well-defined prior probability at all. This ‘hypothesis’ actually describes a family of hypotheses that resemble each other by the fact that they all describe a possible state of nature that makes it coincidentally the case that the phenomena in question are as if light is quantized. Each of these subhypotheses is characterized by a miniscule prior probability. Second, even if we grant that H ₁ is sufficiently clear for analysis, this hypothesis is still far less likely apart from considerations of the evidence than H ₂—i.e., it is more likely that all phenomena involving light's generation, conversion, and interaction with matter are as if light is quantized because of some law than because of chance. Thus, although the likelihoods are equal between the two hypotheses, the prior probabilities are far from it. This fact, rather than any difference in the abilities of the hypotheses abilities to unify the evidence, makes H ₁ dubious relative to H ₂.

Example 4. Lange's fourth example appeals to the ratio (Ω) of the universe's gravitational potential energy (e ₁) to its kinetic energy (e ₂).Footnote ³ H ₁ is the inflationary Big-Bang theory, which unifies the two components of Ω, where $\Omega =1$. H ₂ is an equivalent Big-Bang theory, but for the fact that it unifies the two components of Ω where $\Omega =3.78\times 10^{-27}$. Lange observes that both (H ₁ & e ₁) and (H ₂ & e ₁) will entail e ₂. Accordingly, MM will treat both hypotheses as equally unifying. However, Lange argues that “it is crucial to the epistemic advantage possessed by the inflationary Big-Bang model, in virtue of unifying [e ₁] and [e ₂] …, that $\Omega =1$ is not an arbitrary value” (Reference Lange2004, 211). Lange gives two reasons for this point: First, $\Omega \neq 1$ would either mean that the universe is ‘closed’ or ‘open’, ultimately resulting in either the Big-Crunch or an eternally expanding universe. Second, $\Omega =1$ is a significant value in that it reveals the universe to be in a state of ‘unstable equilibrium’. Set at a value $1\pm \varepsilon $, $\varepsilon > 0$, at t ₀, this value would increasingly diverge from this point of equilibrium with the flow of time. Thus, Lange insists that any accurate account of unification must treat these two hypotheses differently.

Of course, Lange is correct that the two hypotheses are to be treated differently. However, Lange errs when he insists that this difference should be accounted for by a difference in unification between the hypotheses. Solely regarding the relative abilities of the two hypotheses to make the two components of Ω positively relevant, these hypotheses are on equal ground. Thus, they are equally unifying according to the MM account. Where these two hypotheses do diverge in their relative worth is—as in the previous example—at the level of prior probabilities. In our background knowledge, as we approach this example, we have various data favoring a value of $\Omega =1$, rather than $\Omega =3.78\times 10^{-27}$. Consequently, given our background knowledge, the prior probability of H ₁ is significantly higher than the prior probability of H ₂.Footnote ⁴ It seems that Lange has mistakenly argued that the two hypotheses must be unequal in their unifying abilities by appealing to intuitions that are already accounted for by the inequality of prior probabilities.

Example 5. Lange's fifth attempted counterexample is intended to show that a positive value for U can be too difficult for a hypothesis to achieve—i.e., in order for a hypothesis to have the virtue of unification, it is not necessary that the hypothesis must achieve a positive value for U. In this example, H is the hypothesis that “Jones has systemic lupus erythematosus,” e ₁ is the observation that “Jones has pleuritis,” and e ₂ is the fact that “Jones has a malar rash.” Both e ₁ and e ₂ are symptoms of lupus; however, neither guarantees, nor is guaranteed by, the presence of lupus. Lange presents the following probability values: $P(e_{1}) =.055$, $P(e_{2}) =.050$, $P(H) =.035$, $P(e_{1}\vert H) =.500$, $P(e_{2}\vert H) =.650$, $P(H\vert e_{1}) =.318$, $P(H\vert e_{2}) =.455$, and $P(H\vert e_{1}\,\& \,e_{2}) =.700$. Using Bayes’ theorem, it follows from these values that $P(e_{1}\vert e_{2}) =.325$ and $P(e_{2}\vert e_{1}) =.297$. Because $P(e_{1}\vert e_{2}) > P(e_{1}) $ and $P(e_{2}\vert e_{1}) > P(e_{2}) $, the evidence set is positively relevant apart from H. However, given H, e ₁ and e ₂ become statistically independent; thus, $P(e_{1}\vert e_{2}\,\& \,H) =P(e_{1}\vert H) $ and $P(e_{2}\vert e_{1}\,\& \,H) =P(e_{2}\vert H) $. Calculating the U-value for H, we get

U is a negative value, so we should conclude that H disunifies the two symptoms. But, Lange avers, “This is surely mistaken: [H] would supply [e ₁] and [e ₂] with a common cause. Clearly, a physician trying to diagnose a patient afflicted with both pleuritis and a malar rash should (ceteris paribus) prefer a unified explanation to distinct explanations of the two symptoms” (Reference Lange2004, 214).

Lange is right that the physician ought to prefer the common cause explanation of both symptoms. But this does not mean that the explanation thus has the virtue of unification. From the Bayesian standpoint, the reason that the physician prefers the common cause explanation H is that $P(H\vert e_{1}\,\& \,e_{2}) $ is significantly high. It is not significant, unlike many cases, even partly as a result of H's unifying the evidence. In fact, as Lange shows, H not only does not have the virtue of unification but we might say it has the vice of disunification. Contra Lange, this is no fault to MM, because we can easily show exactly why H has a high posterior probability, even while it disunifies the evidence. Bayes’ theorem states

The two bits of evidence e ₁ and e ₂ are statistically independent given H:

(I label this form of the equation λ)

Although $P(e_{1}\,\& \,e_{2}\vert H) $ would be greater than $P(e_{1}\vert H) \times P(e_{2}\vert H) $—and thus greater than it is in our derivation above—if H unified the evidence set, H can still achieve a high posterior probability as a disunifying hypothesis. In this particular example, the fact that each symptom is far more likely to appear for someone who has lupis than for someone who does not is reflected probabilistically by likelihoods that are significantly greater than expectedness (the two ratio terms in λ). These two ratios multiplied together are equal to the ratio attained by dividing the posterior probability of H by the prior probability of H—the degree of confirmation that the evidence gives H. In this example, both ratios have a high positive value approximately equal to twenty. Thus, the reason why any rational physician ought to infer H as the common cause explanation of both symptoms is not that the evidence is unified by invoking such an explanation in this example. Rather, it is because the symptoms are far likelier to occur together—or separately—for a person who has lupus than for someone who does not. This fact counteracts the disunifying effect of H on its evidence; the presence of both symptoms ultimately confirms the diagnosis even though the diagnosis renders the two symptoms less positively relevant.

At times, Lange seems to assume that offering a common cause explanation is equivalent to offering a unifying hypothesis. (‘Common cause’ is intended in a strict Reichenbachian sense so that H is a ‘common cause’ of e ₁ and e ₂ iff H screens off e ₂ relative to e ₁—i.e., $P(e_{1}\vert e_{2}\,\& \,H) =P(e_{1}\vert H) $.) I think that this is a mistake since there are at least two very different ideas of unification at work in today's philosophical literature. Apart from the idea accounted for by MM, there is the idea of unification as generality. Regarding this putative virtue, Elliott Sober writes, “The imperative to find explanations that are general is equivalent to the imperative to find explanations that are unified. … [G]enerality is just unification by another name” (Reference Sober and Stadler2003, 208). According to this notion of unification, a common cause explanation is without exception also a unifying hypothesis because, as a common cause, it is a general explanation of all of the evidence. However, if H is a common cause in this strict Reichenbachian sense, then it cannot unify the evidence in the sense accounted for by MM. H's U-value—where H is the common cause of both e ₁ and e ₂—must be negative:Footnote ⁵

Given that the idea of unification as generality must apply to any H that is a common cause explanation, whereas unification according to MM cannot apply to such an H, it is clear that the two ideas of unification are indeed distinct.Footnote ⁶

Regarding his fifth attempted counterexample, Lange seems to be conflating these two distinct notions of unification when he writes, “[H] would supply [e ₁] and [e ₂] with a common cause. Clearly, a physician trying to diagnose a patient afflicted with both pleuritis and a malar rash should (ceteris paribus) prefer a unified explanation to distinct explanations of the two symptoms” (Reference Lange2004, 214, my emphasis). In terms of unification as generality, Lange's point is well-made: H would be a common cause explanation so it would be a unifying (general) explanation of the evidence. However, Lange's point seems confused given the MM notion of unification: according to this analysis, H would be a common cause explanation so it could not possibly also be a unifying explanation.

3. Conclusion

There are five important distinctions that I have now made through my replies to Lange's five examples. First, unification is not equivalent to explanation. H may unify the evidence without explaining it, and H may explain the evidence without unifying it. The next three distinctions separate unification from a hypothesis's prior probability, posterior probability, and likelihood. A hypothesis may unify its evidence but still not approach a high posterior probability because either its prior probability is low, or because its likelihood is low, or both. Additionally, a hypothesis that has a significant prior probability and/or a significant likelihood may have a very high posterior probability, even if it does not unify its evidence. Fifth, unification according to MM is not unification as generality. There are multiple notions of unification at work in philosophy. Although there is some extensional overlap between these notions—e.g., one hypothesis can be unifying in both the generality and in the MM sense at the same time—they are conceptually distinct.