On the Logical Structure of Best Explanations

Abstract

Standard articulations of Inference to the Best Explanation (IBE) imply the uniqueness claim that exactly one explanation should be inferred in response to an explanandum. This claim has been challenged as being both too strong (sometimes agnosticism between candidate explanatory hypotheses seems the rational conclusion) and too weak (in cases where multiple hypotheses might sensibly be conjointly inferred). I propose a novel interpretation of IBE that retains the uniqueness claim while also allowing for agnostic and conjunctive conclusions. I then argue that a particular probabilistic explication of explanatory goodness helpfully guides us in navigating such options when using IBE.

1. Uniqueness and IBE’s critics

Inference to the Best Explanation (IBE) is a form of uncertain inference that favors the single best potential explanation of some given explanandum. As such, standard articulations of IBE imply the following claim:

Uniqueness. Whenever a reasoner is in possession of a set of potential explanations of some given explanandum, IBE advises that agent to infer at least and at most one of these explanations.

Uniqueness secures the intuitive usefulness of IBE, which seems to have a near ubiquitous presence in everyday and scientific reasoning (Lipton 2004; Douven and Schupbach 2015; Schupbach 2017). However, this claim ostensibly makes IBE vulnerable to multiple lines of attack.

For one thing, Uniqueness opens IBE up to the criticism that it is unable in some cases to avoid legislating inferences to manifestly poor explanations. Uniqueness requires reasoners to infer at least one explanation, problematically including cases in which the best explanation is not good enough. This concern underlies van Fraassen’s (1989, 143) “bad lot” objection. Were it not for Uniqueness’s requirement that one best explanation always be inferred, there would be no obvious concern with cases in which we have only a bad lot of hypotheses from which to choose. McCain and Poston (2019) present a closely related “disjunction objection” to IBE: “[A] particular hypothesis [may be] the best explanation of a given set of evidence even though the disjunction of its rivals is more likely to be true. [… But] it is not rational to believe that H is true when H is more likely to be false than true.” In cases where the most explanatory hypothesis is in some sense not good enough, it is unreasonable to go ahead nonetheless and infer that hypothesis. The rationally appropriate response, rather, would seem to be agnosticism between at least some of the explanatory options.

Scientific instances of such situations and the appropriate agnostic response are easy to find, for example, in cases of causal heterogeneity and multiple realizability (Ross 2020, forthcoming). For example, Parkinson’s disease (PD) is produced by a complex of causal pathways across different patients. Nandipati and Litvan (2016) review a variety of studies linking cases of PD to environmental exposures to pathologic agents found in pesticides and industrial compounds. Lesage and Brice (2009) survey a host of genetic studies, highlighting “more than 13 loci and 9 genes that have been identified” as having some role in PD’s etiology. Such studies reveal monogenic forms of PD as well as forms attributable to various gene–environment interactions. As Ross (forthcoming, 10) summarizes, PD “can be produced by single gene variants, single environmental factors, and combinations of genetic and environmental factors.” Without knowing a patient’s genetics or environmental background, it would be heedless for a clinician to attribute a case of PD to any one specific causal pathway; a disjunctive conclusion between possible pathways would be more prudent. Apparently, Uniqueness compels the clinician in this case incautiously to infer too much.

But Uniqueness also makes IBE vulnerable to the criticism that, in other cases, it forces reasoners to infer too little. Salmon (2001, 67) argues that IBE guides reasoners to infer at most one explanation, even in cases where a multiplicity of compatible explanations are conjointly warranted. The reasonable inference here would rather be to the conjunction of explanatory hypotheses.

Examples of such situations and the appropriate conjunctive response abound, for example, in cases of “multicausality” (Ross forthcoming). Causal pathways to PD may cite particular combinations of gene variants and/or environmental conditions working together. Lesage and Brice (2009, R52) note that the large majority ( $ \gt 90\%$ ) of cases of PD are not monogenic. Rather, most cases seem to “result from complex interactions among genes and between genes and environmental factors.” If the evidence for a particular patient is such that a multicausal pathway involving exposure to pesticides combined with a genetic variant as a susceptibility factor is most explanatory, then the appropriate inference would be to the conjunction of these factors. To the extent that IBE forces the clinician instead to pick at most one factor, it recommends an unreasonably stringent conclusion.

In response to these challenges, IBE’s defenders have given up on Uniqueness. The bad lot objection and related concerns have led philosophers to recast IBE as only guiding explanatory inference in cases where the best explanation is “sufficiently good” (Lipton 2004, 93; McCain and Poston 2019, 5). Concerns like Salmon’s have similarly led philosophers to limit IBE’s application to cases in which the available explanatory hypotheses compete in some sense (Lipton 2004). As I argue elsewhere (Schupbach 2019), such hedges impose absurdly strong limitations on IBE’s applicability.

The remainder of this paper argues that philosophers have been too quick to weaken IBE. I defend a traditional account of IBE, Uniqueness and all, against the above challenges. At the heart of my response is the claim that these criticisms rely upon a questionable interpretation of “best explanation.” An alternate interpretation allows IBE’s proponents both to endorse Uniqueness and sidestep the criticisms.

2. The “best explanation”

Virtually all commentators on IBE—defenders and critics alike—interpret “best explanation” to mean the most explanatory individual hypothesis. Potential explanations are taken to correspond one-to-one with individual hypotheses, the best such explanation then naturally amounting to the hypothesis that best explains the explanandum. Thus, Harman (1965, 89) describes IBE as follows: “In making this inference one infers, from the premise that a given hypothesis would provide a ‘better’ explanation for the evidence than would any other hypothesis, to the conclusion that the given hypothesis is true.” More recently, Lange (2022, 85) describes IBE as an inference form in which “we argue that one hypothesis derives some plausibility over its rivals from the fact that the explanations it would give (if it were true) are better than those its rivals would give (if they were true).” This sample is small but entirely characteristic of the literature.

While this interpretation is all but universally assumed, it is never questioned. However, the interpretation is indeed questionable. Arguably, it is also the actual source of trouble when it comes to the above criticisms. The standard interpretation makes any instance of IBE problematically sensitive to a given individuation of hypotheses. The inferences we may draw become artificially limited by the contingent way in which we have carved up the space of hypotheses. Any such individuation becomes an inferential barrier, blocking our path to conclusions that may describe better explanations.

In the PD example, any particular lot of available explanatory hypotheses can either be too finely or coarsely grained for an optimal explanation. Coarsely individuated hypotheses (e.g., environmental factors or genetic variants) will often not be sufficiently informative to have any substantial explanatory value. If the evidence is best explained by a combination of genetic and environmental causes, then the clinician may want to start conjoining some of these coarsely grained alternatives to construct more informative explanations. Finely grained hypotheses (e.g., long-term, repeated exposure to organochlorines combined with short-term exposure to rotenone and the G2385R mutation in the LRRK2 gene) may be too specific. Given less detailed evidence, the most explanatory diagnosis may be merely to posit that there was, for example, some past exposure to pesticides combined with a genetic susceptibility factor. A less committed explanation such as this is the disjunction of more finely grained alternatives.

These observations suggest a way of defending IBE without giving up Uniqueness. Namely, we might reconsider our interpretation of “best explanation.” There is at least one other natural interpretation available. On this alternative, the “best explanation” can refer to whatever stance apropos the hypotheses is most explanatory (Schupbach 2019, 158–9). Individual hypotheses may provide best explanations, but this need not be the case.

On such an interpretation, explanatory inference is not strangely beholden to the way in which the space of hypotheses happens to be carved up. But relative to such a space, explanatory stances available for inference include, in principle, all Boolean combinations of the hypotheses. Individual hypotheses may be combined in any way that results in better explanations. If multiple hypotheses conjointly offer a more explanatory stance than any of these individually—as in cases of multicausality—then IBE will guide us to infer such conjunctive explanations when “best explanation” is understood in the proposed sense. If an agnostic shrug between individual hypotheses is all the explanation one can properly infer—as in some cases of causal heterogeneity or multiple realizability—then that is the conclusion that IBE, so interpreted, will recommend.

3. Determining the structure of best explanations

This reinterpretation provides IBE with a potential end-around well-known criticisms; however, it also gives rise to a pressing, new challenge. According to this reading, individual hypotheses may be conjoined, disjoined, or otherwise logically combined to formulate candidate best explanations. Accordingly, reasoners must not only compare the explanatory goodness of the hypotheses on the table, but also assess whether any explanatory improvements come by way of logically strengthening or weakening such hypotheses. Both possibilities can seem problematic. Strictly stronger explanations are informationally more complex, and so less probable, than correspondingly weaker alternatives; on the upside, they are also more informative. Strictly weaker, informationally simpler explanatory stances are less informative but inevitably more probable than correspondingly stronger stances.

The new challenge facing supporters of this account of IBE is to offer some clarity on how to negotiate these opposing goals. Ideally, we seek an explication of explanatory goodness that provides a principled balance between informational simplicity (the relative weakness of our explanation’s claims) and informativeness apropos the explanandum. Such an account should guide us in determining tipping points at which explanatory stances are exactly as strong and informative with respect to the explanandum as they should be—any stronger and their inevitable loss of likeliness is not worth any remaining potential gains in informativeness.

In recent work, Glass and Schupbach (2023a,b) develop and defend the following measure of the degree of explanatory goodness that an explanatory hypothesis $h$ has with respect to an explanandum $e$ :

$${\cal E}\left( {e,h} \right) = {\rm{log}}\left( {{{{\rm{Pr}}(e | h){\rm{Pr}}{{(h)}^{1/2}}} \over {{\rm{Pr}}\left( e \right)}}} \right).$$

The remainder of this section compares ${\cal E}$ to alternatives, highlighting ${\cal E}$ ’s particular suitability as an explication of our target notion of explanatory goodness.

Formal epistemologists offer measures of explanatory “power” that provide prima facie plausible alternative explications of our target concept (McGrew 2003; Schupbach and Sprenger 2011; Crupi and Tentori 2012). These all are “relevance measures” (Fitelson 1999, S363), gauging the degree of statistical relevance between any $h$ and $e$ . Any such measure $r\left( {e,h} \right)$ implies the following:

$$r\left( {e,{h_1}} \right) \gt r\left( {e,{h_2}} \right){\rm \ iff\ {Pr}}(e|{h_1}) \gt {\rm Pr}(e|{h_2}).$$

This simple implication of relevance measures provides a strong argument against their application for our purposes. If any relevance measure is used to explicate the notion of explanatory goodness at work in IBE (as we’ve interpreted it), then IBE will virtually always guide us to infer logically stronger explanations. Let ${h_1}$ provide an appealing potential explanation with substantial positive relevance to $e$ : ${\rm{Pr}}(e|{h_1}) \gg {\rm{Pr}}\left( e \right)$ . Now consider any additional ${h_2}$ at all; so long as it isn’t contrary to ${h_1}$ , it can be irrelevant to or even as negatively associated with ${h_1}$ as you like. If $e$ is even slightly more likely given ${h_1}{\rm{\;}}\& \;{h_2}$ than given ${h_1}$ alone, ${\rm{Pr}}(e|{h_1}{\rm{\;}}\& \;{h_2}) \gt Pr(e|{h_1})$ , this account tells us to favor the logically stronger (and possibly exceedingly improbable) explanation. In general, whenever the likelihood of $e$ can be bumped up by strictly strengthening one’s explanatory stance, this logically stronger position will win out in terms of the above inequality. This makes logically stronger, informationally complex explanations far too easy to prefer.

There is a price we pay when we favor a logically stronger conclusion; there are strictly more ways such a stance could be wrong. The problem with accepting any relevance measure as our explication of explanatory goodness is that it essentially ignores this price. Any benefit in accounting for $e$ (no matter how slight) is worth any price we pay by complicating our explanatory stance (no matter how great). This account thus fails for our purposes. Evidently, what is needed is an account of explanatory goodness that tempers considerations of explanatory relevance with penalties for informational complexity.

Plausibly, Bayes’ theorem does exactly this. The ratio ${r_{{\rm{GM}}}}\left( {e,h} \right) = {\rm{Pr}}(e|h)/{\rm{Pr}}\left( e \right)$ is a relevance measure proposed by Good (1968) and McGrew (2003) for gauging $h$ ’s ability to account explanatorily for $e$ ; as such, ${r_{{\rm{GM}}}}$ fails to penalize for complexity. Because increasing informational complexity (increasing logical strength) corresponds to a decreasing probability, an explanatory conclusion’s prior probability provides a straightforward penalization factor for complexity. For example, for logically independent ${h_1}$ and ${h_2}$ , ${h_1}{\rm{\;}}\& \;{h_2}$ is strictly more informationally complex than either individual hypothesis, and so it should be penalized relative to these weaker options. This can be achieved by weighting a hypothesis’s explanatory goodness by it’s probability—since ${\rm{Pr}}\left( {{h_1}{\rm{\;}}\& \;{h_2}} \right) \le {\rm{Pr}}\left( {{h_1}{\rm{\;}}\left[ {{h_2}} \right]} \right)$ . Bayes’ theorem does precisely this:

$${\rm{Pr}}(h|e) = {{{\rm{Pr}}(e|h)} \over {{\rm{Pr}}\left( e \right)}} \times {\rm{Pr}}\left( h \right) = {r_{{\rm{GM}}}}\left( {e,h} \right) \times {\rm{Pr}}\left( h \right).$$

However, using ${\rm{Pr}}(h|e)$ to balance explanatory relevance against informational complexity essentially leads to the opposite problem as that facing relevance measures. If posterior probabilities are used to explicate explanatory goodness, then IBE always guides us to infer logically weaker stances. Let ${h_1}$ provide an appealing potential explanation with some substantial positive relevance to $e$ : ${\rm{Pr}}(e|{h_1}) \gg {\rm{Pr}}\left( e \right)$ . Now consider any additional ${h_2}$ at all. So long as ${h_2}$ is not inconsistent with ${h_1}$ or $e$ , then we get the result ${\rm{Pr}}({h_1} \vee {h_2}|e) \gt Pr({h_1}|e)$ , and thus this account tells us to favor the logically weaker (and possibly maximally uninformative) explanation. Logically weaker positions inevitably win out in terms of the above inequality and so are favored by IBE if we use posterior probabilities to gauge explanatory goodness. This makes logically weaker explanations sure winners, precluding us from ever inferring informative explanations.

Relevance measures ignore informational complexity altogether, while posterior probabilities place extreme weight on complexity such that logically stronger explanations are banned from the start. The notion of explanatory goodness at work in IBE when we adjudicate between explanatory options at different levels of logical strength must accordingly strike a balance between these options. Measure ${\cal E}$ plausibly provides exactly such a balance:

$${\cal E}\left( {e,h} \right) = {\rm{log}}\left( {{{{\rm{Pr}}(e|h){\rm{Pr}}{{(h)}^{1/2}}} \over {{\rm{Pr}}\left( e \right)}}} \right).$$

Indeed, Good (1968, 130) originally developed and defended this measure because it “gives equal weights” to explanatory relevance and informational simplicity (the “avoidance of clutter”).

Note that ${\cal E}$ would amount simply to the logarithm of the posterior probability, but for the fact that the factor penalizing for complexity, ${\rm{Pr}}\left( h \right)$ , is given less weight—being raised to the power $1/2$ instead of $1$ . This is appropriate since, for IBE purposes, posterior probability enforces a problematically extreme such penalty. We can also think of ${\cal E}$ as being equivalent to the relevance measure ${r_{{\rm{GM}}}}$ but for the fact that ${\rm{Pr}}\left( h \right)$ is given non-zero weight. This too is appropriate, since ${r_{{\rm{GM}}}}$ and all other relevance measures fail for our purposes because they do not penalize for complexity.¹

4. IBE’s critics revisited

This section reconsiders the objections to IBE summarized earlier with our interpretation of “best explanation” and ${\cal E}$ in hand. Salmon objects that IBE’s Uniqueness claim prohibits reasoners from inferring multiple explanatory hypotheses in cases of explanatory multiplicity. In response, philosophers give up Uniqueness, greatly limiting IBE’s domain of applicability to cases in which individual hypotheses compete. Our novel interpretation of “best explanation” instead allows us to respond by showing that IBE in its traditional form (Uniqueness and all) can recommend inference to conjunctions of hypotheses in the salient cases. Once we interpret IBE as allowing inferences to stances that may have the logical structure of any Boolean combination of the individual hypotheses on offer, ${\cal E}$ reveals that it is indeed possible for logically stronger, less probable stances to be explanatorily superior. ²

Consider a case in which ${h_1}$ and ${h_2}$ each offer potential explanations of $e$ . The conjunctive stance ${h_1}{\rm{\;}}\& \;{h_2}$ is explanatorily superior to both individual hypotheses under the following condition (Glass and Schupbach 2023a,b):

(1) $${\rm{log}}\left[ {{{{\rm{Pr}}(e|{h_1}{\rm{\;}}\& \;{h_2})} \over {{\rm{Pr}}(e|{h_1})}}} \right] \gt {\rm log}\left[ {{1 \over {{\rm{Pr}}({h_2}|e{\rm{\;}}\& \;{h_1})}}} \right].$$

The left-hand side of (1) is ${r_{{\rm{GM}}}}(e,{h_2}|{h_1})$ ; in words, this is the explanatory relevance that ${h_2}$ has with respect to $e$ in light of already accepting ${h_1}$ . The right-hand side explicates the degree to which ${h_2}$ is penalized for adding more information to our explanation in light of already accepting ${h_1}$ and $e$ . Condition (1) thus clarifies that stronger explanations are to be preferred whenever the explanatory relevance to $e$ that would be added by inferring ${h_2}$ in addition to ${h_1}$ outweighs the price in informational complexity incurred by this move. The point at which this just ceases to be true is the tipping point at which the conclusion is exactly as informative as it should be, since any further logical strengthening of the inferred explanation would not be worth the price in increased complexity (decreased probability).

This is the situation, for example, in diagnoses of PD for which the evidence is sufficiently rich to be satisfactorily accounted for only with a multicausal explanation. Such a conclusion would inevitably be less probable than a less detailed, more agnostic explanation. But that lower probability is worth the gains in explanatory relevance and informativeness in accounting for the evidence.

The bad lot and disjunction objections both argue that Uniqueness forces reasoners to infer too much. In cases where none of our individual hypotheses are explanatorily good (or good enough), the complaint is that Uniqueness still rashly compels us to infer one of these. IBE’s defenders offer the ad hoc response that IBE only applies when there is no such concern—i.e., when the best is good enough, when the lot is not entirely bad, etc. An alternative response made possible by our reinterpretation of “best explanation” instead shows that IBE in its traditional form (Uniqueness and all) does not require us to infer explanatorily poor individual hypotheses.

Consider the main case put forward by McCain and Poston (2019, 2) in presenting their objection:

Let ${h_1}$ be the hypothesis that a fair coin has been chosen, i.e., ${\rm{Pr}}({heads}|{h_1}) = 1/2$ ; ${h_2}$ is the hypothesis that a coin with a strong bias for heads is chosen, e.g., ${\rm{Pr}}({heads}|{h_2}) = 3/4$ ; and ${h_3}$ is the hypothesis that a coin with a strong bias against heads is chosen, e.g., ${\rm{Pr}}({heads}|{h_3}) = 1/4$ . There are only three coins to be chosen, and each has the same probability of being chosen— ${\rm{Pr}}\left( {{h_1}} \right) = {\rm{Pr}}\left( {{h_2}} \right) = {\rm{Pr}}\left( {{h_3}} \right) = 1/3$ . The results of the flip of each coin are independent of previous flips. A coin is selected at random and flipped four times. The results are $e$ : $\langle H,T,T,H\rangle $ .

The most explanatory of the individual hypotheses apropos $e$ seems clearly to be ${h_1}$ ; however, ${\rm{Pr}}({h_1}|e) \approx .47 \lt .5$ . That is, considering inferences only to individual hypotheses, ${h_1}$ provides the intuitively best explanation of $e$ while nonetheless being more likely false than true in light of $e$ . It is this fact that McCain and Poston insist makes ${h_1}$ not explanatorily good enough for inference. They go on to endorse an ad hoc response, simply requiring that IBE only applies in cases where the best explanation is more probable than not.

What becomes of this same case if we adopt our interpretation of “best explanation” along with ${\cal E}$ as our explication of explanatory goodness? Doing so multiplies the explanatory conclusions available for inference, our lot of candidate explanations now including the individual hypotheses ${h_1}$ , ${h_2}$ , and ${h_3}$ along with their various Boolean combinations. Importantly, these new alternatives include agnostic stances like ${h_2} \vee {h_3}$ and ${h_1} \vee {h_2} \vee {h_3}$ . Taking into account these possible stances leads to a different inference than in McCain and Poston’s discussion. First, ${\cal E}$ assesses the most explanatory hypothesis ${h_1}$ as having negative explanatory value:

\begin{align*}{\cal E}\left( e{,{h_1}} \right) & = {\rm{log}}\left( {{{{\rm{Pr}}(e|{h_1}){\rm{Pr}}{{({h_1})}^{1/2}}} \over {{\rm{Pr}}\left( e \right)}}} \right) \\ &= {\rm{log}}\left( {{{{{0.5}^4} \cdot 1/{3^{1/2}}} \over {1/3 \cdot {{0.25}^2} \cdot {{0.75}^2} + 1/3 \cdot {{0.5}^4} + 1/3 \cdot {{0.75}^2} \cdot {{0.25}^2}}}} \right) = - .089. \end{align*}

Nonetheless, ${h_1}$ unsurprisingly performs far better than the other hypotheses: ${\cal E}\left( {e,{h_2}} \right) = {\cal E}\left( {e,{h_3}} \right) = - .339$ . Interestingly, ${h_1}$ scores better even than the stance that remains agnostic between the other two individual hypotheses: ${\cal E}\left( {e,{h_2} \vee {h_3}} \right) = - .188$ . However, there remains one (and only one) alternative stance that has non-negative explanatory value: ${\cal E}\left( {e,{h_1} \vee {h_2} \vee {h_3}} \right) = 0$ . The “best explanation” then according to ${\cal E}$ is the maximally agnostic ${h_1} \vee {h_2} \vee {h_3}$ , acknowledging that the evidence is not sufficiently rich to warrant any informative explanatory conclusion whatever.

Far from being more likely false than true, this stance has unit probability, ${\rm{Pr}}({h_1} \vee {h_2} \vee {h_3}|e) = 1$ . However, ${\cal E}$ also rightly reveals that this maximally uninformative conclusion is the best of bad explanatory options, having no positive explanatory value. In fact, it is easy to show for all cases that a fully agnostic stance between alternatives necessarily has zero explanatory value over any explanandum according to ${\cal E}$ . This is completely appropriate, lending formal backing to the common observation that tautologies cannot explain anything (neither an explanandum nor its negation).

Since ${\rm{Pr}}({h_1} \vee {h_2} \vee {h_3}|e) \gt .5$ , this case no longer serves McCain and Poston’s purposes as a counterexample. But the fact that this stance can at once be so probable and explanatorily impotent starts to suggest a deeper issue with their objection. Specifically, this result reveals an important disconnect between a stance’s explanatory value and its probability; ${h_1} \vee {h_2} \vee {h_3}$ couldn’t score better in terms of probability, but it couldn’t be more worthless in terms of explanatory value. This disconnect is also suggested in our response to Salmon’s objection, in which we observed that probability can rightly be sacrificed in explanatory inference for the sake of greater explanatory relevance. The upshot is that there is no simple connection between explanatory value (the determining factor in explanatory inference) and probability. If this is right, then, contrary to the account of IBE that McCain and Poston end up defending, we cannot simply identify “sufficient explanatory goodness” with probability exceeding $.5$ —or any other probabilistic threshold.

Through the lens of our alternative interpretation of IBE, McCain and Poston’s coin case—far from constituting a counterexample—helpfully clarifies a rational response to cases involving “bad lots” of explanatory hypotheses. When none of the individual hypotheses in our lot of potential explanations have any positive explanatory value with respect to the explanandum, agnostic stances between such hypotheses become explanatorily appealing. In the worst case, when we are truly working with a purely bad lot of such hypotheses, we can still do no worse than acknowledge that we are in the weakest of explanatory positions by concluding the disjunction of all of these. ${\cal E}$ provides the guide here, either directing us to take the fully agnostic, explanatorily vacuous stance or to more informative options that provide us with positive explanatory value when such exist.

Real-life examples like the diagnosis of a particular patient with PD will sometimes fit this description. Relative to the evidence of such a case $e$ , a disjunction of causal hypotheses (e.g., ${h_1} \vee {h_2}$ ) is explanatorily better than the individual etiologies on offer ( ${h_1}$ and ${h_2}$ ) under the following condition (Glass and Schupbach 2023a,b):

(2) $${\rm{log}}\left[ {{1 \over {{\rm{Pr}}({h_1}|\left( {{h_1} \vee {h_2}} \right){\rm{\;}}\& \;e)}}} \right] \gt {\rm log}\left[ {{{{\rm{Pr}}(e|{h_1}{\rm{\;}}\& \;\left( {{h_1} \vee {h_2}} \right))} \over {{\rm{Pr}}(e|{h_1} \vee {h_2})}}} \right].$$

The left-hand side of (2) explicates the degree to which a commitment specifically to ${h_1}$ would be penalized for adding more information to our explanation in light of already accepting the disjunctive stance ${h_1} \vee {h_2}$ along with $e$ . The right-hand side is ${r_{{\rm{GM}}}}(e,{h_1}|{h_1} \vee {h_2})$ —i.e., the explanatory relevance that committing to ${h_1}$ would gain us with respect to $e$ compared to merely inferring ${h_1} \vee {h_2}$ . This condition thus makes precise the sensible idea that we should opt for more agnostic, weaker explanations so long as the gains in explanatory relevance we could acquire by strengthening our conclusions would not be worth the cost incurred by inferring such an inevitably less probable, more specific conclusion.

Other interesting results may be observed by extending McCain and Poston’s example. Let’s say they flip their coin four more times, with the result being the new evidence set $e{\rm{'}}$ : $\langle H,T,T,H,H,T,H,T\rangle $ . In this case (as in any case), the maximally agnostic ${h_1} \vee {h_2} \vee {h_3}$ continues to have zero explanatory value. However, some of the alternative stances now have positive explanatory goodness. Most notably, the best explanation in this case is provided by ${h_1}$ with ${\cal E}\left( {e{\rm{'}},{h_1}} \right) = 0.025$ . By contrast, ${h_2}$ , ${h_3}$ , and the agnostic stance ${h_2} \vee {h_3}$ all appropriately score worse relative to $e{\rm{'}}$ , with ${\cal E}\left( {e,{h_2}} \right) = {\cal E}\left( {e,{h_3}} \right) = - .474$ and ${\cal E}\left( {e{\rm{'}},{h_2} \vee {h_3}} \right) = - .323$ . Importantly, this extended case also provides no counterexample à la McCain and Poston, as ${h_1}$ is more likely true than false relative to $e{\rm{'}}$ : ${\rm{Pr}}({h_1}|e{\rm{'}}) = .612$ .

The natural follow-up question to this last point is whether our approach provides us with a general way around McCain and Poston’s alleged counterexamples. Is the “best explanation” in our sense always more probably true than false? In fact, no; there exist cases for which the stance with maximal ${\cal E}$ nonetheless has probability $ \lt .5$ conditional on the explanandum. ³

However, this fact is both unsurprising and unconcerning in light of the work set out in this paper. We have already suggested why when diagnosing the “deeper issue” with McCain and Poston’s objection. Recall the disconnect between a stance’s explanatory value and its probability; there is no straightforward relation between these such that, for example, we can simply assume that the most explanatory stance must be more probable than not. To think otherwise is to give improper weight to simplicity over explanatory informativeness. A blanket requirement that best explanations must rise above a certain probabilistic threshold ignores the explanatory value that more complexity (less probability) can bring. In seeking best explanations, reasoners should sometimes be willing to sacrifice probabilistic assurance that their stance is not false for the sake of endorsing more informative explanatory positions. Rather than forcing a contrived link between explanatory value and probability, the present account provides a principled balance between simplicity and informativeness.