2. IBE’s problem of uncertain evidence

Let us start with a familiar problem of uncertain evidence. Bayesian conditionalization (BC) says that when you learn some evidence $E$ , your new subjective probability in any hypothesis $H$ , ${P_{\rm{n}}}\left( H \right)$ , should equal your previous conditional probability in $H$ given $E$ , i.e., ${P_{\rm{o}}}(H|E)$ . Here, “learning” $E$ means becoming absolutely certain that $E$ is true, i.e., setting ${P_{\rm{n}}}\left( E \right) = 1$ .Footnote ² This implication of BC is generally considered implausible in and of itself, for surely we should distinguish between many evidential propositions and patently self-evident truths such as $1 + 1 = 2$ . Moreover, requiring certainty of any empirical proposition is highly problematic within the Bayesian framework, where such extreme probabilities cannot ever be revised through later applications of BC.

A common response to this Bayesian problem of uncertain evidence, influentially suggested by Jeffrey (Reference Jeffrey1965), is to replace BC with a more general updating rule. Jeffrey conditionalization (JC) says that when your subjective probabilities in some partition of evidential propositions ${E_1}, \ldots, {E_n}$ changes in any way, your new probability for $H$ should be a weighted sum of your previous conditional probabilities of $H$ given each ${E_i}$ , where each weight is your new probability in ${E_i}$ , i.e., ${P_{\rm{n}}}\left( H \right) = \mathop \sum \nolimits_{i \le n} {P_{\rm{n}}}\left( {{E_i}} \right){P_{\rm{o}}}(H|{E_i})$ . JC nicely allows for each evidential proposition in the partition to be uncertain, since we can have ${P_{\rm{n}}}\left( {{E_i}} \right) \lt 1$ for all $i$ , while still giving us BC as a special limiting case in which some evidential proposition ${E_i}$ in the partition is assigned probability 1. Thus, while JC may itself not be entirely unproblematic (see, e.g., Weisberg Reference Weisberg2009a; Trpin Reference Trpin2020), it is safe to say that the possibility of obtaining uncertain evidence has motivated a profound revision of the Bayesian framework.

By contrast, standard accounts of IBE have not yet accommodated this possibility of obtaining uncertain evidence. The only extant discussion of uncertain evidence with reference to “IBE” appears in Trpin and Pellert’s (Reference Trpin and Pellert2019) account of how JC may be combined with the idea, due originally to van Fraassen (Reference van Fraassen1989) and later developed by Douven (Reference Douven2022), that the hypotheses which provide better explanations of some evidence should be given bonus probability points as one updates on that evidence. Trpin and Pellert show how this can be done in an elegant way by generalizing Douven’s IBE-inspired updating rule, EXPL, which is itself a generalization of BC, and argue that this new probabilistic updating rule is preferable to JC in a number of simulations. What the literature does not contain, however, is any hint at how uncertain evidence may be handled in standard accounts of IBE, i.e., those according to which IBE is a form of defeasible non-deductive inference (rather than a mere variation on a probabilistic updating rule such as BC).

This is an important lacuna in the literature, for—as I’ll now argue—such standard accounts of IBE face a structurally similar problem concerning uncertain evidence, which in turn calls for a correspondingly profound revision of these accounts. As before, the problem stems from the fact that some or all of one’s total evidence $\mathbb{E}$ may be uncertain for various reasons. In many cases, such evidential uncertainty undermines the inferrability of a given hypothesis $H$ that provides the best explanation of $\mathbb{E}$ , for $H$ may only provide a better explanation than an alternative $H{\rm{'}}$ on the assumption that this uncertain evidence is indeed accurate. However, as we shall see, standard accounts of IBE as involving an inference to $H$ from $\mathbb{E}$ are not even capable of representing such uncertainty about the evidence—let alone its implication for the inferrability of a given hypothesis.

For example, consider an everyday case discussed by van Fraassen (Reference van Fraassen1980, 19–20): “I hear scratching in the wall, the patter of little feet at midnight, my cheese disappears—and I infer that a mouse has come to live with me.” Setting aside van Fraassen’s (and others’) concerns about the cogency of IBE, it seems prima facie reasonable to make the inference from ( ${E_{\rm{s}}}$ ) scratching in the wall, ( ${E_{\rm{p}}}$ ) pattering of little feet, and ( ${E_{\rm{d}}}$ ) disappearance of cheese, to the hypothesis ( ${H_{\rm{m}}}$ ) that a mouse has come to live with me. After all, ${H_{\rm{m}}}$ arguably provides the best explanation of ${E_{\rm{s}}}$ , ${E_{\rm{p}}}$ , and ${E_{\rm{d}}}$ taken together. So far so good. But what if you are not so sure about ${E_{\rm{d}}}$ , i.e., that the cheese actually disappeared, e.g., because you don’t quite remember whether you ate the cheese yourself the night before? Surely, such uncertainty about the evidence should be taken into account in considering whether, or the extent to which, ${H_{\rm{m}}}$ can be inferred from ${E_{\rm{s}}}$ , ${E_{\rm{p}}}$ , and ${E_{\rm{d}}}$ in the situation in question.

This is especially apparent if there is another available hypothesis that might provide an equally good, or indeed better, explanation of the remaining evidence, i.e., the evidence of which you are certain. For example, it is not hard to imagine a case in which the scratching and the pattering, i.e., ${E_{\rm{s}}}$ and ${E_{\rm{p}}}$ , would be best explained by ( ${H_{\rm{s}}}$ ) the presence of a squirrel outside your house (rather than a mouse inside it). In that case, since you are uncertain about the crucial piece of evidence ${E_{\rm{d}}}$ that discriminates between this alternative hypothesis ${H_{\rm{s}}}$ and the original mouse hypothesis ${H_{\rm{m}}}$ , it seems at best dubious to infer ${H_{\rm{m}}}$ —even though there is a clear sense in which it provides the best explanation of the entirety of your evidence (including evidence about which you are uncertain). Indeed, it would seem to be equally dubious to infer the alternative hypothesis ${H_{\rm{s}}}$ in such a situation, even though it provides the best explanation of the evidence of which you are certain. After all, you are not certain that ${E_{\rm{d}}}$ is false either—rather, you are uncertain whether ${E_{\rm{d}}}$ is true or false—and if ${E_{\rm{d}}}$ were true then ${H_{\rm{s}}}$ would arguably not be the best explanation of your evidence.Footnote ³

In sum, then, standard accounts of IBE, on which one may infer a hypothesis $H$ just in case it would provide the best explanation of one’s total evidence $\mathbb{E}$ , are simply unsuited to handle cases of uncertain evidence. The underlying reason for this should be clear at this point. On these accounts, a given piece of potential evidence ${E_i}$ is either included in, or excluded from, the set of total evidence $\mathbb{E}$ of which a given hypothesis $H$ must provide the best explanation if $H$ is to be inferrable by IBE. If ${E_i}$ is uncertain, it occupies a sort of in-between space, neither clearly included nor clearly excluded from $\mathbb{E}$ . It is unclear at present how this in-between space could or should be conceptualized in accounts of IBE; nor is it clear what sort of claims IBE should allow one to infer from evidence that partly occupies such an in-between space.

One possible response to this problem of uncertain evidence is to jettison standard accounts of IBE altogether, e.g., by replacing them with a purely probabilistic framework in which hypotheses are never really inferred at all. Before we take such drastic measures, however, let us consider whether it may be possible to modify—or, rather, generalize—standard accounts of IBE so as to handle the problem in a more accommodating manner.

3. Evidentially robust IBE

The key idea behind the solution I wish to explore is that, in cases of evidential uncertainty, it may still be that the same hypothesis provides the best explanation regardless of whether uncertain pieces of evidence are included in the set of evidence to be explained. To put the point differently, it may be that a given hypothesis’s status as the best explanation is robust across all the different possibilities that are left open by one’s uncertainty about the evidence; in that case, the evidential uncertainty should not matter to the inferrability of $H$ .

To flesh out this idea, let us model an agent’s total evidence at a given time as a set of evidential propositions, $\mathbb{E} = \left\{ {{E_1}, {\rm\ldots}, {E_m}} \right\}$ . Given some standard for what makes an evidential proposition certain (or, equivalently, uncertain), there is a subset ${\mathbb{E}_{\rm{c}}} \subseteq \mathbb{E}$ which contains only evidential propositions of which one is certain. Now, let’s say that $\mathbb{E}_k^{\rm{o}}$ is an open evidential combination if and only if ${\mathbb{E}_{\rm{c}}} \subseteq \mathbb{E}_k^{\rm{o}} \subseteq \mathbb{E}$ . So an open evidential combination has every certain evidential proposition as a member, and none, some, or all of the uncertain evidential propositions. For example, in our variation on van Fraassen’s mouse case, supposing that ${E_{\rm{s}}}$ and ${E_{\rm{p}}}$ are certain but ${E_{\rm{d}}}$ is uncertain, the open evidential combinations are simply $\left\{ {{E_{\rm{s}}},{E_{\rm{p}}},{E_{\rm{d}}}} \right\}$ and $\left\{ {{E_{\rm{s}}},{E_{\rm{p}}}} \right\}$ . (Had ${E_{\rm{p}}}$ also been uncertain, the open evidential combinations would also have included $\left\{ {{E_{\rm{s}}},{E_{\rm{d}}}} \right\}$ and $\left\{ {{E_{\rm{s}}}} \right\}$ .)

With this notion of open evidential combinations in hand, my suggestion is—to a first approximation—that IBE should be modified to state that one may infer a hypothesis if it provides the best explanation of any open evidential combination. More precisely:

• ${\bf{IB}}{{\bf{E}}_{{\bf{ER}}}}$ * : If $H$ would provide a better explanation of any open evidential combination $\mathbb{E}_k^{\rm{o}}$ than any other available hypothesis, then one may defeasibly infer $H$

The thought here is that regardless of which of the open evidential combinations turns out to be accurate, the inferred hypothesis would provide the best explanation of it. In the limiting case where there is no uncertainty about any evidence, i.e., $\mathbb{E} = {\mathbb{E}_{\rm{c}}}$ , the only open evidential combination would be the total evidence itself, i.e., $\mathbb{E}_k^{\rm{o}} = \mathbb{E}$ . In that special case, ${\rm{IB}}{{\rm{E}}_{{\rm{ER}}}}$ * reduces to ${\rm{IB}}{{\rm{E}}_{{\rm{ST}}}}$ . In this sense, ${\rm{IB}}{{\rm{E}}_{{\rm{ER}}}}$ * is a generalization of standard accounts of IBE.

However, ${\rm{IB}}{{\rm{E}}_{{\rm{ER}}}}$ * is still too restrictive in some cases of evidential uncertainty. Consider cases in which there is no single hypothesis that provides the best explanation of every open evidential combination. Indeed, our variation of van Fraassen’s mouse inference is a case in point: ${H_{\rm{m}}}$ provides the best explanation of $\left\{ {{E_{\rm{s}}},{E_{\rm{p}}},{E_{\rm{d}}}} \right\}$ , while ${H_{\rm{s}}}$ provides the best explanation of $\left\{ {{E_{\rm{s}}},{E_{\rm{p}}}} \right\}$ . So there would be no hypothesis $H$ that could be inferred via ${\rm{IB}}{{\rm{E}}_{{\rm{ER}}}}$ *. However, note that in such cases there may nevertheless be some other claim that is robust across these explanatory hypotheses, in that it is implied by each one of them. For example, ${H_{\rm{m}}}$ and ${H_{\rm{s}}}$ both imply ( ${H_{\rm{r}}}$ ) that there is a rodent in or near your house. This logically weaker claim, implied by each of the best explanations of the two open evidential combinations, should surely be inferrable by our generalization of ${\rm{IB}}{{\rm{E}}_{{\rm{ST}}}}$ .

To accommodate this thought, we may modify ${\rm{IB}}{{\rm{E}}_{{\rm{ER}}}}$ * to say that one may infer a claim $C$ if it is implied by each hypothesis that provides the best explanation of some open evidential combination:

• ${\bf{IB}}{{\bf{E}}_{{\bf{ER}}}}$ : If $C$ is implied by each hypothesis ${H_i}$ that provides a better explanation than any other available hypothesis of some open evidential combination $\mathbb{E}_k^{\rm{o}}$ , then one may defeasibly infer $C$ .

In the limiting case where there is no uncertainty about any evidence ( $\mathbb{E} = {\mathbb{E}_{\rm{c}}}$ ), ${\rm{IB}}{{\rm{E}}_{{\rm{ER}}}}$ reduces to the view that one may infer a claim only if it is implied by the hypothesis that provides the best explanation of one’s total evidence. Since any hypothesis implies itself, this is itself a generalization of ${\rm{IB}}{{\rm{E}}_{{\rm{ST}}}}$ , albeit a very natural one (see Lipton Reference Lipton2004, 63–4; Dellsén Reference Dellsén2017, 20–21). ${\rm{IB}}{{\rm{E}}_{{\rm{ER}}}}$ clearly delivers the desired verdict in the variation of van Fraassen’s mouse case in which ${E_{\rm{d}}}$ is uncertain, viz. that ${H_{\rm{r}}}$ may be inferred (see Figure 1).

Figure 1. ${H_{\rm{r}}}$ is implied by both ${H_{\rm{m}}}$ and ${H_{\rm{s}}}$ , each of which provides the best available explanation of an open evidential combination.

Indeed, it also delivers plausible verdicts in further variations on the case in which, say, ${E_{\rm{p}}}$ is also uncertain. In that case, the open evidential combinations would include, in addition to $\left\{ {{E_{\rm{s}}},{E_{\rm{p}}},{E_{\rm{d}}}} \right\}$ and $\left\{ {{E_{\rm{s}}},{E_{\rm{p}}}} \right\}$ , also $\left\{ {{E_{\rm{s}}},{E_{\rm{d}}}} \right\}$ and $\left\{ {{E_{\rm{s}}}} \right\}$ . Now suppose—as seems plausible—that ${H_{\rm{m}}}$ and ${H_{\rm{s}}}$ still provide the best explanations of these open evidential combinations, e.g., because ${H_{\rm{m}}}$ best explains $\left\{ {{E_{\rm{s}}},{E_{\rm{d}}}} \right\}$ and ${H_{\rm{s}}}$ best explains $\left\{ {{E_{\rm{s}}}} \right\}$ . In that case, ${H_{\rm{r}}}$ would still be inferrable via ${\rm{IB}}{{\rm{E}}_{{\rm{ER}}}}$ (see Figure 2).

Figure 2. ${H_{\rm{r}}}$ is implied by both ${H_{\rm{m}}}$ and ${H_{\rm{s}}}$ , each of which provides the best available explanation of two open evidential combinations.

With that said, things would be different if some entirely different hypothesis, e.g., the hypothesis ( ${H_{\rm{p}}}$ ) that you are being pranked by a member of your family, provided the best explanation of one of these open evidential combinations, e.g., $\left\{ {{E_{\rm{s}}}} \right\}$ . In that case, ${\rm{IB}}{{\rm{E}}_{{\rm{ER}}}}$ would not license an inference to either ${H_{\rm{r}}}$ or ${H_{\rm{p}}}$ , although it would allow you to infer their disjunction ${H_{\rm{r}}} \vee {H_{\rm{p}}}$ , which of course is implied by ${H_{\rm{m}}}$ , ${H_{\rm{s}}}$ , and ${H_{\rm{p}}}$ . As this illustrates, ${\rm{IB}}{{\rm{E}}_{{\rm{ER}}}}$ will sometimes only license inferences to quite modest conclusions, such as disjunctions of genuinely explanatory hypotheses, especially when one is uncertain about several evidential propositions. Is this a problem for ${\rm{IB}}{{\rm{E}}_{{\rm{ER}}}}$ ? I don’t think so. On the contrary, when there is more uncertainty about the evidential propositions from which one is inferring, it seems entirely appropriate to hold off on making ambitious inferences unless and until the evidential situation changes, either by one’s becoming more certain about the previously uncertain evidence, or—which is presumably more common—by obtaining other evidence that some of the relevant hypotheses cannot plausibly explain at all. For example, supposing that ${H_{\rm{p}}}$ does indeed provide the best explanation of $\left\{ {{E_{\rm{s}}}} \right\}$ , and that this is an open evidential combination, it does seem reasonable not to infer ${H_{\rm{r}}}$ unless and until you have gathered further evidence to rule out ${H_{\rm{p}}}$ , e.g., by interrogating your family members. So while ${\rm{IB}}{{\rm{E}}_{{\rm{ER}}}}$ is a quite cautious inference rule in situations of evidential uncertainty, it is arguably cautious in just the right way.

4. Einstein on the Mt. Wilson experiments

Thus far, I have motivated ${\rm{IB}}{{\rm{E}}_{{\rm{ER}}}}$ with variations on a case drawn from everyday life. But does ${\rm{IB}}{{\rm{E}}_{{\rm{ER}}}}$ also help us understand how hypotheses are inferred from uncertain evidence in scientific practice? In this section, I briefly discuss one case which suggests that it does—although of course further work is needed to evaluate this in a more systematic fashion.

Einstein’s special theory of relativity famously discards with the luminiferous ether posited by classical theories of electromagnetism. In so doing, it elegantly explains the famous result, due originally to Michelson and Morley (Reference Michelson and Morley1887), that no effects of an “ether drift”—i.e., movement of the Earth relative to the ether—were observed in increasingly meticulous experiments. Although special relativity also provided rather compelling explanations of other phenomena, such as the speed of light in water flowing in and against the light’s direction of travel (Fizeau experiments), it is safe to say that ether drift experiments were extremely important to establish the plausibility of special relativity in its early years, as Einstein acknowledged (see, e.g., Fölsing Reference Fölsing1998, 219). Moreover, the fact that special relativity elegantly explains the lack of an observed ether drift is often taken to show that Einstein used something like IBE in arguing for his theory (e.g., Douven Reference Douven2002; Janssen Reference Janssen2002).

In the 1920s, however, the significance of earlier ether drift experiments were cast into doubt by several new experiments performed at a much greater altitude than before, viz. at the top of Mt. Wilson near Pasadena (Hentschel Reference Hentschel1992). These experimental efforts were led by the highly regarded experimentalist Dayton Miller, who had not only collaborated with Morley on influential replications of the original ether drift experiment in 1905, but was also the president of the American Physical Society. In a series of papers, Miller reported small but positive interference effects of the sort Michelson, Morley, and himself had failed to observe at lesser altitudes (e.g., Miller Reference Miller1925; Miller Reference Miller1933). These experiments seemed to lend support to versions of classical electrodynamics on which the effects of ether drift could only really be observed at very high altitudes.

Miller’s results caused a major upheaval in the scientific community at the time. Einstein himself remained relatively skeptical, however, expressing doubts both privately and publicly about the reliability of Miller’s results (Hentschel Reference Hentschel1992, 603–608; see also Norton Reference Norton2024, 27–28). But Einstein nevertheless took the apparent results sufficiently seriously to visit Miller specifically to discuss them in person. It seems, then, that Einstein was uncertain about Miller’s results from Mt. Wilson in a way that he was not regarding various other experiments relevant to his theory, e.g., the Michelson and Morley and Fizeau experiments. Accordingly, we could say—imposing the terminology of section 2—that for Einstein there were at least two open evidential combinations relevant to special relativity at the time: one consisting of all the empirical results available to Einstein at the time, and one consisting of all those results except for Miller’s results from Mt. Wilson.

Einstein was convinced that it would not be possible to modify special relativity so as to provide a plausible explanation of ether drift. The reason for this was Einstein’s insistence that ether drift would contradict the constancy of the speed of light, which in turn forms one of the two fundamental principles of his theory. The internal coherence of special relativity would therefore be irredeemably lost by any accommodation of Miller’s results within the theory. Accordingly, while special relativity remains the best explanation of the open evidential combination in which Miller’s results are excluded, Einstein seems to have ruled out an appropriately modified version of special relativity as the best explanation of the more inclusive open evidential combination in which Miller’s results are included. Since there was thus, by Einstein’s lights, no weaker version of special relativity implied by the best explanations of both evidential combinations, no such version of special relativity could be inferred via ${\rm{IB}}{{\rm{E}}_{{\rm{ER}}}}$ while these evidential combinations are both open. This accords with Einstein’s insistence that “[i]f the results of the Miller experiments were to be confirmed, then relativity theory could not be maintained” (Einstein Reference Einstein1926; quoted in, and translated by, Hentschel Reference Hentschel1992, 606).

Of course, this situation did not last forever, and ${\rm{IB}}{{\rm{E}}_{{\rm{ER}}}}$ also sheds light on how the situation was resolved. Subsequent analyses of Miller’s results suggested that they were due to temperature changes that caused unequal expansion of certain rods in the experimental setup, leading to an unexpected experimental error. Although Einstein seems to have suspected that something like this might explain Miller’s results, there was no way to know this for certain until several years later (Shankland et al. Reference Shankland, McCluskey, Leone and Kuerti1955). At that point, however, ${\rm{IB}}{{\rm{E}}_{{\rm{ER}}}}$ suggests that special relativity could once again be defeasibly inferred, since the evidential uncertainty about Miller’s experiments had been resolved in such a way that alternatives to special relativity would no longer provide best explanations of any open evidential combination. Special relativity could, then, reassume its position as the default theory in its domain.