evidence, explanation, and the empirical status of scientific realism

Evidence, Explanation, and the Empirical Status ofScientific Realism

Igor DouvenDepartment of Philosophy, Erasmus University Rotterdam

[email protected]

Abstract

There is good reason to believe that, if it can be decided at all, the realism debate mustbe decided on empirical grounds. In an earlier paper, the author has proposed a methodfor empirically deciding the debate that relies on Glymour’s bootstrapping account ofconfirmation. The present paper proposes a variant method that can make do with the lesscontroversial Bayesian confirmation theory.

1. Introduction. The scientific realism debate concerns the extent of our epistemic accessto the world. According to antirealists, this access is limited to what is observable, that is,what can be seen by the unaided human eye. Realists deny this, and most even believe thatmodern-day science has already to a large extent succeeded in charting the unobservable. Itwill not have been missed by anyone interested in this debate that, over the past twenty yearsor so, its tone has become considerably less heated, and that claims made by either side tendto be less imperialist now than they were in the beginning of the 1980s. For instance, vanFraassen, the major representant of the antirealist position, in some passages of his [1989]seems to be granting that it may be rational to endorse realism, even though he earlier hadalleged the position to involve “empty strutting and posturing” for which he could only “ex-press disdain” (van Fraassen [1985:255]). In the other camp, Kukla [1998, Ch. 12] even moreoutspokenly suggests that realism and antirealism are both rational positions.

If Kukla (ibid.) is right, the parties had to end up in this apparent state of mutual toler-ance. According to him, “the differences between realists and antirealists are irreconcilable”(p. 154) because, he thinks, there are “no considerations of fact or logic that can—or should—persuade proponents of either side to switch” (p. 153). Indeed, while I am unaware of anya priori argument to the effect that there can be no a priori argument for either position, thepossibility of settling the question whether we have epistemic access to the unobservablewithout having any particular knowledge about the world we inhabit, seems remote. More-over, the fact that no one so far has been able to provide an a priori argument for either realismor antirealism provides itself something of an a posteriori justification for the thought that thedebate is not decidable on a priori grounds. And at least prima facie the prospects for ana posteriori resolution of the debate seem hardly less bleak, given that realists and antirealistsdisagree over two of the—or even the two—most fundamental questions pertaining to anykind of empirical research.1

1By an a posteriori resolution I mean one that empirically settles the debate in a way that is acceptable to bothparties. The disagreement over the two fundamental questions referred to is, of course, not even a prima facieimpediment to the more modest task of settling the debate in a way acceptable to one party only (and Psillos [1999,Ch. 4] may be right that this more modest task is one still worth undertaking).

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One is the question what the range of accessible evidence is. For example, micrographsthat a realist may describe as being “pretty nice pictures of DNA” (as Giere does, in Callebaut(ed.) [1993:171]), and thus may regard as providing direct proof of our access to the unob-servable, the antirealist will tend to conceive of as simply being “new phenomena that mustalso be saved by our theories, and suffice to refute theories to be discarded” (van Fraassen[2001:154 f]). For the antirealist the particular arrangements of dots on the micrographs mayor may not show what the realist claims they show—all that can be ascertained is whetherthey accord with what our theories imply about such arrangements, or so she holds.

The other is the question what the correct principles of scientific reasoning are: grantedwhat our evidence is, what may we infer from it? Most pertinent in this respect is the contro-versy over the methodological status of explanatory connections between bodies of evidenceand hypotheses, and in particular over the legitimacy of Inference to the Best Explanation(IBE), a rule realists typically do and antirealists typically don’t accept.2 This leads an-tirealists to reject the well-known “miracle argument,” among others, which is an empiricalargument for realism, but one that crucially hinges on IBE.3

In the present paper I want to argue that, while the difficulties that face an empiricalapproach to the realism debate are not to be discounted, they are not insurmountable either.The strategy to be presented below for resolving the debate is a variant of one to be found inDouven [2002a], [2003]. A major advantage of the new over the old strategy is that, whereasthe latter heavily relies on Glymour’s [1980] account of bootstrap confirmation, the formercan make do with the more widely held Bayesian confirmation theory. However, at leastprima facie the new strategy does come with a cost, too: While the old strategy accepted theantirealist strictures on evidence and the role of explanation, the new one assumes answersto the aforementioned questions that diverge from both the standard realist and the standardantirealist answers. As I shall argue, though, this is, in effect, not really a cost at all, for these“non-standard” answers are more natural and plausible than the standard ones anyhow, andshould be acceptable to realists and antirealists alike.

A Bayesian approach to the problem of scientific realism is not new. In fact, I start bydiscussing a Bayesian strategy for resolving the realism debate proposed by Dorling. As willbe seen, however, this strategy is unsuccessful; it fails to do justice to some of the subtletiesof, in particular, van Fraassen’s brand of antirealism (§ 2). In § 3, I briefly rehearse myearlier bootstrap test for realism which will be recast in a Bayesian format in § 6. In §§ 4and 5, respectively, I explain and motivate the answers to the questions regarding evidenceand explanation which the strategy presupposes.

2. Dorling’s Bayesian Strategy and Why It Fails. Dorling [1992] focuses on local real-ism/antirealism debates, that is, debates concerning the proper epistemic attitude toward aparticular scientific theory. His suggestion is that these can be settled by simple Bayesianconditionalization on the available evidence. He tries to demonstrate this with the aid ofseveral examples, one of which I briefly consider here.

In this example (pp. 366–369), “TR” denotes some particular scientific theory not solelyabout observables, and “TP” denotes the set of observable consequences of TR. Dorling thenconsiders two philosophers (scientists?), one of whom is supposed to be a realist, the otheran antirealist; both are Bayesian learners in the sense that they accommodate new evidence

2See, e.g., van Fraassen [1980], [1989, Ch. 6], Laudan [1981], Fine [1984], Psillos [1996], [1999], and Ladymanet al. [1997] for a sample of arguments pro and con IBE.

3For increasingly refined versions of this argument, see Putnam [1975], Boyd [1981], [1984], [1985], and Psillos[1999, Ch. 4].

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by dint of the rule of conditionalization, that is, the rule according to which on the receipt ofnew evidence one ought to set equal one’s new probability for any hypothesis to one’s oldprobability for that hypothesis conditional on the evidence received. Their distinct epistemicattitudes are supposed to be reflected in the differences in the initial probabilities they assignto TR: the realist assigns it a probability of .6, the antirealist a probability of .2. Since TP isimplied by TR, both assign a conditional probability of 1 to the former given the latter. Theconditional probability of TP given the negation of TR is less straightforward, but Dorlingassumes that the realist and antirealist agree that it is .2.

We are now to suppose that our two philosophers obtain enough evidence for TP’s correct-ness to make them virtually certain of it. What effect would that have on their probabilitiesfor TR? Applying Bayes’ rule, we find that the realist will now assign a probability of (ap-proximately) .9 to TR and that the antirealist will assign it a probability of (approximately) .6.4

A first, rather unsurprising, observation Dorling makes about this result is that both the real-ist and the antirealist are more confident in TR now than they were before they received theevidence for TP. More surprising may be that the antirealist’s new confidence in TR has in-creased to such an extent that she now has more confidence in TR than in its negation, which,in Dorling’s view, means that she has converted to realism with respect to TR (p. 369).

It seems that in this particular case the realism debate has been settled in favor of therealist. While the example is rather abstract, Dorling claims, and to some degree makesplausible, that many local realism debates in the history of science fit it in all relevant respects.

Note that it suffices that some local realist debates have been, or can be, settled in favorof the realist in order to refute antirealism; for in that case we do have epistemic access tothe unobservable, even if only to a rather modest extent perhaps. And although the samewould not suffice as a defense of some of the more ambitious versions of scientific realism,like for instance Boyd’s [1981], [1984], [1985], Devitt’s [1991], and Niiniluoto’s [1999],according to which scientific theories are typically approximately true, a defense of thesestronger versions along the lines indicated by Dorling would appear possible as well; theonly difference between (merely) refuting antirealism and defending an ambitious realismhas to do with the number of local realism debates that must be settled in favor of the realist.

Unfortunately, I do not think Dorling’s strategy is viable in the first place. It may be thatin his example he has managed to correctly represent some antirealists, but, although sheseemingly is among Dorling’s targets (p. 362), it would be a flagrant misrepresentation of themodern, sophisticated antirealist (such as van Fraassen’s constructive empiricist) were we toidentify her with the antirealist Dorling puts on the scene. The latter’s ontological claim “issimply the negation of [the realist’s ontological claim]” (p. 363); for example, the antirealist“reject[s] the existence of atoms” (p. 367). That, however, is not at all what a sophisticatedantirealist does. It will hardly need recounting that her point rather is that, since, first, forevery theory that postulates unobservables there are empirically equivalent rivals (i.e., rivalsthat are not distinguishable from it on the basis of purely observational data) and, second,rational or justified judgements about the truth or falsity of any given theory can only be basedon observational data (this premise is sometimes called Knowledge Empiricism; hereafterKE for short), theory choice is radically underdetermined—which is to say that, unless atheory having non-observable consequences is refuted by the observations, there is no way

4A tacit and not entirely uncontroversial assumption in Dorling’s argument seems to be that being virtually certainof a proposition allows one to conditionalize on it. But, first, I believe that, at least under circumstances which areseemingly realized in Dorling’s example, this practice is unassailable; see Douven [2002b]. Secondly, Howson[2000:198–201] shows that the argument can be modified in such a way that it avoids conditionalizing on TP.

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of knowing its truth-value, and thus also no way of knowing that the theory is false.5 Theantirealist therefore counsels agnosticism, and not disbelief, as the proper epistemic attitudewith respect to what our theories tell us about the unobservable part of the world.

How does this affect Dorling’s argument? This is made perfectly clear in the followingpassage from van Fraassen’s [1989:193 f], which in fact anticipated Dorling’s argument:

Consider ¼ the hypothesis that there are quarks ¼ . The scientist has initially somedegree of belief that this is true. As evidence comes in, that degree can be raised, to anyhigher degree. That is a logical point: if some proposition X has positive probability,conditionalizing on other propositions can enhance the probability of X .

The mistake in this argument is to assume that agnosticism is represented by a lowprobability. That confuses lack or suspension of opinion with opinion of a certain sort.To represent agnosticism, we must take seriously the vagueness of opinion¼ .

Van Fraassen then goes on to argue that a person’s agnosticism concerning a proposition H isto be identified with her assigning a vague or interval-valued probability [0, p] to H, where p isthe probability of H’s least probable observable consequence(s), and that conditionalizing onother propositions can at most have the effect of removing the upper limit on that interval,or, as one might also put it, it can only increase the vagueness of H’s probability. Thusin Dorling’s argument, TP’s becoming certain or nearly certain would for a sophisticatedantirealist à la van Fraassen at most effect an increase of the upper bound on her probabilityfor TR. But that would leave her as agnostic about TR as she was. In particular, she cannot besaid to have converted to realism with regard to TR. It thus appears that Dorling’s defense iswithout any force against a van Fraassen–type antirealism.

Van Fraassen’s way of modelling agnosticism is not the only one nor necessarily the best;see Hájek [1998] and Monton [1998] for some criticisms. While the point against Dorling’sdefense of scientific realism seems to stand on any reasonable construal of agnosticism, theBayesian strategy for resolving the realism debate to be presented in this paper is premisedon a specific way of modelling agnosticism that is more or less suggested in Monton [1998].According to Monton, agnosticism is compatible with assigning sharp probabilities to hy-potheses concerning the unobservable. The antirealist should just

point out that, according to [her position], the sort of evidence it would take to signif-icantly raise the probability of [a theoretical] hypothesis will never be obtained. Anytheory which entails the existence of quarks is empirically equivalent to many other the-ories, many of which do not entail the existence of quarks. Any evidence that is obtainedwill raise the probability of each of the empirically equivalent theories by an equal fac-tor. The [antirealist] scientist’s prior probability distribution across the various theoriesis such that, no matter what evidence is obtained, the probability that quarks exist willcontinue to remain low. (Monton [1998:211])

One way to understand this proposal is as a kind of recommendation to the antirealist toconsider for any theoretical hypothesis to which she assigns a non-zero prior some rival hy-potheses that are empirically equivalent to it, and to assign to at least one of these rivals a priorequal to that assigned to the former hypothesis.6 And patently, if two hypotheses’ H1 and H2being empirically equivalent is so defined as to imply that, for any proposition E reporting the

5This is rough. See for more precise statements of the argument, as well as for discussion of it, van Fraassen[1980], Laudan [1990], Laudan and Leplin [1991], Earman [1993], Leplin [1997], Douven and Horsten [1998],Kukla [1998], [2001], Psillos [1999, Ch. 8], Douven [2000], and Devitt [2002].

6Strictly speaking, all that is required for Monton’s purposes is the following, weaker principle: Given empiri-cally equivalent hypotheses H1,¼, Hn, there is no i Î {1,¼, n} such that p(Hi) > Ú j H j , with 1 6 j 6 n and j ¹ i,and p(×) the agent’s prior probability function.

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result of an observation (or the results of various observations), p(E | H1) = p(E | H2), thenMonton’s recommendation, if followed by the antirealist, ensures that, whatever evidenceshe may obtain, she will never come to assign a probability exceeding .5 to any theory thatis (entirely or partly) about unobservables; for then we have, by Bayes’ theorem, that for allsuch E,

p(H1 | E)p(H2 | E)

=p(H1)p(E | H1)

p(H2)p(E | H2)= 1.

It thus appears that, on this proposal, none of the antirealist’s probabilities need be vague inorder for her to be protected against conversion along the lines Dorling envisaged.

Notice that, given a theoretical hypothesis, the requirement to consider rivals empiricallyequivalent to it can always be met; if in no other way, then by letting one or more “as-ifrivals”—hypotheses which state that the observable part of reality is in all respects as if theformer hypothesis were true but which make at least one assertion incompatible with it—playthe part of empirically equivalent rivals.

In his [1998] reply to Monton’s paper, van Fraassen does not comment on the aboveproposal. I thus take it that he also does not object to it. And for the concerns of the presentpaper that suffices. Still, it is worth stressing that Monton’s way of modelling agnosticismhas a rather important advantage over van Fraassen’s. In particular, the former has at least inone respect much greater flexibility than the latter. Suppose you believe that for each theoryin the realm of fundamental physics there exist a great many empirically equivalent rivals. Bycontrast, you believe that, although theories in population biology have empirically equivalentrivals, too, they have much fewer of them than have theories in fundamental physics. If youhappen to be an antirealist, then (we may assume) you will take the foregoing to be sufficientreasons for remaining agnostic about theories in fundamental physics and, respectively, abouttheories in population biology. Nevertheless, you would seem to make perfectly good senseif you were to deny that you have exactly the same epistemic attitude vis-à-vis the theoriesin fundamental physics as you have vis-à-vis those in population biology. In particular, wewould have no difficulty understanding what you were saying if you said that you are moreskeptical about the former theories than you are about the latter. Still, on van Fraassen’saccount, for any of these theories, whether in fundamental physics or in population biology,your probability will be vague from 0 to that theory’s least probable observable consequence.So it would be possible that you assign exactly the same—vague—probability to a theory infundamental physics as you do to one in population biology even though, in an intuitive sense,you are much more skeptical about the former than about the latter. Clearly, on Monton’saccount the intuitive difference is representable—you can assign a (much) lower probabilityto the former than to the latter—without in the least compromising your intention to remainagnostic about both theories.

3. Bootstrapping Scientific Realism. In Douven [2002a], [2003], I developed a strategyfor testing scientific realism that is less straightforward than Dorling’s but, as far as I could(and can) see, does not beg any antirealist issues. The strategy capitalizes on Hacking’s[1981] well-known argument for the veridicality of our current types of microscopes (a hy-pothesis I will henceforth refer to by “V ”) and on the no less well-known tacking argumentfor the reliability of IBE (the hypothesis that IBE is reliable I will refer to by “R”). What Iargued in that paper was that, although when taken separately Hacking’s argument appearsto beg the question of the range of evidence against the antirealist and the tacking argumentthe question of the methodological status of explanation, they can be combined so as to yielda procedure for testing scientific realism that does not beg any questions against any party

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to the realism debate. Because the Bayesian procedure to be presented in § 6 employs someelements central to, and is meant to be an emendation of, the earlier-proposed procedure, Ihere briefly rehearse the latter.

First consider Hacking’s exceedingly straightforward argument for V. In it, he starts byciting evidence to the effect that very different types of microscope give remarkably similaroutputs when given the same input; he then claims that it is just too much to believe that thisis due to some coincidence; and from this he concludes that it is reasonable to believe thatthe different types of microscope are all veridical.7

As several authors have argued, quite convincingly to my mind, the argument is question-begging for implicitly relying on IBE, according to which the best explanation of some knownfact is typically true.8 For what the argument really must be, according to these authors, isthat we have reason to believe that the different microscopes figuring in Hacking’s argumentare veridical because, if true, that would best explain the fact that they all gave a similaroutput when they were given similar inputs.9 If so, then the antirealist will not accept theargument, given that she does not accept IBE.

The tacking argument for the reliability of IBE basically says that there are good em-pirical grounds for believing that IBE is reliable. We found out—the argument goes—thatthe rule mostly works; very often when we adopted some hypothesis because it provided thebest explanation for some known fact, the hypothesis later turned out to be true. Of course,crucial to the realism debate is whether the rule is reliable also when applied to the unobserv-able, like, for instance, when we postulate some unobservable entities because their existencewould best explain the phenomena.10 According to the tacking argument, we have empiricalreasons to believe that IBE is reliable in those cases, too, for—says the argument—variousunobservable entities that at some point in time had been postulated for strictly explanatoryreasons were later discovered by means of microscopes or other advanced “observation” in-struments.11

Like Hacking’s, this argument appears question-begging. As intimated before, the an-tirealist holds that our epistemic access to the world is limited to the observable, where by“observable” she means “observable by the unaided human senses.” So, clearly, she will

7Hacking’s argument actually only involves certain, and definitely not all, modern types of microscope. Thereis no reason to believe the argument cannot be extended to include other types of microscope, however. In fact, iteven seems possible to extend it to “observation” devices other than microscopes, such as X-ray diffractometers, forinstance.

8This is more or less the textbook version of IBE; see van Fraassen [1989, Ch. 6] for a critique of it. A morerefined version of IBE, and one that is immune to van Fraassen’s critique of the textbook version, is offered inKuipers [2000, Ch. 7]. For simplicity, I will assume the textbook version of IBE throughout the present paper; thetest to be presented in § 6 is, as far as I can see, not dependent on any specific version of IBE.

9See, for instance, van Fraassen [1985:298], Devitt [1991:112], and Reiner and Pearson [1995:64].10Or, to use Psillos’ [1996] suggestive terminology, crucial is whether the rule is reliable also when applied

“vertically,” not just when applied “horizontally.”11This strategy has been pursued by, among others, Harré [1986], [1988] and Bird [1998:160]. Kitcher’s [2001]

“Galilean strategy” also argues for the reliability of IBE on inductive grounds. But he seems to think that it suffices topoint to the reliability of IBE when applied to observable things and then shift the burden of proof to the antirealistby raising the question, “Why should there be an important difference in the reliability of the method when wecan no longer check the consequences by some independent means?” (p. 177 f). However, this strategy seemsto be based on a mistake. According to Kitcher (p. 177), “When [antirealists] insist on [the thesis that we canonly check methods of justification that lead to conclusions whose truth values can be directly ascertained just byinvestigating observables] they must hold that the world is so adjusted that a perfectly good method turns unreliablewhen it is applied below the threshold of our (contingent) powers of observation.” But, clearly enough, one canremain agnostic about whether otherwise reliable methods are reliable also when applied to the unobservable andstill maintain that the only methods we can check are those that yield conclusions only about observables. So inparticular it is not incumbent on the antirealist to argue for the unreliability of certain methods when they are appliedto the unobservable.

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object to the premise that certain unobservables have been discovered to exist by means ofmicroscopes. That the spots on micrographs biologists in the 1950s identified are due to thepresence of viruses is a hypothesis about unobservables, and as with any such hypothesis,the (van Fraassen–type) antirealist will want to remain agnostic about it. Consequently, tospeak of the discovery of viruses, or of any other unobservable entities, would beg a centralantirealist tenet.

To obtain from these question-begging arguments a non-question-begging method fortesting scientific realism, I made use of Glymour’s [1980] theory of bootstrap confirmation.This theory aims to do justice to the Duhemian claim that the confirmation of any singlehypothesis is in an important sense always relativized, for it is always confirmation relativeto one or more auxiliary hypotheses; but at the same time it aims to maintain the possibilityof absolute confirmation of complexes of hypotheses, or theories. The following definitioncaptures the central idea of Glymour’s account (“ßß” stands for the generalized conjunctionoperator):

Definition 3.1 (Bootstrap Confirmation) Let theory T have axioms H1,¼, Hn. Then evi-dence E bootstrap confirms T iff T Ç {E} 0 ^ and for each i Î {1,¼, n} the followingconditions hold:

1. there is an S Ì {H1,¼, Hn} such that Hi /Î S anda. E confirms Hi relative to ßß S; andb. there is possible—but non-actual—evidence E ¢ such that E ¢does not confirm (or

even disconfirms) Hi relative to ßß S;2. there is no S¢ Í {H1,¼, Hn} such that E disconfirms Hi relative to ßß S¢.

Thus, on this account, confirmation of the separate axioms of a theory relative to other axiomsof that theory amounts, under certain conditions, to unrelativized confirmation of the theoryas a whole.12

How can bootstrap confirmation help to empirically resolve this debate? Consider thetheory T with axioms V and R, and say that our total current evidence, E, comprises the dataHacking adduces in his argument for V as well as all available reports of (alleged) discover-ies of unobservable entities previously postulated on explanatory grounds (and comprises noother data that could be of relevance to either V or R). It would seem that, on any intuitivelyacceptable theory of (simple, non-bootstrap) confirmation, E must confirm V relative to R:Given that, as is agreed upon by all hands, the data Hacking obtained are best explained byassuming our microscopes to be veridical, on the assumption that IBE is a reliable rule ofinference, those data confirm V. Likewise, E confirms R relative to V ; on the assumption thatmodern types of microscopes are veridical, the alleged discoveries of unobservables are gen-uine discoveries of such entities and thus show, or at least lend support to the hypothesis, thatIBE is reliable also when applied to the unobservable. Notice that we now have already gonesome way toward a bootstrap confirmation of T : Condition 1.a of Definition 3.1 is satisfiedfor both V and R and, given that T has but two axioms, thereby Condition 2 of that definitionis satisfied automatically. Accordingly, it only remains to be shown that Condition 1.b issatisfied.

This condition requires that the assumption of R in testing V does not “trivialize” the testby guaranteeing confirmation of V, whatever the evidence, and that, in the same sense, theassumption of V in testing R does not trivialize this second test. As to the former, Hacking

12Depending on the exact theory of (simple, non-bootstrap) confirmation that is assumed, Condition 2 is superflu-ous in the face of the requirement that the theory and evidence be jointly consistent. For further comments on anddiscussion of Definition 3.1 the reader is referred to Douven [2002a] and Douven and Meijs [2003].

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might have obtained very different outputs from the different microscopes he used in his ex-periments, even though they were given the same or similar inputs. If that had happened, theevidence would not have confirmed V relative to R; arguably, such evidence would discon-firm V relative to R, as it would seem best explained by assuming V to be false. Hence, byassuming R we are not guaranteeing confirmation or even precluding disconfirmation of V.As to the other part of the test, it might have turned out that only very few of the putativeunobservable entities that were accepted on purely explanatory grounds retained their placesin our theories after the introduction of, for instance, the electron microscope. And if so, thatwould have cast considerable doubt on R, not just notwithstanding the assumption of V buteven especially on that assumption (absent the assumption, there is still the possibility that theentities once postulated were discarded because faulty microscopes led us to wrongly believeother types of entities are responsible for the phenomena the former entities were previouslythought to explain). So here the auxiliary employed does not protect against disconfirmationeither. Thus, Condition 1.b is satisfied as well.

Hence, our total current evidence bootstrap confirms T, i.e., it bootstrap confirms in tan-dem the hypothesis stating that IBE is a reliable rule of inference and the hypothesis statingthat modern microscopes are veridical.

How exactly can testing V and R help to empirically resolve the realism debate? First ofall, if V is true then of course antirealism is false; for in that case our microscopes do giveus access to at least part of the unobservable. Hypothesis V is also more indirectly relevantbecause, if it is true, the premise KE of the argument from underdetermination cannot be true.For in that case we may have more to go on in assessing a theory than purely observationaldata; data obtained by means of microscopes or similar instruments would, under the givensupposition, be relevant, too. Similarly, the truth of R is directly relevant because so manydefenses of realism rely on IBE, and also indirectly relevant because it, too, undercuts KE:if IBE is a reliable rule of inference, then clearly it would seem prudent if, in determiningwhich of a number of rival theories to believe, we take into account not only the availableobservational data but also consider how well the various theories do qua explanations.

Does the converse hold as well, that is, is any evidence against V or R (or both) alsoevidence against scientific realism? While it is logically possible that we have epistemicaccess to the unobservable even if V and R are both false, I suspect that any realist will admitthat her reasons for believing that we actually do have such access in some way or otherdepend on at least one of V and R and presumably even on both. Thus, even though refutingor disconfirming V or R, or both, does not directly refute or disconfirm scientific realism,it is likely to undermine the realist’s reasons for being a realist. Note, incidentally, thatrefutation or disconfirmation of either V or R does not require any complicated constructions.Already in our daily lives we might have discovered, and might still discover, that inferringto the best explanation scores hardly better, or even worse, than guesswork; and experimentssuch as Hacking’s could easily have yielded, and might still yield, negative results, whichpresumably would have made, or would make, even the staunchest realist doubtful about theveridicality of our microscopes.

Thus, the foregoing seems to provide us with the wherewithal for deciding the realismdebate empirically. Still, it has appeared to me that many are wary of the strategy heredepicted because of its reliance on the bootstrapping account. And although I am not awareof any argument showing that the account is fundamentally flawed,13 it would nevertheless

13Most if not all of the critique bootstrap confirmation has received (like, e.g., that of Christensen [1983], [1990],and Edidin [1983]) seems to concern either the fact that Glymour’s original presentation countenanced (what Earmanand Glymour [1988:261] call) macho-bootstrapping, i.e., that an axiom served as its own auxiliary, or the particular,“Hempelian” format of Glymour’s presentation. However, it appears that Glymour’s theory can do perfectly well

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be good to have available a strategy for testing scientific realism that does not involve aconfirmation theory as controversial as Glymour’s. In § 6, I hope to offer such a strategy. Itpresupposes specific conceptions of evidence and of the methodological role of explanationwhich are to be presented first.

4. The Range of Accessible Evidence. What is the range of evidence accessible to us?What propositions expressible in our language can state evidence? Which propositions canbe basic or fundamental beliefs for a person? Which propositions can receive probability 1as a direct effect of experience and which can have their probability altered only mediately,because some other proposition receives probability 1? I take these to be more or less equiva-lent ways of stating the same question—though, patently enough, some assume very differentphilosophical backgrounds. In this section, I aim to develop an answer to it that in my opinionis more plausible than either the customary realist or the customary antirealist answer. First Ibriefly remind you of the latter answers.

While he refrains from discussing the question of the range of evidence in any detail,it is clear that, in contrast to earlier empiricists such as Mach, Neurath, and Russell, vanFraassen does not want to restrict the range of accessible evidence to what can be expressedby statements reporting personal impressions (or sensations or sense data), but countenancesas evidence also reports of observable phenomena. On the other hand, he does not wantto be so liberal as most realists are, who believe reports from experiments or tests usingmicroscopes and comparable instruments, with their typical references to entities, events,and processes invisible to the naked eye, should be taken literally and should be regarded,so taken, as reporting evidence; as intimated earlier, van Fraassen wants to remain agnosticabout the correctness of such reports.

In the eyes of many, van Fraassen’s position on this point is incoherent.14 “True,” theysay, “there is always some risk that a report that goes beyond the strictly observable is in-correct. But there is already a risk involved in taking a strict observation report as statingevidence; since the human perceptual apparatus is not infallible, the correctness of such re-ports is not a matter of absolute certainty either.” However, it seems that van Fraassen hasan at least formally adequate response to this objection. As he says in a related context, “itis not an epistemological principle that one might as well hang for a sheep as for a lamb”([1980:94]). Indeed, I know of no argument to the effect that one’s willingness to take somerisk by countenancing unaided observation as a source of evidence commits one to takingthe additional risk involved in, for instance, countenancing the typical research report of theworking microbiologist as such a source.

The response is formally adequate, I said. At the same time, I find it philosophicallyunsatisfactory because dogmatic and unnatural. It can hardly be denied that there is an addi-tional risk in countenancing reports going beyond the observable as evidence; it has happenedoften enough that scientists had to recall what they had earlier claimed they had seen throughtheir microscopes. I also agree with van Fraassen that this warrants treating such reports dif-ferently from “pure” observation reports. But should we, because of the extra risk, treat theformer in a manner wholly different from how the latter are treated? In my opinion, which

without macho-bootstrapping (see Earman and Glymour [1988]); note that, accordingly, the first clause of Defini-tion 3.1 expressly demands that, in a test of a given hypothesis, that hypothesis not be among the auxiliaries. Asto the second kind of criticisms, already Glymour [1980:127] points out, and van Fraassen [1983] shows more ex-plicitly, that bootstrap confirmation is not tied to that format. See for more on this Douven [2002a] and Douven andMeijs [2003].

14See, e.g., various of the contributions to Churchland and Hooker’s (ed.) [1985] collection (especially Church-land’s, Giere’s, Musgrave’s, and Wilson’s); also Devitt [1991] and Psillos [1999].

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I will try to make plausible below, the answer is no. Of course my partial agreement withvan Fraassen implies that in my view the realist position on the question of accessible evi-dence, which acknowledges—as it must—that our current “observation” instruments are notcompletely trustworthy yet permits us to take the outputs of such devices as providing, forinstance, “pretty nice pictures of DNA” (to repeat Giere’s words), is not satisfactory either.As a matter of fact, I find it no less dogmatic and unnatural than van Fraassen’s.

To begin to see what an undogmatic position, and one more natural than either of theforegoing, would be, consider how ordinary people react when they experience that theirmemories betrayed them, that something they seemed to remember with great clarity wasactually false, like, for example, when they were certain they had been at some party butthen later discovered that they simply couldn’t have been there. I think that no reasonableperson ever concluded on the basis of such failures that she were better to remain agnosticabout what she seems to recollect from memory; nor, however, is it likely that, after shehas been confronted with its fallibility, such a person goes on trusting her memory in thesame way or to the same extent as she did before that event. What failures of the sort atissue are likely to occasion in her, I believe, is a sense of caution vis-à-vis her memories.I, for one, do trust my memory, but due to experiences of the aforementioned kind I do nottrust it for the full 100 % but only for, say, 95 %. That does not make me an agnostic aboutmy I seem to remember, although it does mean that for me my memories no longer havethe infallible status I once thought could be accorded to them. A similar story can be toldwith respect to eyesight, which we know is (for most people) quite, but just not completely,reliable. My eyes’ less-than-perfect reliability, and the concomitant possibility that whatI believe I see is going on around me is not really going on around me, or at least not as Ibelieve it is going on, are things I permanently reckon with; but while they have an undeniableimpact on the epistemic role of my perceptual beliefs, they do not make me an agnostic aboutwhat I seem to be seeing. And I submit that, from an intuitive viewpoint, the story to betold about microscopes is not essentially different either. As with memory and eyesight, thenatural response of those routinely working with microscopes to registered past failures in thepractice of microscopy seems to be to be cautious about, to not fully trust, what is obtainedby means of microscopes—presumably even very cautious, given the relatively large numberof such failures; but they’re not agnostics about micrographs.

That to be cautious about what, for instance, our memories seem to tell us about the pastis very different from being agnostic about that as well as from treating it epistemically in away as if we had never noticed the fallibility of memory, only tells us what is not involvedin being cautious about our memories, etc. In order to be clearer about what is involvedin it and, in particular, what exactly it means for the evidential status of, for instance, ourmemories to be cautious about them, let us ask this: If I have come to know that my memoryis not fully dependable, and therefore do not fully trust my recollections from memory any-more, is there still a sense in which they can be regarded as constituting evidence about mypast? Likewise, if I am only 40 % convinced of the veridicality of modern types of micro-scopes, then can I or can I not regard the following claim as reporting evidence for (possiblyamong other things) the existence of the bacterium Erwinia and the actuality of the unob-servable process of crosswall formation: “Micrographs showing stages of cell division forsynchronically grown Erwinia showing invagination of cytoplasmic membrane and forma-tion of crosswall septum” (Atlas [1986:310]; caption of a series of micrographs)? From theliterature on the realism debate one gets the impression that at least the answer to the secondquestion must be either a simple yes or a simple no. But clearly, the former—realist—answerentirely ignores what seem to be perfectly legitimate doubts about our current microscopes;the second—antirealist—answer for no discernible reason takes these doubts to warrant an

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epistemic attitude much more extreme than we are naturally inclined to adopt in the face ofsuch doubts. So neither answer seems satisfactory. Is there a third? If so, what is it? Andgiven an answer to the latter question, can evidence still bear on what (else) I believe orshould believe (still assuming a Bayesian framework)? If so, how?

Rather unsurprisingly, I believe the answer to the first question to be positive. The an-swer to the second question involves a distinction between a notion of evidence understoodin a sense broad enough for the citation just given to count as an evidence report—and notas a theoretical hypothesis about what is responsible for the dots on the micrographs, as vanFraassen would have it—and a “narrower” notion of evidence which brackets any ontologicalimplications in the unobservable domain an evidence report, broadly understood, may have.If E is an evidence report broadly understood, then by “e” I shall mean the claim E strippedof its ontological implications in the unobservable domain; in other words, e claims that allphenomena are as if E (if E carries an ontological commitment only to observables, then,of course, E = e). For example, if E is the proposition expressed by the above citation con-cerning the micrographs, then e is the proposition expressed by something like the followingsentence: “Micrographs showing what seem to be stages of cell division for synchronicallygrown Erwinia¼ .”15

Now say that you are given these micrographs and that you are less than fully confidentabout the veridicality of the type of microscope with which they were produced. If you werea dogmatic realist or a dogmatic antirealist, you would describe them as constituting evi-dence E or evidence e, respectively. But how should you describe them if the descriptionis to reflect your uncertainty about the veridicality of the microscope? The following wouldcertainly seem a reasonable response: “I’m not sure what we have here; if I were our col-league X [known as a dogmatic realist], I would say ‘E’, but if I were colleague Y [a dogmaticantirealist], I would say ‘e’.” Let us regard evidence reports of this type as being sui generisand denote them as “E/e,” etc. Still, calling them evidence reports does not help to showhow, as a Bayesian, you can learn anything from them.

As a Bayesian, you learn from evidence by conditionalizing on it. But it is not imme-diately obvious how you are to conditionalize on “uncertain” evidence such as E/e or, whatamounts to the same, what your probability for A given E/e should be, for any proposition A.Note that it would be wrong to think that you should simply conditionalize on e, given thatyou are certain about that (for it is implied by both E and e): the rule of conditionalizationwould apply only if you were to assign probability 1 to e and to no stronger proposition,and you, after all, are unsure about whether e is the strongest evidence you have obtained orwhether that rather is E (becoming sure of e is not the same thing as becoming sure that e isone’s total evidence). Neither is it an option to conditionalize on the disjunction E Þ e. Giventhat, for all propositions E, E implies e—if E is the case, then all phenomena will be as ifE is the case—it holds that (E Þ e) º e, so that p(A | E Þ e) = p(A | e), for any A. Hence, thisstrategy would land one automatically in the camp of the dogmatic antirealist, which is notwhat we want.16

15If one wants, one can make finer distinctions than between just “broad” and “narrow” evidence. For instance,one could further distinguish between narrow evidence as understood here and a notion of (even narrower) evidencethat would be acceptable to philosophers such as Mach and Neurath. It will be obvious how to adapt everything thatis being said in the text below if one wants to distinguish between more than two sorts of evidence.

16Nor can the problem be solved by conditionalizing not on EÞe but on EÞ(eßE): since E implies e, it also holdsthat (E Þ (e ß E)) º e, and thus that p(A | E Þ (e ß E)) = p(A | e) (here and elsewhere, a bar above a letter functionsas the negation operator). However, since quite similar worries as are aired here directed Jeffrey to the generalizedrule of conditionalization that now goes by his name (cf. his [1983, Ch. 11]), one would expect that rule to be ofhelp in solving the problem at hand. I believe that this is indeed the case. Exactly how it can be of help I discussin Appendix A. Yet I will below appeal to the (mathematically very similar) theory of expert functions in order to

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Yet I believe that you can learn from the micrographs and, in general, from uncertain evi-dence. I suggest that we take guidance here from the theory of expert functions as developedby, among others, Gaifman [1986], van Fraassen [1989], and Cooke [1991]. To illustrate thebasic idea behind this theory, suppose you want to know whether you are likely to develop aheart condition within the next ten years, and that, to this end, you consulted two cardiolo-gists. According to one of them, you have a .05 chance of developing a heart condition withinthe next ten years; according to the other, that chance is .1. Then to regard these cardiologistsas a panel of experts on the matter of your developing a heart condition within the next tenyears is to make your personal probability for that event to happen a mixture, or weightedaverage, of the cardiologists’ probabilities .05 and .1, where the weights reflect your relativeconfidence in each of the cardiologists; for example, if you consider the first as being twiceas competent on the matter as the second, you will set your degree of belief for developinga heart condition in the next ten years equal to (2/3)(.05) + (1/3)(.1) » .067. The followingdefinitions make this idea both general and precise:17

Definition 4.1 (Expert Function) Let p(×) be person S’s personal probability function. Thenq(×) is an expert function for S concerning family F of propositions iff pIA | q(A) = xM = x forall propositions A in F.

Notice that it follows from this and the standard ratio definition of conditional probabilitythat if q(×) is an expert function for S concerning F, then for all A, B Î F : p(A | B) = q(A | B).

Definition 4.2 (Panel of Experts) Let S1,¼, Sn have personal probability functionsp1(×),¼, pn(×). Then S1,¼, Sn constitute a panel of experts for S on F iff q(×) is an expertfunction for S in the sense of Definition 4.1 and there is a weight function w(×) such that forall A Î F: q(A) = Ún

i=1 w(Si)pi(A).

Here w(×) is a weight function precisely if, first, for all i with 1 6 i 6 n: 0 6 w(Si),and, second, Ún

i=1 w(Si) = 1. In principle this function could stand for anything, but in theinteresting cases it measures S’s confidence that the respective experts are right about thepropositions in F.

A consequence of Definition 4.2 worth noting is that, while S’s probability for A con-ditional on B (for arbitrary A, B Î F) will be some average of the various pi(A | B)’s, itin general will not be the average as weighted by the weight function w(×) but an averageweighted by some other weight function wB(×). The reason for this is simple: B may provideevidence for or against the authorities of the individual experts.

I propose to model learning from uncertain evidence along these lines. To specify theproposal, suppose you are given the micrographs putatively showing the process of crosswallformation in Erwinia from our example. While you are unsure about how to put the evidenceyou obtain from them, you can ask yourself what probability you would assign to somegiven proposition A conditional on receiving the micrographs if you were to endorse thedogmatic realist standpoint vis-à-vis the question of the range of evidence and, respectively,what probability you would assign if you were to endorse the dogmatic antirealist standpointvis-à-vis that question. We’ll suppose that you are able to answer these questions, and denotethe resulting “counterfactual” probabilities by “r(A | E/e)” and “a(A | E/e).” Notice thatthis is much like having consulted two experts who hold opposite views about how to put

tackle our problem. This is mainly for reasons of unity: the same theory can be applied to the question involving therole of explanatory considerations, and I see no (natural) way to apply Jeffrey conditionalization to that.

17These definitions are taken from van Fraassen [1989:198, 202 f], the first more or less literally.

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the evidence in this case. At any rate, this is how, on the current proposal, you should viewthe situation, that is to say, you should regard the “imagined-dogmatic-realist-in-you” andthe “imagined-dogmatic-antirealist-in-you” as constituting a panel of experts and thus make,in accordance with the theory of expert functions, your probability for any proposition Aconditional on E/e, that is, p(A | E/e), a weighted average of r(A | E/e) and a(A | E/e).

What should the weights be? It seems that, to the extent you believe microscopes to beveridical, you must think it correct to describe the micrographs in the way a dogmatic realistwould do, and thus also to assign probabilities conditional on receiving the micrographs asyou would do were you a dogmatic realist, that is, to set p(A | E/e) = r(A | E/e) for all A;and that, to the extent you believe microscopes not to be veridical, you must think it correctto describe the micrographs in the way a dogmatic antirealist would do, and thus also toassign probabilities conditional on receiving the micrographs as you would do were you adogmatic antirealist, that is, to set p(A | E/e) = a(A | E/e). In other words, it seems that theweights should be determined by your probability for V, though, for the reason explained twoparagraphs earlier, the weights ought to be your probabilities for V and V conditional on E/e(so that the evidence may affect your confidence in the imaginary experts). This yields thefollowing formula for determining conditional probabilities on uncertain evidence:

p(A | E/e) = p(V | E/e)r(A | E/e) + p(V | E/e)a(A | E/e).(1)

For ease of reference, I add the observation that, as a simple consequence of (1), if (i) p(V |E/e) > p(V ), and (ii) r(A | E/e) > a(A | E/e), then

p(A | E/e) > p(V )r(A | E) + p(V )a(A | e).(2)

This inequality will be one of the two main pillars of the methodology for empirically resolv-ing the realism debate to be described in § 6.

It should be noted that (1) cannot be entirely general, for it cannot be utilized to determinethe probability of V itself conditional on some item of uncertain evidence. This might seemto pose a problem, given that the formula does assume that we can somehow determinesuch conditional probabilities. However, I think it should be largely uncontroversial that,on pain of circularity, in assessing the probability of V there can be no role for evidencewhich in a clear sense already presupposes V, if only partly or to some extent—as is thecase for our notion of uncertain evidence—and that, accordingly, we ought to assume thatp(V | E/e) = p(V | e) for all uncertain evidence E/e and all probability functions p(×).18

A comment on the use of counterfactual probabilities: While it must be acknowledgedthat it is not self-evident that we have, or can come up with (if needed), counterfactual proba-bilities such as r(A | E/e) and a(A | E/e), in making the assumption that we do have them, orat least can come up with them, I am in good company, as counterfactual probabilities playa key role in some of the more prominent solutions to the so-called problem of old evidence,like for instance those put forward by Horwich [1982] and Howson [1984], [1985]. In these,we are asked to consider what probability we would have assigned to some hypothesis had wenot known some particular piece of evidence for it. That seems to be not essentially differentfrom what an application of (3) demands from us.

18In fact, if given some uncertain evidence E/e one’s probability for V determines how to weigh the “E-part”and the “e-part” of that evidence against one another (as seems plausible, whether this is done according to (1) oraccording to some other principle), then a regress would seem to threaten if uncertain evidence were allowed a rolein assessing the probability for V. After all, if a piece of uncertain evidence were to change one’s probability for V,one would, in light of one’s new probability for V, not have weighed E and e correctly and thus would, presumably,have to reassess one’s probability for V now weighing the parts differently, which, again may lead to a change inone’s probability for V, and so on.

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In sum, there is an undogmatic and natural alternative to both the standard realist andthe standard antirealist position as regards the question of the range of accessible evidence.In particular, the alternative does not impel one to set aside one’s doubts concerning certainmechanisms one has utilized to obtain (putative) evidence nor to give more import to thesedoubts than one is naturally inclined to do, but lets reflect the degree of one’s doubtfulnessabout a mechanism in the evidential status of the items generated by it. And, as we have seen,it is possible for the Bayesian to learn from the uncertain kind of evidence involved in thisalternative, namely via equation (1).19

5. The Methodological Role of Explanatory Considerations. Much the same story aswas just told about evidence can be told about the role of explanatory factors, and specifi-cally about that of IBE, in scientific methodology. Most if not all of the literature related tothe realism debate gives the impression that realists categorically believe in IBE while antire-alists are just as categorical in their rejection of the rule. However, in view of the argumentsthat so far have been brought forward both pro and con IBE, neither categorical belief in IBE,nor categorical rejection of it, seems warranted. Though the tacking argument may give somesupport to the claim that IBE is reliable, certainly from the perspective of someone who isfully convinced of the veridicality of current microscopes, the evidence present versions ofthe argument are able to cite can hardly be said to be so ample that the argument makes a con-clusive, or even fairly conclusive, case for IBE. And while van Fraassen [1989, Ch. 6] claimsto show quite generally that the rule of IBE cannot be followed rationally, the arguments heproffers for this claim all seem to fail.20 Better arguments, either in favor of or against IBE,may be ahead of us; in fact, the test to be presented in the next section will, hopefully, help usto such an argument (whether in favor of or against IBE will depend on what the data turn outto be). But what are we to do in the absence of such an argument? Follow the realist in her(seemingly rather blind) faith in IBE? Or instead follow the antirealist in her wholesale rejec-tion of it? Given that I am neither fully convinced that the rule is reliable, nor, that it is not, Iam uncomfortable with either option. In following the realist I would, intuitively, accord toomuch epistemic weight to explanatory considerations, but in following the antirealist, I wouldaccord them too little. Instead, I should like to accord them a weight which reflects the degreeto which I am confident that such considerations are truth-conducive. Just as in the case ofevidence, I believe that an undogmatic, “middle” position regarding the methodological roleof explanatory factors, and of IBE in particular, can be developed that enables one to do justthat.

Before this position can be described, I should clarify an issue about the relationshipbetween IBE and Bayesianism—the confirmation theory we are assuming—that otherwisecould easily lead to misunderstandings. Suppose you believe, as realists tend to do, thatexplanatoriness is a mark of truth. Does that mean you cannot be a faithful Bayesian? I haveheard several people assert that van Fraassen’s critique of IBE in his [1989, Ch. 6] shows that,indeed, there is no room for explanatory considerations within a Bayesian framework. Butthat is not at all what it shows (nor does van Fraassen claim otherwise). At most it shows thatthere cannot be a probabilistic rule for belief change that assigns bonus (or malus) points forexplanatory force (or the lack thereof).21 Even if this is granted, however, it is still perfectly

19To be entirely precise, this is possible for the Bayesian who is willing to make use of expert functions andcounterfactual probabilities, elements that, while not at the core of Bayesian epistemology perhaps, are certainly notforeign to it either.

20Cf. Douven [1999], [2002a], Okasha [2000].21In Douven [1999] I argue that van Fraassen’s critique does not even show this much. See also Psillos [2004].

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possible to give explanatory considerations their due in a Bayesian framework, namely, indetermining prior probabilities.22 For instance, if a person believes that hypothesis H would,if true, best explain evidence E, then nothing in the Bayesian framework prohibits her fromassigning a prior probability of 1 to H given E, or, in other words, from setting her priorfor H ßE equal to that for E alone. Similarly, if H and H ¢ both accommodate E, but H betterexplains E than H ¢ does, then nothing prohibits her from assigning a greater probabilityto H ß E than to H ¢ß E.

That a Bayesian can consistently accord weight to explanatory considerations does notmean that any Bayesian will think she must accord weight to them. Colorably, how shethinks about this will depend on how confident she is that explanatory considerations aretruth-conducive. If she is convinced, like Harman, that they are truth-conducive, then she willmost likely agree with him that prior probabilities are to be based on explanatory considera-tions. If, on the other hand, she shares the antirealist’s point of view and regards explanatoryconsiderations as being at best of pragmatic concern, and unrelated to truth, she will not wantto let them have any say in determining priors. But what if one’s view on this matter, likemine and, no doubt, that of a great many other people, cannot be quite identified with eitherthe realist’s or the antirealist’s? What if one believes to some degree, and to some degreeonly, that explanatory considerations are truth-conducive?

The following proposal parallels the earlier one for learning from uncertain evidence: Forconcreteness, suppose you want to determine your probability for hypothesis H conditionalon evidence E, where you believe that there is some explanatory connection between H and E(for instance, that H is the best potential explanation for E), and where, in order not to com-plicate matters, we assume E to be “certain” evidence. Then try to imagine what probabilitiesyou would assign to H given E were you to categorically endorse respectively eschew IBE.We’ll suppose that you can come up with these counterfactual probabilities, and denote themby “R(H | E)” and “A(H | E),” respectively. This is again much like having a panel ofexperts in the sense of Definition 4.2, consisting of a “realist” expert who, in determiningprobabilities, gives full weight to explanatory considerations and an “antirealist” expert who,in doing the same, disregards them completely. And here too it is part of my proposal thatyou proceed by making your actual probability for H conditional on E a weighted averageof the counterfactual conditional probabilities R(H | E) and A(H | E). The obvious choiceof weights this time are your probabilities for the reliability of IBE and its negation, both, ofcourse, conditional on E (for the reason explained in the comment on Definition 4.2). Thus,on this proposal, your conditional probability for H given E will be

p(H | E) = p(R | E)R(H | E) + p(R | E)A(H | E).(3)

I state for later reference an obvious consequence of (3), namely that, if (i) p(R | E) > p(R),and (ii) R(A | E) > A(A | E), then

p(A | E) > p(R)R(A | E) + p(R)A(A | E).(4)

This is the second main pillar of the methodology to be utilized in § 6.Two remarks on the foregoing: First, for basically the same reason (1) could not be

entirely general, (3) cannot be entirely general either; in particular, it cannot be used to de-termine p(R | E), for any E. But as with (1), this does not really pose a problem. For it isagain uncontroversial, I think, that in assessing the probability of R on some given piece of

22As various authors have pointed out; see, e.g., Day and Kincaid [1994] and Harman [1999]. Harman [1999:114]even goes so far as to generally assert that “probabilities are not themselves basic but are derivative from considera-tions of inference to the best explanation.”

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evidence E one should, on pain of circularity, disregard any possible explanatory connectionsbetween R and E; that is to say, even someone who fully subscribes to IBE should not as-sume the rule to be reliable in testing that very assumption.23 And if so, that means that indetermining your probability for R conditional on some E, you should set R(R | E) equal toA(R | E), in which case it follows from (3) that p(R | E) = R(R | E) = A(R | E).

And second, we have restricted (3) to “certain” evidence, but what if there is an explana-tory connection between a given hypothesis and some uncertain evidence—as surely therecould be—and one wants to be undogmatic both about the question of the range of evidenceand about that of the methodological significance of explanatory considerations? The follow-ing may be a plausible way to combine (1) and (3):

p(A | E/e) = p(V ß R | E/e)R(A | E/e) + p(V ß R | E/e)A (A | E/e),(5)

where R(A | E/e) and A (A | E/e) are the probabilities you would assign to A conditionalon E/e if you were a dogmatic realist (respectively, antirealist) both about evidence andabout the role of explanation. However, since in the remainder of the paper we will only beconcerned with testing R and V, for our purposes (1) and (3) suffice entirely. As was pointedout, in testing V we should refrain from utilizing uncertain evidence, and in testing R weshould refrain from taking into account any possible explanatory connections between R andwhatever evidence we have. Accordingly, in testing neither of the two hypotheses is there acall for something like (5).

Hence, there is no need to follow either the realist or the antirealist in their extreme po-sitions regarding the question of the methodological status of explanatory considerations.By (3), we can determine our conditional probabilities in a way which gives weight to ex-planatory considerations proportional to our confidence in IBE.

6. A Strategy for the Undogmatic. This section describes a strategy for testing the realismdebate that is open to anyone who is undogmatic (in the sense of the previous sections)both about the issue of the range of evidence and about that of the role of explanation inmethodology. The strategy is basically the bootstrap strategy from § 3 all over again, but thistime without the controversial bootstrapping part. Recall that bootstrapping was employed totest the conjunction of V and R and that as one part of our evidence we assumed the resultscited in Hacking’s paper about microscopy and as another we assumed the reports of allegeddiscoveries by means of microscopes and similar instruments of entities previously positedfor purely explanatory reasons. We had to interpret this latter evidence “narrowly,” that is, asreporting alleged discoveries of unobservables, in order not to beg any antirealist issues. Thereports could then nevertheless be brought to bear on our hypothesis R, namely by using Vas an auxiliary, which in its turn was supported by the data from Hacking’s paper through thehelp of R, all in a manner formally correct according to the theory of bootstrap confirmation.As will be seen, the answers to the questions of evidence and of the methodological role ofexplanatory considerations developed in the previous sections allow us to test V and R in away that avoids the usage of bootstrapping. In particular, it will be shown that under certainspecifiable conditions the undogmatic antirealist may come to assign a probability greaterthan .5 to both V and R and thus may, as we saw at the end of § 3, come to have reason

23According to some authors, it is all right to use IBE in defending IBE; see, e.g., Psillos [1999, Ch. 4] forsome particularly forceful arguments for that claim. However, so far I’m unconvinced; see Douven [2001]. Note,moreover, that if explanatory considerations were to have a role in determining R’s probability given some evidence,then a regress problem would threaten similar to the one we said would threaten if uncertain evidence were to havea role in determining the probability of V ; cf. note 18.

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for abandoning her position. It is left to the reader to verify that exactly the same strategywill, under different conditions, lead an undogmatic realist who initially assigns probabilitiesgreater than .5 to both V and R to lower her probabilities for those hypotheses to a valuebelow .5.

As adumbrated in § 2, agnosticism is here assumed to be modelled in the manner sug-gested by Monton. Thus, the antirealist assigns sharp probabilities to all hypotheses, includ-ing “theoretical” ones, though in assigning prior probabilities she will for any theoreticalhypothesis to which she assigns a positive probability assign an equal probability to at leastone other theoretical hypothesis that is empirically equivalent to the former. This has conse-quences for her priors for V and R, for while these hypotheses do not stipulate unobservables,they in a clear sense both are (also) about unobservables: V says that whatever unobservableentities we seem to see by means of our microscopes and similar instruments are really thereas seen; and R says that IBE is a reliable rule of inference, also in cases in which what weinfer by means of it has ontological implications in the unobservable domain. Hence, foreach of V and R there is—at least—the empirically equivalent rival that the hypothesis is em-pirically adequate yet false. The antirealist’s prior probabilities for them will thus be sharp,but not exceed .5.

I first show how the strategy works by means of an example and then state a mathematicalresult generalizing the example.

Example 6.1 Consider an undogmatic antirealist whose probabilities we represent by thefunction p(×), and suppose that p(V ) = p(R) = .3. Further suppose she comes to know theevidence referred to in the tacking argument for IBE. Since she is undogmatic as regardsthe question of the range of evidence, and since the evidence was obtained by means of mi-croscopes, for her it is uncertain evidence; so let us denote it by “E/e.” What impact willlearning E/e have on her probability for R? First, it is clear that the evidence broadly inter-preted supports R to a (much) greater degree than does the same evidence when bracketed,and hence, we can safely assume, what she would assign to R conditional on E/e were sheto side with realists dogmatic about the range of evidence will exceed what she would assignwere she to side with antirealists dogmatic about that issue; that is, as we shall again put it,r(R | E/e) > a(R | E/e). Secondly, the evidence, whether taken as discoveries of unobservableentities or merely taken as seeming discoveries of such entities, by no stretch of the imagina-tion could be said to be negatively relevant to V ; so we may assume that p(V | E/e) > p(V ).Then it follows from inequality (2) that

p(R | E/e) > p(V )r(R | E/e) + p(V )a(R | E/e).(6)

Plausibly, even when narrowly interpreted, E/e will at least somewhat support R—that thedata are seemingly in accordance with R can hardly bear negatively on that hypothesis—but let us just say that a(R | E/e) > p(R) and focus our attention on the remaining factorin (6), r(R | E/e). In § 3, we saw that the dogmatic antirealist refuses to accept the conclusionof the tacking argument because she does not accept the realist’s purported evidence to theeffect that, due to technological advances, some unobservable types of entities have becomeepistemically accessible to us in essentially the same way as observable types of entities areepistemically accessible to us. But if, as seems to be the case, this is the dogmatic antirealist’sonly reason not to go along with the realist in considering the evidence cited in the tackingargument as already being reasonably good evidence for R, then we should expect our un-dogmatic antirealist to assign a probability to R conditional on a broad interpretation of whatshe takes to be the evidence (namely E/e) that will be more or less what the “typical” realist’s

17

probability for R is on E, that is to say, it should be expected to be fairly high. Suppose this isthe case, and that r(R | E/e) = .6. Then the antirealist’s probability for R after she has cometo know E/e, that is, pE/e(R), will be equal to or greater than (.3)(.6) + (.7)(.3) = .39, and thuswill have increased by at least .09.

Next suppose that the antirealist learns about the reports of Hacking’s experiments inmicroscopy. Call this evidence “E ¢.” Clearly, E ¢ is narrow, bracketed evidence (i.e., E ¢ =e¢), consisting, as it does, of reports of micrographs showing similar arrangements of dotsobtained from different types of instruments when these were given similar inputs. Whateffect will learning E ¢ have on the antirealist’s probability for V ? Since she is undogmaticconcerning the methodological role of explanatory considerations, and since there is a clearexplanatory connection between E ¢ and R, she will, when confronted with E ¢, ask herselfwhat probability she would assign to V given this evidence were she fully convinced of thereliability of IBE, that is, what value she ought to give to RE/e(V | E

¢), and, respectively, whatprobability she would assign were she fully convinced of the rule not being reliable, that is,what value she ought to give toAE/e(V | E

¢). Since—everybody agrees—V best explains E ¢,someone fully convinced that IBE is reliable will conclude from E ¢ that V is at least verylikely.24 Accordingly, we may expect the antirealist’s value for RE/e(V | E ¢) to be at leastclose to 1. Arguably, from the opposite perspective, that is, the perspective of someone whodisregards any explanatory connections, the evidence will still support V to some extent, butjust say that AE/e(V | E ¢) > pE/e(V ), which, we assumed above, is greater than or equalto p(V ) (= .3). Further note that E ¢ at a minimum will not be negatively relevant to R, that is,pE/e(R | E

¢) > pE/e(R). Then it follows from inequality (4) that

pE/e(V | E¢) > pE/e(R)RE/e(V | E

¢) + pE/e(R)AE/e(V | E¢).(7)

That is to say, pE/e(V | E ¢) ' (.39)(1) + (.61)(.3) = .537. Thus, if the antirealist comes toknow E ¢, her new probability for V, i.e., pE/e,E ¢ (V ), will have increased by (approximately) atleast .237 as compared to her initial probability for V.

Obviously, the antirealist may search for further “successes”25 of IBE, similar to the onescited in the tacking argument. Suppose she finds some; it is “discovered,” say, that prions (in-fectious proteins which have been postulated to explain a number of neurological disorders,like, e.g., Creutzfeldt–Jacob’s disease) and other unobservable entities for which to date welack any microscopic evidence (however broadly that notion is interpreted) exist. Since these“discoveries” will involve the use of microscopes or similar instruments, our undogmaticantirealist must take the evidence to be uncertain evidence; let us thus denote it by “E */ e*.”Then compute once more the lower bound on the antirealist’s probability for R in the light ofthese new data, in the same way as we did in (6); that is, we want to know for what valuesof pE/e,E ¢ (R | E

*/ e*) the following holds (it is readily seen that the conditions that make (2)apply are satisfied):

pE/e,E ¢ (R | E*/ e*) > pE/e,E ¢ (V )rE/e,E ¢ (R | E

*/ e*) + pE/e,E ¢ (V )aE/e,E ¢ (R | E*/ e*).(8)

Surely E*/ e*, when taken broadly, will count as further evidence for R, so we may supposethat rE/e,E ¢ (R | E

*/ e*) > r(R | E/e), which was .6; let us hence say that rE/e,E ¢ (R | E*/ e*) = .65.

Further, since it is hard to see how E*/ e*, when taken narrowly, could bear negatively on R,we may assume that aE/e,E ¢ (R | e

*) > pE/e,E ¢ (R). With what we already had, we now find that,

24To be convinced that IBE is reliable is not necessarily to be convinced that it unfailingly leads to the truth, ofcourse.

25That is to say, successes or seeming successes, depending on whether or not we interpret the data broadly.

18

after E*/ e* has become known to the antirealist, her probability for R will be greater than orequal to (approximately) .53.

The antirealist could go on in this way, doing further experiments along the lines ofHacking’s and searching for more “successful” applications of IBE of the sort found earlier.Then, if the data turned out favorably, that would have her further shore up her probabilitiesfor V and R. But we can already now note that the antirealist’s probabilities for V and R bothexceed .5. ì

Of course this is sufficient to show what I promised to show, to wit, that an antirealist,despite her initial agnosticism regarding V and R, may come to assign probabilities to thesehypotheses that exceed .5. However, in the example we made several assumptions. Andwhile some of them seemed to be self-evident—such as, for instance, that data of “discov-eries” of previously postulated entities bear more heavily on R when they are interpretedbroadly than when they are interpreted narrowly—it would be of some interest to know howmuch depended on, for instance, our assumption about the antirealist’s priors for V and R,the one about her probability for R conditional on certain data concerning “discoveries” ofpreviously postulated entities, or the one about the order in which the data become known toher, assumptions which do not seem so evident. The following generalization partly answersthis question by stating a set of mostly rather minimal, yet jointly sufficient, conditions foran antirealist to come to hold probabilities for V and R exceeding .5. It is a partial answeronly because I am not able to prove that the conditions are also necessary. In fact, I would beunsurprised if there were some other Bayesian or, more generally, probabilistic strategy forobtaining the same result given some other conditions.

Let p(×) be the antirealist’s probability function at time t. The example assumed theantirealist to have prior probabilities of .3 for V and R, but it suffices to require that at t theybe positive, that is:

(9) p(V ) > 0; (10) p(R) > 0.

I further assume the following:

(11) There is a sequence S of evidence reports containing all and only the evidencerelevant to V and/or R the antirealist comes to know from time t onwards. Eachelement of S is either a piece of uncertain evidence—for obvious reasons, reportsof this kind are denoted by “Ei / ei” (i Î N)—or a piece of evidence that is pos-sibly explanatorily connected to V —these reports are denoted by “E ¢i ” (i Î N).It is assumed that Ei / ei becomes known before E j / e j if i < j, and similarly forE ¢i and E ¢j, but there are no further assumptions concerning the order in which thereports become known.

So, for instance, the beginning of S might look thus:

(*) S = YE1/ e1, E2/ e2, E ¢1, E3/ e3, E ¢2, E ¢3, E ¢4, E4/ e4,¼].A point about notation: If among the first m + n elements of S there are m evidence

reports of the sort we indicate by “E ¢i ” and n of the sort we indicate by “Ei / ei,” then pmn (×) is

the probability function p(×) updated, consecutively, on the first m + n elements of S, in theorder in which they appear in the sequence; similarly for pm

n (× | ×). For example, if S should

begin as in (*), then p02(×) is the result of updating p(×) on the first two elements of S, and

p43(×) the result of updating it on the first seven elements.

19

The following assumptions are made about the probabilistic relations between R and theelements of S. First, similar to what was assumed in the example, we assume that none ofthe Ei / ei is, when narrowly interpreted, negatively relevant to R, that is, for all E ¢m, En / en Î S:

amn-1(R | En / en) > pm

n-1(R).(12)

It should further be obvious that for all E ¢m, En / en Î S (provided amn-1(R | En / en) ¹ 1, ofcourse)

rmn-1(R | En / en) > a

mn-1(R | En / en).(13)

That is to say, we assume that reported “discoveries” of previously postulated unobservablesweigh more heavily in favor of R when taken as real discoveries than when taken as seemingdiscoveries. Note that it follows from (12) and (13) that, for each En / en Î S, rmn-1(R |En / en) > p

mn-1(R) (unless p

mn-1(R) should be 1). Further it seemed reasonable to assume that

the evidence of Hacking’s experiments was not negatively relevant to R. Correspondingly, Ihere assume that, for all E ¢m, En / en Î S,

pm-1n (R | E ¢m) > pm-1

n (R).(14)

Finally, in the example it was assumed that already the evidence mentioned in the tackingargument would, when broadly understood, bring the antirealist’s probability for R above .5.But it is actually enough to make the much weaker assumption that, with En / en Î S andE ¢m Î S,26

limm,n®¥

rmn-1(R | En / en) > .5.(15)

Parallel assumptions are made concerning the probabilistic relations between V and theelements of S:

(16) Am-1n (V | E ¢m) > pm-1

n (V );

(17) Rm-1n (V | E ¢m) > Am-1

n (V | E ¢m);

(18) pmn-1(V | En / en) > p

mn-1(V );

(19) limm,n®¥ Rm-1n (V | E ¢m) > .5.

All assumptions are supposed to be quantified in the same way as the corresponding assump-tions concerning R. Needless to say, a similar proviso as was made regarding (13) mustbe made for (16). It may also be noted that it is not assumed that Rm-1

n (V | E ¢m) » 1 forany E ¢m, En / en (in contrast to what we did in Example 6.1). This is just in order to make theresult as strong as possible.

We then have the following (for a proof, see Appendix B):

Proposition 6.1 If (9)–(19) hold, then

1. there are E ¢k, El / el Î S such that for all E ¢m Î S with m > k and all En / en Î S withn > l: pm

n (R) > .5; and

26Though I expect the following to be obvious, it won’t hurt mentioning that, although the antirealist must, asan agnostic about theoretical hypotheses, assign a prior probability to R that is no greater than .5, her agnosticismdoes not prevent her from having conditional probabilities for R, or any other theoretical hypothesis, that are greaterthan .5. In fact, some must be, already as a matter of logic: a theory of probability that implies something differentthan p(A | B) = 1 for any A, B such that B ¢ A is clearly absurd, whether or not A is a theoretical hypothesis.(On van Fraassen’s way of modelling agnosticism, this means that the agnostic cannot only have vague conditionalprobabilities for any theoretical hypothesis.)

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2. there are E ¢k¢ , El¢ / el¢ Î S such that for all E ¢m¢ Î S with m¢ > k¢ and all En¢ / en¢ Î S with

n¢ > l¢: pm¢l¢ (V ) > .5.

In other words, given favorable data, and in the long run, a person whose degrees of be-lief function satisfies conditions (9)–(11) will come to assign probabilities above .5 to bothV and R. How long the run must be for that to happen will depend on her priors for V and R,the rate at which rmn-1(R | En) and Rm-1

n (V | E ¢m) come to assume values above .5, and theorder in which the data roll in. But, as is exhibited by Example 6.1, the long run need not belong at all.

In the present context the most significant feature of this result may be that among theconditions (9)–(19) there are none which preclude agnosticism regarding V or R (one can as-sign low but positive probabilities to these propositions without being agnostic about them).Prima facie this might seem puzzling. For didn’t we say that her agnosticism regarding ahypothesis (as modelled per Monton’s proposal) secures the agnostic from ever assigning aprobability greater than .5 to that hypothesis? So then shouldn’t the antirealist’s agnosticismregarding V and R secure her from ever assigning probabilities greater than .5 to these hy-potheses? But, in effect, it is not hard at all to understand why in Example 6.1 and under themore general conditions given later she does come to assign probabilities greater than .5 toboth V and R, despite her initial agnosticism. Two things are important to note here.

First, that the antirealist starts by assigning equal priors to R and at least one empiri-cally equivalent rival to R is sufficient to ensure that no evidence that is acceptable as suchto the dogmatic antirealist—bracketed, narrow evidence, that is—will ever raise the agent’sprobability for R to a value above .5, for no bracketed evidence can discriminate betweenempirically equivalent theories. However, evidence broadly understood can. While empiri-cally equivalent theories are indistinguishable by purely observational data, they tell differentstories about the unobservable, and evidence broadly understood, by going beyond the purelyobservational, may finger one of these stories as the correct one.27 For example, among onetheory’s non-observational consequences may be that the process of crosswall formation (aprocess invisible to the naked eye) is real, while its rivals deny this; and micrographs, broadlyinterpreted, may show that process to be real indeed. In the case of R, no bracketed evidencecan show that explanatory factors are a reliable guide to the truth about the unobservable, butevidence broadly understood can do so, or at least can lend support to that claim. And, theway we proposed to model an undogmatic person’s learning from uncertain evidence, brack-eted and “unbracketed” (i.e., broad) evidence can both have an impact on her probabilityfor R.

Second, if explanatory factors are taken into consideration, even bracketed evidence iscapable of discriminating between empirically equivalent rivals; this is what does the workin the second and, more generally, even-numbered steps of the strategy. Since V is the bestexplanation of the data reported in Hacking’s paper, V ’s probability given these data forsomeone convinced that the best explanation of known data is typically true (that is, for

27More formally, we said in § 2 that if H1 and H2 are empirically equivalent, then p(E | H1) = p(E | H2) for allpropositions E whose truth we can ascertain by purely observational means, that is, as we can now put it, for all Esuch that E = e. But it is not implied by the concept of empirical equivalence, nor has anyone ever suggested, that ifH1 and H2 are empirically equivalent, it follows also for all broad evidence E that p(E | H1) = p(E | H2). Couldn’tone redefine the notion of empirical equivalence so that the latter would follow, too? Of course one could. But, first,given the current state of microscopy, that would already now starkly marginalize the problem of underdetermination(and in the future might even further marginalize it); in the end, it might leave us to be antirealist only about whatsome realists have claimed, on independent grounds, we should be antirealists about anyway (see, e.g., Jones [1991]).And second, there is already much controversy over whether empirically equivalent rivals do exist for most theoriesor even for some theories; given a stronger notion of empirically equivalent, the burden that rests on the antirealistto make plausible that indeed there always or mostly are such rivals would even become heavier.

21

someone convinced of R) will be 1 (or close to 1); that of any empirically equivalent rivalmust then be 0 (or close to 0). And, the way we proposed to model learning that is undogmaticwith respect to the question of whether explanatory factors are to be taken into account inassigning probabilities, the probability an undogmatic person would assign to V conditionalon data such as are reported in Hacking’s paper were she fully convinced of IBE’s reliabilityis partly determinative of her actual conditional probability.

In closing, I want to address a worry concerning our way of modelling agnosticism thatmight arise in response to the foregoing. It is easy to see that the testing strategy proposed inthis section would not have worked had agnosticism been modelled in the manner suggestedby van Fraassen; for instance, already if in inequality (6) we take p(V ), and hence also p(V ),as having an interval value that contains 0, then p(R | E/e) must be vague over an intervalcontaining 0 as well; and this would in turn suffice to make pE/e(V | E

¢) vague, and so on.28 Itthus might be asked whether the fact that on Monton’s proposal antirealism is refutable whileon van Fraassen’s it is not, or at least not in any discernible way, is not itself sufficient reasonfor the antirealist to reject Monton’s proposal and model her agnosticism by dint of vagueprobabilities. To start with, however, if the antirealist were to prefer van Fraassen’s mannerof modelling agnosticism merely because it ensures that her position cannot be shown falseby any evidence, then that would seem quite ad hoc. Much more importantly, there wouldbe an obvious tension between such a response and the general empiricist outlook whichunderlies van Fraassen’s antirealism in the philosophy of science. After all, it would be apeculiar type of empiricist who would want to make sure in advance that no evidence couldever give her second thoughts about her philosophical tenets regarding science.

7. Conclusion. If the foregoing is correct, then it is possible to resolve the realism debatein a broadly Bayesian framework. Of course, doing so requires (at least as far as this papergoes) that one be undogmatic regarding both the question of the range of accessible evidenceand the methodological status of explanatory considerations. However, as I hope to haveshown, the undogmatic answers to these questions given in §§ 4 and 5 can be motivatedon independent grounds and at any rate are more reasonable than the ones more commonlyencountered in the literature. Thus, that the strategy offered in this paper is committed tothem does not compromise it in the least.

Acknowledgments. Versions of this paper have been presented at a colloquium on scien-tific realism at Erasmus University Rotterdam in January 2003 and at the Ecole NormaleSupérieure, Paris, in June 2003. I thank the audiences on those occasions for stimulatingquestions and remarks and I thank especially Daniel Andler for inviting me to Paris. I amalso greatly indebted to Fred Muller, Jan-Willem Romeyn, and Christopher von Bülow forproviding me with helpful comments.

Appendix A: Learning from Uncertain Evidence via Jeffrey Conditionalization

In this appendix I say a few words about a way of learning from uncertain evidence differentfrom that exposed in § 4, this one involving Jeffrey conditionalization instead of the theoryof expert functions.

28For by the standard multiplication and addition rules for intervals we have that x × [0, y] = [0, xy] and [0, x] +[0, y] = [0, x + y].

22

The rule of Jeffrey conditionalization was not proposed by way of answer to the questionwhich propositions can potentially be taken as evidence, but rather as a solution to the prob-lem that, even if it is clear which propositions can be thus taken, there sometimes seems tobe no proposition at all that could be regarded as characterizing our perceptual experiences.To use one of Jeffrey’s examples (Jeffrey [1983:165 f]), a glimpse of a cloth by candlelightmay raise an agent’s probability that the cloth is green without making her certain that it is.In fact, there may be no proposition of which the event made her entirely certain, and yet itwould seem that the agent did learn something in that event. Since strict conditionalizationonly applies in cases in which some proposition’s probability is raised to 1, and in order to becapable of representing the kind of learning that occurs in the cloth example, Jeffrey proposedto generalize the rule of conditionalization. Let {Ei} be a countable collection of propositionswhich partition logical space and which are to be thought of as being directly affected bythe agent’s experience (in the cloth example, they are plausibly thought of as propositionsabout the cloth’s color). Further let p(×) and P(×) denote the agent’s pre-experience and post-experience probability function, respectively. Then the change from the former to the latteraccords with Jeffrey’s rule precisely if for all propositions A it holds that

P(A) = âi

P(Ei)p(A | Ei).(20)

If one of the Ei’s gets probability 1 then, of course, Jeffrey’s rule reduces to strict condition-alization.

Although, as I said, Jeffrey’s problem was not precisely the problem we were concernedwith in § 4, the two are evidently related in that both involve situations in which, intuitively, anagent does learn something new, but what she learns is not the truth of some proposition. Andit does seem possible to apply Jeffrey’s rule to our problem, even though such an applicationis not entirely straightforward. The reason that it cannot be applied straightforwardly is thatJeffrey’s rule operates on a partition of propositions, and, for any proposition E, it and its“bracketed” version are not mutually exclusive. But this problem can be overcome: Givensuch a pair of propositions, we can always form the partition {E, Eße, e} and apply the ruleto that. This yields, for all such partitions and all propositions A:

P(A) = P(E)p(A | E) + P(Eße)p(A | Eße) + P(e)p(A | e).(21)

Here p(×) and P(×) are the agent’s probability functions before and after she has been given themicrograph(s) or spectrograph(s) or whatever else it is she describes as providing uncertainevidence E/e.29

For present concerns, we may assume that the agent gets a good view, in excellent lightingconditions, of the micrographs or whatever, and thus that P(e) = 0. Equation (21) thenreduces to

P(A) = P(E)p(A | E) + P(Eße)p(A | Eße).(22)

We cannot say in general what values the agent should assign to P(E) and P(Eße). Clearly,it would be incorrect to think that, because P(e) = 0, she can simply conditionalize on e andthus set P(E) equal to p(E | e), and similarly for P(Eße); as we said earlier, the rule of strictconditionalization would apply only if the agent were to assign probability 1 to e and to no

29As in the case of the expert functions strategy, it should be obvious that, and how, the present strategy can begeneralized. For instance, if the agent is unsure about whether E is her evidence or e or the even weaker propositionthat all her experiences are as if E were the case—which we might denote by “e”—the rule could be applied to thepartition {E, Eße, eße, e} in essentially the same way as it is now applied to the simpler partition.

23

stronger proposition, and she is just not sure whether e really is the strongest evidence shehas obtained. However, in cases in which E reports evidence, broadly construed, producedby a microscope or some other instrument that, if V holds true, is veridical, we can establishsome interesting connections between the agent’s values for P(E) and P(Eße) on the basis ofher probability for V.

Let E be a report of the aforementioned kind. Then the following holds analytically:

p(E | eßV ) = 1.(23)

(To see this, recall the example from § 4: if the micrographs seemingly show the processof crosswall formation, and the microscope by which they were produced is veridical, thenthey do show the process of crosswall formation, with certainty.) Further let us make theassumption that, if p(×) and P(×) are your probability functions before and after you havebecome certain of E/e, respectively, then

P(E) > p(E | e).(24)

The assumption is hardly more than a triviality. For if, contrary to the assumption that you areundogmatic vis-à-vis the question of the range of evidence, you were an dogmatic antirealist,you would, after you had obtained what you now refer to as evidence E/e, assign probability 1to e. In that case your new probability for E, i.e., P(E), would be set equal to p(E | e) (strictconditionalization would apply, after all). If, on the other hand, you were, contrary to fact,a dogmatic realist, your new probability for E would, as a matter of course, equal 1. Nowit seems arguable that, given that you are undogmatic, and unsure about whether to assignprobability 1 only to e or also to the stronger E, the probability P(E) should strike somebalance between p(E | e) and 1 (determined, presumably, by your probability for V ). Whatseems definitely true, however, is that P(E) should not be lower than what it would be if youwere a dogmatic antirealist. For our purposes, this weak assumption suffices.

It is now easy to show that P(E) > p(V ) on condition that e is irrelevant or positivelyrelevant to p(V ), that is,

p(V | e) > p(V ).(25)

For the following chain of implications can be seen to hold (the second and last hold on thebasis of equations (23) and (24), respectively):

(26) p(V | e) > p(V ) Ûp(V ß e)

p(e)> p(V ) Û

p(V ß E ß e)p(e)

> p(V ) Û

p(V ß E | e) > p(V ) Þ p(E | e) > p(V ) Þ P(E) > p(V ).

And thus also P(E) 6 p(V ) and, a fortiori, P(Eße) 6 p(V ).From this and (22) we immediately infer:

Proposition A.1 Let p(×), P(×), and V be as before. Then for all E for which (25) holds andfor all A, we have:

P(A)ìïïïíïïïî

> p(V )p(A | E) + p(V )p(A | Eße) if p(A | E) > p(A | Eße),6 p(V )p(A | E) + p(V )p(A | Eße) if p(A | E) < p(A | Eße),= p(A | E) if p(A | E) = p(A | Eße).

24

So, there is a further way in which the Bayesian agent can learn from uncertain evi-dence: given evidence E/e, she could simply apply Jeffrey conditionalization to the partition{E, Eße, e}. And for the kind of cases in which E/e reports evidence obtained by means ofsome instrument V claims to be veridical, Proposition A.1 establishes some systematic rela-tionships between the agent’s probability for V and the impact which the evidence has on herbelief state.

It seems worthwhile to investigate both the way of learning from uncertain evidence pre-sented here and the one presented in § 4 in much greater detail than has been done in thispaper, and to compare them with one another. Here let me just note that, for the purposesof this paper, the two methods, while they do not give us the exact same results, do in theend amount to the same. To see that they do not give quite the same results, suppose thatp(V | E/e) > p(V ) and that, for some A, we have p(A | E) > p(A | e). Then Proposition A.1will give a lower value as lower bound on P(A) (or, equivalently, p(A | E/e)) than inequal-ity (2) of § 4 will—for, clearly, p(A | e) > p(A | Eße).30 As to the point that for the purposesof this paper it would have made no difference if I had used Jeffrey conditionalization insteadof the theory of expert functions in order to model learning from uncertain evidence, consideragain Example 6.1. Using Proposition A.1 to determine a lower bound on p(R | E/e) givesus, instead of (6), this:

p(R | E/e) > p(V )p(R | E) + p(V )p(R | Eße).(27)

It must be that p(R | Eße) < p(R | e); to err on the safe side, let us say that p(R | Eße) = 0.Then, given the further assumptions we made at this point in the example, we find that, afterthe antirealist has come to know E/e, her probability for R will be at least .18 (i.e., it mayhave decreased relative to her prior probability, which was .3). Proceeding to the second stepof the example, and making the same assumptions as we did in the example, we now find thather new probability for V after she has come to know both E/e and E ¢, equals at least .426.In the third step, again for safety assuming that pE/e,E ¢ (R | E*ß e*) = 0, and again usingProposition A.1 instead of inequality (2), we get that pE/e,E ¢ (R | E*/ e*) ' .277. Here theexample stopped, but we could go on in the same way, and, as one can easily check, if wedid so go on, always assuming favorable data and making basically the same probabilisticassumptions, the antirealist’s probabilities for both V and R would soon come to have lowerbounds greater than .5.

Appendix B: Proof of Proposition 6.1

I assume the same notational conventions as before.

Proposition 6.1 If assumptions (9)–(19) of § 6 hold, then

1. there are E ¢k, El / el Î S such that for all E ¢m Î S with m > k and all En / en Î S withn > l: pm

n (R) > .5; and

2. there are E ¢k¢ , El¢ / el¢ Î S such that for all E ¢m¢ Î S with m¢ > k¢ and all En¢ / en¢ Î S with

n¢ > l¢: pm¢l¢ (V ) > 5.

30On condition that p(Eße) = 0, of course, but this condition is met given that p(A | E) > p(A | e).

25

Proof: One first verifies that, because of assumptions (13), (14), (17), and (18), we caninvoke equations (2) and (4) to calculate lower bounds on, respectively, p

mn-1(R | En / en) and

pm-1n (V | E ¢m-1), for all m, n. Then note that, because, for all m, n, by (12), amn-1(R | En / en) >

pmn-1(R), and by (14), pm-1

n (R | E ¢m) > pm-1n (R), we can set

pmn-1(R | En / en) = pm

n-1(V )rmn-1(R | En / en) + pm

n-1(V )pmn-1(R)(28)

for all m, n in order to obtain a function that “really” only gives a lower bound on pmn-1(R |

En / en). Given equation (28), pmn-1(R | En / en) is a convex combination of rmn-1(R | En / en) and

pmn-1(R) for all m, n, and so, given that it follows from assumptions (12) and (13) that, for all

m, n, we have rmn-1(R | En / en) > pmn-1(R), it holds for all m, n that

pmn-1(R | En / en) > pm

n-1(R).(29)

By conditionalization, we have

pmn-1(R) = pm

n-2(R | En-1/ en-1).(30)

And from (14), (29), and (30) it follows that pmn-1(R | En / en), as defined by (28), is monoton-

ically non-decreasing. Because, clearly, it is also bounded, it converges.

Similarly, one notes that the function defined by

pm-1n (V | E ¢m) = pm-1

n (R)Rm-1n (V | E ¢m) + pm-1

n (R)pm-1n (V )(31)

“really” gives a lower bound on pm-1n (V | E ¢m) for all m, n. It is then proved in the same way

as above that pm-1n (V | E ¢m) is monotonically non-decreasing and hence also converges.

We next show that the function defined by (28) converges to a value greater than .5. Considerthat

pmn-1(R | En / en) = pm

n-1(V )rmn-1(R | En / en) + pm

n-1(V )pmn-1(R)

= pmn-1(V )r

mn-1(R | En / en) + I1 - pm

n-1(V )M pmn-1(R)

= pmn-1(V )r

mn-1(R | En / en) + pm

n-1(R) - pmn-1(V )p

mn-1(R).

(32)

Rearranging gives

pmn-1(R | En / en) - pm

n-1(R) = pmn-1(V )r

mn-1(R | En / en) - pm

n-1(V )pmn-1(R).(33)

That pmn-1(R | En / en) is convergent implies that

limm,n®¥

Apmn-1(R | En / en) - pm

n-2(R | En-1/ en-1)E = 0,(34)

and hence, given (30), also that

limm,n®¥

Apmn-1(R | En / en) - pm

n-1(R)E = 0.(35)

From (33) and (35), it follows that

limm,n®¥

Apmn-1(V )r

mn-1(R | En / en) - pm

n-1(V )pmn-1(R)E = 0,(36)

26

which is the case precisely if either

limm,n®¥

pmn-1(V ) = 0(37)

or

limm,n®¥

pmn-1(R) = lim

m,n®¥r

mn-1(R | En / en).(38)

But because of (9), (18), and the fact that pm-1n (V | E ¢m) is monotonically non-decreasing,

(37) cannot be the case, so that (38) must be. Since that holds for what actually is only a lowerbound on the value of pm

n (R), for all m, n, and since, by (15), limm,n®¥ rmn-1(R | En / en) > .5,

Proposition 6.1.1 must be true.

The proof of Proposition 6.1.2 is analogous.

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