Explanation and the New Riddle of Induction

EXPLANATION AND THE NEW RIDDLE OFINDUCTION

BY BARRY WARD

I propose a novel solution to Goodman’s new riddle of induction, one on which aspects of scien-tific methodology preclude significant confirmation of the Grue Hypothesis. The solution appealsto intuitive constraints on the confirmation of explanatory hypotheses, and can be construed as afragment of a theory of Inference to the Best Explanation. I give it an objective Bayesian formali-sation, and contrast it with Goodman’s and Sober’s solutions, which make appeal to both meth-odological and non-methodological considerations, and those of Jackson, Godfrey-Smith, andWhite, on which explanatory considerations play a very different role.

I. FORMULATING THE PROBLEM

Goodman’s new riddle of induction1 derives from the intuition that posi-tive instances confirm. Let ‘grue’ apply to all things examined before t(where t is some future time) just in case they are green, and to otherthings just in case they are blue, and consider the hypotheses:

(Green) All emeralds are green

(Grue) All emeralds are grue

Contemporarily observed green emeralds provide positive instances ofboth, but it is prima facie paradoxical to suppose they confirm (Grue).Thus, the riddle: on what basis should our confirmation theory discrimi-nate between these cases?

Goodman’s solution depends on accounting (Green) lawlike and (Grue)not, where only lawlike hypotheses can be confirmed by their instances.In addition, he takes the relevant notion of confirmation to require

1 N. Goodman, Fact, Fiction, and Forecast (3rd edition), (Indianapolis and New York:Bobbs-Merrill, 1973).

The Philosophical Quarterly Vol. 62, No. 247 April 2012ISSN 0031-8094 doi: 10.1111/j.1467-9213.2012.00044.x

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confirming that unexamined emeralds conform to the generalisation; it isnot enough to merely confirm the generalisation’s truth. But this divisionof hypotheses into the lawlike / confirmable and non-lawlike / unconfirm-able is untenable. As Wesley Salmon observed,2 if I pluck several figsfrom my tree and find them tasteless, that confirms the prediction thatthe remaining, unplucked figs are also tasteless. We have genuine confir-mation in Goodman’s sense, but ‘all the figs on this tree are tasteless’ isnot lawlike.

However, one does not have to endorse Goodman’s error to recognise,with Hume, the central role of nomic claims in the formation of expecta-tions. Emeralds in particular might originate in different ways. For all weknow at the beginning of our researches, those formed in geological pro-cesses that involve different dynamics of temperature and pressure, or thepresence of different trace elements, might have different colours.3 Conse-quently, if we sample for emeralds without regard for whether we aresampling across the range of nomologically possible emeralds — i.e., atleast across a diverse range of formation conditions, perhaps by examin-ing emeralds formed in different types of strata, or through synthesis inthe laboratory — uniformly finding green ones is just not good evidencethat the generalisation is true. Absent special knowledge that certain no-mologically possible types of emeralds have never, do not, and will not,actually occur, confirmation of the generalisation demands confirmationthat anything constituted as an emerald must invariably be green i.e., con-firmation of the law. This is just inductive commonsense. A scientificresearcher who held the generalisation significantly confirmed while lack-ing evidence regarding a diversity of nomologically possible emeraldswould be held incompetent. So, notwithstanding the over-privileging oflaws in Goodman’s own confirmation theory, a proper treatment of hisparadox must focus on the confirmation of the laws associated with theabove generalisations.

That is the most important reason for making nomic confirmation ourprimary focus. But, for what it’s worth, there is good prima facie evidencethat (Green) standardly expresses a nomic claim, not a mere generalisa-tion. Intuitively, it concerns the emerald’s natural colour i.e., the colourcaused by the emerald’s constitution. We do not falsify it by painting an

2 W.C. Salmon, ‘Four Decades of Scientific Explanation’ pp. 3–219, at p. 49, in W.C.Salmon and P. Kitcher (ed.) Minnesota Studies in the Philosophy of Science, 13, (Minneapolis:University of Minnesota Press, 1989).

3 Numerous types of gemstones have different colours in virtue of the presence of differ-ent trace elements, and many elements have differently coloured allotropes whose forma-tion crucially depends on temperature and pressure e.g., phosphorous and sulphur.

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emerald blue. And we would surely deny it, if we had good theoreticalreasons for believing that some types of emeralds are not green, whetherthey are actualised or not. Thus, the intended hypothesis might be moreaptly expressed as ‘it is a law that all emeralds are (naturally) green.’4

(Grue) also is plausibly nomic. If we somehow contrived to paint blue allthe emeralds unexamined by t, that would no more vindicate (Grue) thanit would refute (Green). So, it also concerns natural colour. And it is intui-tive that it would be refuted if we had good theoretical reasons for believ-ing that emeralds unexamined by t would be green, even if no suchemeralds actually existed.

However, it is important to emphasise that our focus on the nomic isnot crucially motivated by the ordinary language usage of these sentences.The fundamental point is that confirmation of the nomic claims regardingemeralds plays an integral role in the confirmation of the correspondinggeneralisations. If we proceed directly to the latter, we bypass the riddle’sproper formulation and solution.

II. FINDING AN ASYMMETRY

Our task is to find a principled and relevant basis for discriminatingbetween the hypotheses. Exploiting the qualitativeness of the predicatesinvolved has a prima facie appeal: ‘green’ is qualitative, but ‘grue’ is not,since its meaning involves reference to a particular temporal location, t.Goodman, of course, considered this proposal, and his response (pp. 79–80) was devastating. Introducing the predicate ‘bleen’ as applying to allthings examined before t just in case they are blue and to other thingsjust in case they are green, we see that ‘green’ applies to things examinedbefore t just in case they are grue and to other things just in case theyare bleen. Thus, which predicates count as qualitative depends on ourchoice of primitives, and it is very hard to see a non-question-beggingbasis for making that choice.

This symmetry is the root of the deep and distinctive difficulty ofGoodman’s paradox, and in the context of mere generalisations, it is hardto see how it can be evaded. If we substitute ‘grue if examined before t

4 One technicality should be noted. Since (Green) is not intended to hold for caseswhere we paint an emerald or otherwise interfere with its natural colour, it apparentlyincorporates some species of Ceteris Paribus (CP) clause. I cannot address concerns regard-ing the tenability of CP laws here, but see my ‘The Natural Kind Analysis of Ceteris Pari-bus Law Statements’, Philosophical Topics, 35 (2007), pp. 359–380. In the following, I restrictmy discussion of confirmation to cases where the CP clause is satisfied.

EXPLANATION AND THE NEW RIDDLE 367


and bleen otherwise’ for ‘green’ in ‘all emeralds are green’ we obtain anequivalent generalisation, one with the same truth conditions. Substituting‘green if examined before t and blue otherwise’ for ‘grue’ in ‘all emeraldsare grue’ yields another such pair. We have two pairs of equivalent gener-alisations and no basis for favouring one pair over the other. So, we haveno basis for favouring ‘green’ over ‘grue’ and (Green)’s confirmation overthat of (Grue).

Once we recognise law confirmation as our primary focus, however,things are significantly different. In the law context, substitution of coex-tensive expressions may yield distinct laws.5 Indeed, the laws may differeven when the expressions are necessarily coextensive. From here, I shalluse ‘(Green)’ and ‘(Grue)’ to refer to the law claims. We obtain the law(Green*) by substituting ‘grue if examined before t and bleen otherwise’for ‘green’ in (Green):

(Green) All emeralds are green.

(Green*) All emeralds are grue if examined before t and bleen otherwise.

Both laws entail the generalisation that all emeralds are green. How-ever, they differ in their explanations of this regularity. According to(Green) the time of examination is irrelevant to an emerald’s colour.But according to (Green*), an emerald, X, that I have just examined isgreen because it is grue and was examined before t. An emerald, Y,not examined before t, is green because it is bleen and was not exam-ined before t. So, according to (Green*), time of examination is explan-atorily relevant to the colour of each emerald. Such a law — onEinstein’s theological characterisation — would be the work of a godthat is not merely subtle but malicious in that it conceals this depen-dence from observation by specifying that emeralds examined beforeand after t have different grolours contrived just so that emeralds neces-sarily manifest as green. Nevertheless, the supported explanations andthe laws are distinct. Thus, without getting into the details of the confir-mation theory just yet, confirmation of (Green) and (Green*) need notgo hand in hand.

‘Grue’ is associated with two law statements:

(Grue) All emeralds are grue.

(Grue*) All emeralds are green if examined before t and blue otherwise

5 Fred Dretske makes the related observation that the class of laws is not closed undersubstitution of coextensive expressions in ‘Laws of Nature’, Philosophy of Science, 44 (1977),pp. 248–268, at p. 250.

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As with (Green) and (Green*), despite the fact that they are related bysubstitution of coextensive expressions, these laws may be distinct. How-ever, there is a crucial semantical difference between ‘green’ and ‘grue’.My use of ‘green’ need not be mediated by my use of grolour terminol-ogy. I can apply the term, even if the predicates ‘grue’ and ‘bleen’ arenot in my vocabulary, as indeed is the case for the great majority ofhumanity. By contrast, Goodman introduces ‘grue’ by explicating it interms of ‘green’, ‘blue’, and ‘time of examination’, and we neither have apre-existing concept of grueness that is independent of our colour con-cepts, nor are we told anything about grueness that would provide thebasis for one. The only grasp we have on grueness is that grue things aregreen if examined before t and blue otherwise. So, if it is a law that allemeralds are grue, it must be a law that all emeralds are green if exam-ined before t and blue otherwise. Like (Green*) and (Grue*) which renderthe explanatory dependence explicit, (Grue) also entails an explanatorydependence on time of examination i.e., (Grue) at least entails (Grue*).So, there is an asymmetry that holds between (Green) and the other threelaws: it is the only one that does not support an explanatory dependenceon time of examination. This is the asymmetry which shall ultimatelyresolve the paradox.

But first, we must clarify our position on ‘grue’ and (Grue). The impor-tant claim is that (Grue), despite the absence of explicit reference to timeof examination, nevertheless supports the corresponding explanatorydependence. This is justified by a claim about the conditions for correctlyapplying ‘grue’. However, it does not depend upon an assumption thatGoodman defines ‘grue’ as ‘green if examined before t and blue otherwise.’Goodman (pp. 79–80) speaks only of explaining grue in terms of greenand blue, or conversely explaining green in terms of grue and bleen. So,we could, for instance, treat ‘grue’ as a natural kind term. The factremains, our sole guide to its correct use is that it applies to things thatare green and examined before t and blue otherwise. In the terms of anatural kind semantics6 we would read this application condition as theexplanatory stereotype for ‘grue’, the feature of the putative paradigms tobe explained by whatever property grueness turns out to be. Grueness isthat property that explains being green if examined before t and blueotherwise. Thus, it remains the case that, in virtue of the manner inwhich the meaning of ‘grue’ is specified, (Grue) supports the explanatorydependence made explicit in (Grue*).

6 See, for instance, H. Putnam, ‘Explanation and Reference’ in G. Pearce and P. May-nard (ed.) Conceptual Change, (Dordrecht: Reidel, 1973), pp. 199–221.



Detailed semantical stories aside, an emerald qualifies as grue only if itis green if examined before t and blue otherwise. This is an ineliminablefeature of Goodman’s paradox: for the same data to confirm (Grue) and(Green) we must be willing to account green emeralds examined before tas grue, and inductive projections using (Grue) are paradoxical preciselybecause we account grue emeralds unexamined by t as blue. And in thelaw context, those application conditions support the associated explana-tory dependence on time of examination. By contrast, we don’t have tounderstand green emeralds as being grue if examined before t and bleenotherwise. Thus, in the law context, greenness does not automatically sup-port such a dependence. So, the asymmetry is robust.

We shall use it to argue that only (Green) should be significantly con-firmed. The other three should be given scant credence. Does thisdemand, counterintuitively, that we deny the generalisation ‘all emeraldsare grue if examined before t and bleen otherwise’ despite its having thesame truth conditions as the generalisation ‘all emeralds are green’? No,given the specification of the meanings of the predicates, we take that grue-some generalisation as true if and only if all emeralds are green. What(Green*)’s rejection motivates is denying ‘grue’ and ‘bleen’ the status ofnatural properties. How we should ultimately understand the notion of anatural property is a metaphysical issue that lies beyond the scope of thispaper. However, our solution will speak to their epistemology, in effectproviding a constraint on confirmation that a property is natural.

III. AN INFORMAL SOLUTION

(Grue)’s support of an explanatory dependence on time of examination isan issue that has been broached elsewhere. Rosemarie Rheinwald tookthe Grue hypothesis to entail that if an emerald is examined before t, itchanges colour, but as Catherine Elgin pointed out, this is not so.7 How-ever, Elgin further claimed that even allowing that the hypothesis is law-like and supports the counterfactual ‘if an emerald that is in fact blue hadbeen examined before t, it would have been green’, does not entail thatits having been examined before t would have caused it to be green.Here, I disagree. Certainly, the law does not imply that any emerald everchanges its colour. But neither this point nor the well-known deficienciesof naive counterfactual analyses of causation make it reasonable to deny

7 R. Rheinwald, ‘An Epistemic Solution to Goodman’s New Riddle of Induction’, Syn-these, 95, 1 (1993), pp. 55–76, at p. 65, and C. Elgin, ‘Outstanding Problems: Replies to ZiFCritics’, Synthese, 95, 1 (1993), pp. 129–140, at pp. 130–132.

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that the law supports causal dependencies on time of examination andthe associated explanations.

The causal claims are indeed strange. Given Goodman’s specifica-tion of ‘grue’, an emerald first examined in 2011 will, according to(Grue), have been green for as long as it existed prior to that, yet itsgreenness is explained by its examination. A currently existing emeraldthat will not be examined until after t has been blue since its origin invirtue of that fact. So, these explanations involve backward causation,but that provides no basis whatsoever for denying that (Grue) supportsthem.

It is a regrettable, potentially confusing feature of the paradox asposed by Goodman that the gruesome nomic claims have this feature,and we could exploit it to resolve the paradox, claiming that hypothesesthat support backward-directed explanations are unworthy of confirma-tion on those grounds alone. However, backward-directedness is not thefundamental problem with (Grue). The law that all emeralds aregrue**, where ‘grue**’ applies to things that are blue for all times priorto examination and green from time of first examination, only supportsthe normal, non-backwards, explanation that being examined causesnaturally blue emeralds to become green. Yet its confirmation by obser-vations of green emeralds is similarly paradoxical, since they intuitivelylend no support to the claim that yet unexamined emeralds are cur-rently blue. The solution that follows, and the prominent solutions withwhich I contrast it, treat this variant hypothesis and (Grue) in exactlythe same way. The backward directedness of the (Grue)-explanations isincidental to the puzzle Goodman posed, and, from here, we disregardit.

So, to the confirmation theory. (Grue) supports a pair of explanations.For an emerald X examined before t:

X is green, because it was examined before t.

For an emerald, Y, not examined before t:

Y is blue, because it will not be examined before t.

These explanations are intimately related to subjunctives and counterfac-tuals that (Grue) supports. For the former:

If X were examined before t, it would be green

If X had not been examined before t, it would have been blue.



For the latter:

If Y had been examined before t, it would have been green,

If Y were not examined before t, it would be blue.

Intuitively, significantly confirming an explanation demands evidence forthe conditionals that underwrite it. In particular, confirming the formerexplanation demands we determine that:

Contemporarily, we cannot obtain evidence for (ii). Hence, we cannot sig-nificantly confirm the posited dependence of colour on time of first exam-ination. So, we cannot significantly confirm (Grue) or any law thatsupports that explanation. (Green) does not posit a contemporarily unveri-fiable dependency. So, we have a confirmational asymmetry between(Green) and (Grue).

The intuition invoked above is compelling, independent of any consid-eration of Goodman’s paradox. In general, to significantly confirm a law,we must confirm that the posited difference-makers make a difference.And absent well-confirmed, more general laws that subsume the law inquestion, this demands data which shows that varying the explanans vari-able yields the posited variations in the explanandum variable. This iswhat we test when we perform controlled experiments. Without worryingabout its precise formulation for the moment, let us call this necessarycondition on the confirmation of a law, ‘Confirmation of Difference-Mak-ing’, (CD)’ for short.8

(CD) also fits our intuitions regarding when we might reasonably con-firm (Grue). Imagine we have several processes for manufacturing emer-alds. Prior to t, all yield only green emeralds; from t on, only blue. Themost exacting scrutiny of the post-t processes cannot differentiate themfrom the respective pre-t processes. We build multiple emerald-productionmachines in different environments and vary every background parameterwe can, to no avail. Ultimately, we are compelled to accept that emeraldcolour is subject to some gruesome dependence, whether it is time of pro-duction or time of first examination. We subsequently dig up emeralds

(i) Some emerald examined prior to t is green, and

(ii) Some emerald unexamined prior to t is not green.

8 Incidentally, (Green*) is immune to significant confirmation, given (CD). The lawimplies that green emeralds examined before t are green because they are grue. Signifi-cantly confirming this explanatory dependence demands we observe an emerald that is firstexamined after t and blue. However, since (Green*) demands that all emeralds unexam-ined by t are bleen, such data would just refute the law.

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that are older than t, but have not been examined until now. They areblue. We find emeralds from the same strata that were dug up by pre-thumans, and they are green. We are now confident that emerald colourcounterfactually depends on time of first examination, and that it isexplained by time of first examination, and we can significantly confirm(Grue). (Grue)’s confirmation requires more than (CD) demands, butmeeting its demand seems crucial.

(CD) precludes (Grue)’s significant confirmation, but not (Green)’s.However, it is important to make it clear that (Green)’s confirmation isnot rendered problematic by the general picture being sketched here. Inparticular, in section I we characterised law confirmation as demandingwe sample a diversity of nomologically possible types of emeralds, and wemight suspect there is a tension between this general idea and the claimthat (Green) can be significantly confirmed by currently available data. Ifwe cannot research the colour of emeralds first examined after t, how canwe be said to have properly explored the diversity of nomological possibil-ities? (CD) has a role to play here too.

Good science does not ignore the possibility of hitherto undetectedexplanatory dependencies and blithely proclaim the simplest law signifi-cantly confirmed by the slenderest of data. When we test a law, we typicallycome with background beliefs regarding which parameters might notimplausibly be relevant based on the evidence of difference-making we haveto hand, and where possible, we vary those parameters. In the case of emer-alds, we have some reason to suppose the presence of trace elements orother variations in formation conditions might matter, since they sometimesdo for other solids. We test for their relevance when we research emeraldsformed in different conditions, and such testing is mandatory if we are tosignificantly confirm (Green). In addition to researching parameters whoserelevance is not implausible, we may take a punt on ones that we have littleor no prior reason to suppose might be relevant, although that is more likelyto happen in the context of discovery when we are largely in the dark, fish-ing around for anything that might help explain a particular effect.

On the other hand, we are not required to search endlessly for possibleundetected dependencies. When we have no reason to suppose that varia-tions in some parameter we pluck from the air would disconfirm thecounterfactual robustness of a regularity, its significant confirmation doesnot require evidence regarding variations in that parameter. The obliga-tions that regulate our practice in regard of data-collection are not soabsurdly stringent. Unwillingness to check whether the boiling point ofwater is affected by my mother being located at the North Pole is not adereliction of scientific duty.



Absence of evidence is not decisive evidence of absence, but as per(CD), it is a reason to not significantly confirm such dependencies. It is areason to give scant credence to the possibility that being examined beforet is relevant to the colour of emeralds, and so it is a reason to hold that therestriction of our sample to emeralds examined before t does not compro-mise its representativeness. The constraints that regulate scientific data col-lection — and I do not claim to have characterised them in detail — donot demand evidence regarding emeralds unexamined before t for (Green)to be significantly confirmed. So, (CD) in tandem with those constraintsunderwrites the legitimacy of (Green)’s contemporary confirmation.

Two last points are worth making. First, there are resolutions of theparadox that appeal to background beliefs that might themselves seemequally problematic in light of the paradox. For instance, Elliot Sober’sview, which we discuss later, is that we confirm all emeralds are green,because we have a prior commitment to an object’s colour superveningon its physical constitution. It should be clear that our account is notproblematic in this way. We are not making a brute appeal to back-ground beliefs regarding which parameters might plausibly be relevantwhen considering the representativeness of our sample. On our view,these beliefs are methodologically constrained, at least by (CD).

Second, once we favour a solution that does not discriminate againstepistemically possible laws on non-methodological grounds, commitmentto (CD) is well-nigh mandatory. We inevitably confirm any general lawdespite an infinity of epistemically possible explanatory dependenciesthat are inconsistent with its truth. If significant confirmation demandedevidence that spoke directly against all such dependencies, rather thanthe absence of evidence in their favour, it would be unobtainable. Amethodological solution demands that (CD), or something very similar,is operative in scientific practice, since its rejection is tantamount toinductive skepticism.

IV. FORMALISING THE SOLUTION

We shall first give (CD) a more precise formulation in terms of two con-straints:

(SCL) If a law posits / supports a difference-making explanatory dependence that is

not significantly confirmed, then the law is not significantly confirmed.

(SCED) Significant confirmation of a difference-making explanatory dependence

requires evidence of at least two instances that manifest the difference-making (i.e.,

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two distinct sets of values of the explanans variables and two corresponding, dis-

tinct, confirming values of the explanandum variable), provided that the depen-

dence is not a restricted case of a broader explanatory dependence that is

significantly confirmed.

The difference-making explanatory dependence supported by (Grue) isexpressed by the mappings:

Time of first examination < t ? Green

Time of first examination � t ? Blue

Disregarding for a moment the proviso in (SCED), the constraints workin tandem to preclude significant confirmation of (Grue). Since we do nothave an instance corresponding to the second value of the explanans vari-able, (SCED) dictates the explanatory dependence is not significantly con-firmed. Hence, (SCL) dictates the law is not significantly confirmed.

The proviso allows for the fact that an explanatory dependence maybe significantly confirmed solely in virtue of the confirmation of a broaderdependence. For instance, Coulomb’s law specifies that particles of chargeq1 and q2, separated by distance r, exert a force on each other of magni-tude F = q1q2/r

2, where F is repulsive if positive and attractive if negative.It supports broader dependencies of the force on the values of the chargesthan those supported by ‘for all charges with magnitudes between73.0000001 and 73.0000002 Coulombs, at separation r = 1.00003 meters,F = q1q2/(1.00003)

2’. And on pain of something very close to inductiveskepticism, we must be able to significantly confirm such narrower depen-dencies without actually observing any instances in the specified range.So, we need the proviso.

Given its inclusion, (SCED) allows the possibility that (Grue) could besignificantly confirmed without direct evidence of a difference-making pairof emeralds, if we had, for instance, good evidence that other gemstoneshad gruesome dependencies that were subject to some systematic relationbetween gemstone-type and grolour that entailed (Grue). Of course, infact, no such hypothesis is well confirmed.

We could certainly spend some time discussing ‘broader explanatorydependence’. However, a weak constraint on the notion will suffice forour purposes: a broader dependence must entail the narrower. Since theproviso demands that the broader dependence is significantly confirmed,hypotheses generated by simply ‘cutting-and-pasting’ (Grue) and a genu-inely well-confirmed hypothesis will not prevent (SCED) from precluding(Grue)’s confirmation. For instance, while ‘all rubies are red and all emer-alds are grue’, understood as a nomic claim, does entail (Grue), it does



not satisfy the proviso. Evidence that confirms the first conjunct does notsignificantly confirm the second, and hence, the entire conjunction is notsignificantly confirmed. What, in general, distinguishes broader explana-tory dependencies from mere concatenations of distinct dependencies isan important question, but our solution does not require an answer.

(SCED) is intended to be weak. It is a necessary condition for significantconfirmation, but I doubt its meagre demand is ever sufficient. Yet, itmight appear too permissive. Suppose we have a law, y = f (u, v), that sup-ports a dependence of y on two explanans variables, u and v. To confirmthe posited dependence of y on u, (SCED) only requires two data pointsfor which y1 = f (u1, v1) and y2 = f (u2, v1). However, (SCED) also allows thatwe could take these two instances as confirming the dependence of y onthe pair (u, v). This might seem to permit gruesome confirmations of thekind we wish to avoid. Let u be the gemstone-type, and v the time. (SCED)is consistent with the significant confirmation by contemporary data of thef specified below:

f(emerald, t < 3,000 A.D.) = green

f(emerald, t � 3000 A.D.) = blue

f(ruby, t < 3,000 A.D.) = red

f(ruby, t � 3000 A.D.) = blue

Contemporary observations of a green emerald and a red ruby providetwo instances that manifest the specified difference-making with respect tothe first explanans variable, and it is consistent with (SCED) that suchdata significantly confirms the dependence of color on (gemstone-type,time) specified by f. This might seem problematic. However, even though(SCED) permits that the dependence on the first variable may be signifi-cantly confirmed, and that the dependence on the 2-tuple (gemstone-type,time) may be significantly confirmed, it does not allow that the depen-dence on the second variable has been significantly confirmed: we do nothave two instances that manifest difference-making with respect to time ofobservation. Hence, crucially, since (SCL) demands that a law is not sig-nificantly confirmed if any posited explanatory dependence is not signifi-cantly confirmed, the law that specifies the gruesome dependence, f, isnot significantly confirmed by such data. The two constraints worktogether to preclude significant confirmation of the gruesome law.

We should consider one other type of example, due to Donald David-son.9 Specifying ‘emeruby’ as applying to emeralds first examined before t

9 D. Davidson, ‘Emeroses by Other Names’, Journal of Philosophy, 63 (1966), pp. 778–779.

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and rubies not examined before t, and ‘gred’ as green and examinedbefore t and red otherwise, ‘all emerubies are gred’ is a counterexampleto Goodman’s theory of projectibility, and variations on this theme, wheresome gruesome predicate is placed in the antecedent position, have raisedtrouble for other proposals10. However, on our analysis, such examplesrelevantly differ from the (Grue) case. Our solution concerns hypothesesthat posit a difference-making explanation, but ‘all emerubies are gred’does not. It does not tell us, for instance, that emeralds not examinedbefore t are red. It tells us nothing about them, and nothing about rubiesexamined before t.11 Similarly, ‘all emerubies are green’ does not supporta difference-making explanation; it simply asserts that all emerubies sharethe one colour. Placing a gruesome predicate in the antecedent positionrestricts the scope of these hypotheses to odd, gerrymandered classes ofobjects, and a rational agent should presumably not significantly confirmthese claims based solely on observations of contemporary emeralds, butthey pose no threat to our solution. Of course, we can construct examplesthat combine both problems using gruesome predicates with different val-ues of t. Defining ‘emeruby’ as applying to emeralds first examined before3,000 A.D. and rubies examined after 3,000 A.D., and ‘gred’ as green iffirst examined before 2,500 A.D. and red otherwise, ‘All emerubies aregred’ now supports difference-making explanations. But this is unpro-blematic. In addition to whatever reasons for unconfirmability it shareswith Davidson’s example, (SCED) and (SCL) preclude its significant con-firmation prior to 2,500 A.D.

It is worth noting that our approach shows some promise for treatinglaws with gruesome antecedents. Given our confidence in (Green) and thelaw that all rubies are red, we confidently endorse the truth of ‘all emeru-bies are gred’, but it should not be confirmed qua law. A generalisation ofour approach delivers this verdict. ‘Emeruby’ insofar as it pertains toemeralds, arbitrarily restricts the scope of the law to emeralds examinedbefore t, and we have no evidence this restriction makes a difference i.e.,that emeralds examined before t differ from other emeralds in regard oftheir natural colour. Just as (SCED) and (SCL) demand evidence of differ-ence-making for each explanatory dependence within the scope of a law,it seems reasonable that significant confirmation of a law demands evi-dence that the features used to delimit its scope make a difference with

10 P. Godfrey-Smith, ‘Goodman’s Problem and Scientific Methodology’, Journal of Philos-ophy, 100, (2003), pp. 573–590, at p. 576, exploits one to critique the solution proposed in F.Jackson, ‘Grue’, Journal of Philosophy, 77 (1975), pp. 113–131.

11 Also, it does not satisfy the proviso in (SCED). It is entailed by the well confirmedconjunction of laws ‘all emeralds are green and all rubies are red’.



regard to the property projected. Indeed we can make the relationshipwith (SCED) and (SCL) more intimate by considering any broader lawthat covers the colour of emeralds for which being examined before t is adifference-maker. By (SCED) and (SCL) we should not significantly con-firm the broader dependence, and it seems reasonable that this go hand-in-hand with unwillingness to significantly confirm a law that, in virtue ofits restricted scope, marks being examined before t as salient to colour.The same point applies mutatis mutandis to the restriction of the law’s scopeto rubies first examined after t.

Of course, many things are green, not just emeralds. How then, do wereconcile the putative lawfulness of (Green) with the demand that beingan emerald is the property that makes a difference regarding colour? Weshould not. It is enough that being an emerald is a property that makes adifference. Similarly, it is a law, albeit a parochial one, that objects at theEarth’s surface are subject to a gravitational acceleration of g = 9.81ms�2, notwithstanding the fact that we could derive a more general lawsubsuming all objects with radii and mass densities appropriately relatedto yield a surface gravitational acceleration of g. Even if a deeper lawidentifies a unique explanans that marks the difference between greenthings and non-green things in general, emerald is one type of realiser ofthat explanans, and we may reasonably conjecture that a suitable sense ofdifference-maker captures this, vindicating (Green)’s confirmability as an,admittedly parochial, law.

We must place our treatment in the broader context of a confirmationtheory. I shall take a Bayesian approach. Now, there has been considerablediscussion of whether Inference to the Best Explanation (IBE) and Bayesian-ism can be sensibly combined. How does this bear on our account? First,some terminological issues. Bayesianism does not speak of belief, but ofdegrees of belief, and hence, does not directly accommodate inferring thetruth of a hypothesis. So, if we are to be Bayesians, the ‘I’ in ‘IBE’ should betaken with a grain of salt. Also, if the best available explanation is manifestlypoor, confidence in its truth will be irrational. So the ‘B’ should not be takenwithout qualification. The ‘E’ holds up pretty well, though. The core issuehere is whether explanatory considerations can play a fundamental role in aBayesian confirmation theory. Granting that IBE, suitably clarified, neithercommits us to a non-probabilistic epistemology nor to significantly confirm-ing bad explanations, we can usefully categorise our constraints as a smallfragment of a theory of IBE.

More substantively, how do we mesh these constraints with themachinery of Bayesianism? Bayes’s theorem states that the posterior prob-ability of a hypothesis, L, in light of evidence, e, and background beliefs,

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K, is P(L|e & K) = P(e|L & K).P(L|K)/P(e|K), where P(L|K) and P(e|K)are, respectively, the prior probabilities of L and e given K, and P(e|L &K) is the likelihood of e given L & K. Combining (SCL) and (SCED), wehave:

(SCL*) Let L be a law statement, K a specification of background beliefs, and P a

degree of belief distribution that does not assign significant probability to any

explanatory dependence of which a dependence supported by L is a restricted case.

Then, if L supports a difference-making explanatory dependence for which K does

not include evidence of at least two instances that manifest the difference-making,

P(L|K) � 0.

For K to provide evidence of difference-making instances, e1 and e2, wedo not require that P(e1|K) = P(e2|K) = 1, but K must render e1 and e2sufficiently plausible to count as good data. Of course, we do not wish todogmatically preclude L’s confirmation. So, by P(L|K) � 0 it is intendedthat P(L|K) > 0, but sufficiently low that no rational agent would hold thelaw trustworthy across the supported range of nomological possibilities.12

(SCL*) demands that agents like ourselves with background beliefs, K1,that do not include observation reports of emeralds first examined atsome time � t should have P((Grue)|K1) � 0. It also demands that condi-tionalisation on arbitrary data e1, e2,…, en that, conjoined with K1, doesnot include a difference-making pair, will fail to significantly confirm(Grue). Define P* = P(.|K1). Then P*((Grue)|e1 & e2 &…& en) = P((Grue)|K1 & e1 & e2 &…& en), and since K1 & e1 & e2 &…& en does not includea difference-making pair, (SCL*) demands P((Grue)|K1 & e1 & e2 &…&en) � 0, and hence, P*((Grue)|e1 & e2 &…& en) � 0. By contrast, (Green)is readily confirmed by conditionalisation on observations of green emer-alds. Hence, reasonable agents that invariably find green emeralds willhave posterior probabilities for which P*((Green|e1 & e2 &…& en))) is sig-nificant, and hence, significantly exceeds P*((Grue)|e1 & e2 &…& en). Andsince the probability of a hypothesis must be at least that of any thatentails it, (Green)’s significant confirmation guarantees that each unexam-ined emerald’s conformity is also significantly confirmed.

What kind of theory do we have here? As detailed above, our explan-atory considerations are naturally read as constraints on our priors, inparticular our priors for explanatory hypotheses conditional on state-ments of evidence. So, the view is a species of objective Bayesianism,albeit distinct from traditional objective views that invoke principles ofindifference or maximisation of entropy. It also contrasts with the recon-

12 ‘Sufficiently plausible’ and ‘sufficiently low’ are admittedly vague, but acceptably soin this context.



ciliation of subjective Bayesianism and IBE advocated by Peter Lipton13

and others. This is a good thing. If we don’t demand that (SCL*) isobeyed, we don’t solve the riddle. More generally, as Jonathan Weisbergpersuasively argues14, IBE is the loser in a marriage with subjectiveBayesianism, since freedom to pick priors is a license to ignore explana-tory considerations.

Bas van Fraassen has a well-known argument that IBE and Bayesian-ism are incompatible.15 If an agent implements IBE by following a rulethat gives a probability boost to hypotheses that best explain the data inaddition to the boost provided by conditionalising on that data, then sheis clearly not being a good Bayesian, since she favours a distinct updatingrule over conditionalisation. She is also susceptible to a diachronic Dutchbook, and might thereby be convicted of irrationality. However, (SCL*) isnot a rule for updating under the impact of new evidence. Thus, confor-mity to (SLC*) does not conflict with the requirement that we update onnew evidence by conditionalisation, and does not make us vulnerable toDutch Book.

Of course, if you are one of those rare folk for whom contemporarygreen emeralds make (Grue) seem immensely compelling, (SCL*) doescounsel that you adopt a new set of priors. Is this demand objectionable?Conformity to (SCL*) is plausibly implicit in the most mundane examplesof nomic inference, and its violations include the most flagrant examplesof inductive irrationality. Those who, based on the evidence available toyou and me, are convinced that jumping out a high window will result inflight rather than serious injury violate (SCL*), at least if they formulatetheir expectations by appeal to some law. Subjective Bayesians, by con-trast, may happily account such an agent as rational. Conformity to(SCL*) seems like a reasonable addition to our characterisation of scien-tific rationality.

V. COMPARISONS

Comparisons with two kinds of solutions seem pertinent: those that makeexplicit appeal to non-methodological factors, and those that appeal tomethodological factors in the same neighbourhood as ours, invokingcounterfactuals or explanatory considerations. Goodman’s own approach,

13 P. Lipton, Inference to the Best Explanation (Second Edition), (New York: Routl-edge, 2004).

14 J. Weisberg, ‘Locating IBE in the Bayesian Framework’, Synthese, 167 (2009), pp. 125–143.15 B. van Fraassen, Laws and Symmetry, (Oxford UP, 1989).

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set forth in chapter IV of Fact, Fiction, and Forecast, is of the first kind, sinceit rests, in part, on contingent features of the history of scientific and com-monsense inductions. ‘All emeralds are grue’ is not confirmable by itspositive instances, because unlike ‘all emeralds are green’ it is not lawlike.Lawlike hypotheses involve only projectible predicates, and projectibility,in turn, is determined by entrenchment. The predicate ‘green’ is aveteran of successful inductions, and hence, it and all coextensive predi-cates are well entrenched, whereas ‘grue’ is not. But the fact that ‘green’is better entrenched is, for all Goodman’s account has to say, an accidentof history.

We already noted one problem with Goodman’s approach in section I:non-lawlike hypotheses may well be confirmed by their instances. Good-man’s appeal to historical contingencies is also a concern. We keep onkeeping on as we did before, unless hypotheses formulated with projecti-ble predicates fail us, but there is no account of why our ancestors shouldgenerally prefer hypotheses formulated with non-gruesome predicates overtheir gruesome competitors in the first place. For all Goodman says, itjust happens that in each case it is so. Other things being equal, an analy-sis that, like ours, provides a unified explanation of such preferences is tobe favoured.

Elliott Sober16 has provided a subjective Bayesian discussion of theparadox. Using Bayes’s theorem in the form, P(h|e) = P(e|h).P(h)/P(e), andtaking it that e confirms h1 more than h2 if and only if P(h1|e) – P(h1) >P(h2|e) – P(h2)

17, it follows that if h1 and h2 both entail e, then e confirmsh1 more than h2 if and only if h1’s prior probability exceeds that of h2.Thus, green emeralds sampled from the population of emeralds confirm(Green) more than (Grue) if and only if the former has the larger prior.Sober (p. 231, p. 236) writes:

‘If we find ALLGREEN more plausible than ALLGRUE, this is because we hold

various substantive, if hard to articulate, theories about the world. Perhaps we

expect emeralds to be alike in color because we think that they are alike in physical

structure, and we believe that color supervenes on physical structure. Of course,

these convictions involve assumptions about the future. But there is no escaping

such commitments; based solely on our experience of the past, the generalisations

cannot be shown to differ in their probabilities…

16 E. Sober, ‘No Model, No Inference: A Bayesian Primer on the Grue Problem’, in D.Stalker (ed.), Grue! The New Riddle of Induction, (Chicago and LaSalle, Illinois: Open Court,1994), pp. 225–240.

17 There are several useful measures of confirmation, but nothing in this discussion cru-cially depends on Sober’s choice.



It isn’t reason alone (or ‘the scientific method’) that induces an asymmetry here,

but substantive assumptions about the way the world is.’

Sober may well be right about emeralds. Confronted with objects of anovel physical type, we generally take it for granted they are coloured notgroloured. However, this just pushes the problem back. Why do weaccept that objects of a given physical type are coloured? Appeal to theunanimously held belief that colour supervenes on physical structure ishard to square with the subjectivist tenor of Sober’s Bayesian analysis —it’s not as if the evidence would have driven us to consensus if someagents had initially favoured gruesome hypotheses. Our solution explainsthe unanimity: (CD) just as easily explains why we expect objects to becoloured but not groloured, as it explains why we expect emeralds to begreen and not grue.

Nevertheless, if we grant Sober such unanimously held backgroundbeliefs, then his view, unlike Goodman’s, might seem to provide a readyexplanation of our general preference for non-gruesome predicates: sim-ply impute the unanimous background belief that all of an object’s prop-erties supervene on its physical structure. However, since colour andother non-gruesome properties supervene on physical structure only ifphysical structure is relevantly non-gruesome, this solution posits, ratherthan explains, a general preference for non-gruesome properties. Explana-tion must stop somewhere, but this one stops before it explains that pref-erence.

Given the kind of methodological constraints to which our accountappeals, reliance on the scientific method is not be contrasted with mak-ing substantive assumptions about the world: our general inductive preju-dices, partially codified by (CD), embody such assumptions. However, theprima facie appeal of our methodological story is that it provides a unifiedexplanation of our general preference for non-gruesome predicates. It isnot at all clear that non-methodological considerations can do the same.

The second kind of solution originates with Frank Jackson (1975), buthas been developed in distinctive ways by Peter Godfrey-Smith (2003)and, most recently, Roger White.18 These authors address the confirma-tion of (Green) and (Grue) construed as generalisations. Jackson (p. 123)claims the relevant inductions are governed by a counterfactual condition:

‘…certain Fs which are H being G does not support other Fs which are not H

being G if it is known that the Fs in the evidence class would not have been G if

they had not been H.’

18 R. White, ‘Explanation as a Guide to Induction’, Philosophers’ Imprint, 5 (2005), pp. 1–29.

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To use one of Jackson’s examples, if I know that all the observed dia-monds (F) have glinted (G) because they have been polished (H), it isplainly unreasonable to infer from their glinting that other, unpolisheddiamonds will glint. Similarly, since I know that observed emeralds (F)have been grue (G) because they have been observed (H), it is unreason-able to infer that other, unobserved emeralds are grue. By contrast,observed green emeralds would have been green if unobserved, and thecounterfactual condition does not block the commonsense induction.

As earlier noted, Godfrey-Smith has counterexampled Jackson’s condi-tion, and in the same article he counterexamples a revised version proposedby Jackson and Roger Pargetter.19 More fundamentally, Jackson’s solutionis incomplete. Background knowledge that observed emeralds would nothave been grue had they been unobserved and that observed emeraldswould have been green had they been unobserved either presupposes aconfirmational standard for nomic claims that favours green over grue —one that is not provided by Jackson’s counterfactual condition — or, likeGoodman’s and Sober’s, must rest on an appeal to non-methodologicalassumptions. Jackson does not specify a favoured option.

Godfrey-Smith addresses this concern by proposing a division of meth-odological labor. In understanding the paradox and induction more gen-erally, we must recognise at least two quite distinct modes of inference.There is inference regarding the nomic relations that obtain betweenproperties, and there is statistical inference regarding the truth of generali-sations, which is appropriate when we can legitimately judge whether asample is representative of the population from which it is drawn. Theformer mode of inference is prior to the latter, providing knowledge ofthe counterfactuals that underwrite judgments of representativeness. Help-ing himself to the counterfactuals, Godfrey-Smith gives a statistical analy-sis that identifies confirmation of the generalisation ‘all emeralds are grue’as mistaken, because it involves a close relative of the problem of con-founding. We have confounding, when we seek to judge whether a vari-able Y is causally dependent on variable X, but an additional variable, Z,non-randomly associated with, but not causally dependent on X, affectsY. Thus, if smokers have lower rates of heart-disease, because theyexercise more than non-smokers, statistics regarding the rates of heart-disease in the smoking and non-smoking populations will not provide

19 F. Jackson and R. Pargetter, ‘Confirmation and the Nomological’, Canadian Journal ofPhilosophy, 10 (1980), pp. 415–428. Elliott Sober also counters Jackson and Pargetter in ‘Con-firmation and Law-Likeness’, Philosophical Review, 97, (1988), pp. 93–98, and Charles Chiha-ra provides a counterexample to Jackson’s original proposal in ‘Quine and theConfirmational Paradoxes’, Midwest Studies in Philosophy, 6 (1982): pp. 425–452.



evidence for smoking causing heart-disease, even though there is a causalconnection. The amount of exercise is the confounding variable. In theGrue case, granted the counterfactuals employed by Jackson, we knowthat sampled emeralds are grue only because they have been observed.Observation itself plays a role analogous to confounding in causalinference — it includes the observed emerald in the extension of ‘grue’ —undermining the claim that the grueness of our sampled emeralds is rep-resentative of the population from which they are drawn.

White explicitly brings explanatory concerns to bear on the paradox.He claims the fact that all observed emeralds are grue is just the fact thatall observed emeralds are green, and argues that ‘all emeralds are green’better explains that fact (pp. 16–17):

‘…the explanation ‘All observed emeralds are grue, because all emeralds are grue’

really amounts to ‘All observed emeralds are green, because all observed emeralds

are green and the unobserved ones are blue.’ This doesn’t seem like much of an

explanation at all. We have just repeated the fact to be explained and tacked on a

claim about the remaining emeralds being a different color. The first part offers no

explanation, and the second just makes it worse, by raising the question of why we

have failed to see the blue emeralds.’

Of course, granted White’s equivalence, one might equally take as explan-andum ‘all observed emeralds are grue’ and similarly argue that ‘Allemeralds are grue’ provides a superior explanation to ‘All emeralds aregrue if observed and bleen otherwise’ i.e., ‘All emeralds are green’. How-ever, like Godfrey-Smith, White does not intend to address Goodman’sparadox in full, and appeals to the same counterfactuals to underwrite hissolution. Unobserved green things would still have been green had theybeen observed, and unobserved bleen things would have been blue, hadthey been observed. Hence, ‘All emeralds are green’ provides the superiorexplanation to ‘All emeralds are grue’, since only the former renders itunsurprising that we failed to encounter any blue emeralds.

Our proposal addresses the confirmation of the nomic claims, (Green)and (Grue), and since it delivers the intuitive verdict, delivers the counter-factuals on which the above accounts of the generalisations’ confirmationdepend e.g. ‘if this emerald had not been examined before t, then itwould still have been green’. It is tempting to think that it, therefore,complements them. However, that temptation should be resisted. Thecited counterfactual ensures that our sample’s representativeness is notcompromised by its being confined to emeralds examined before t. Butbeing examined before t is not the only parameter that could compromiserepresentativeness. As earlier noted, many factors might not implausibly

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be salient to the colour of emeralds, and it is only by sampling across adiversity of nomologically possible conditions that we can reasonablyaffirm counterfactuals stating that had this or that parameter beenvaried, the colour would have been the same; it is only by confirming(Green) that we confirm that our sampled emeralds are representative.But in confirming the law we automatically confirm the generalisation.So, there’s no second chapter where, counterfactuals in hand, we considerwhich of the generalisations should be confirmed.

VI. IN CONCLUSION

You’d be hard pressed to find anyone who would take observations ofgreen emeralds to confirm (Grue). That they are not significantly confirm-ing is just commonsense. Our solution has the virtue of explaining why. Itis commonsense that significant confirmation of a difference-making expla-nation demands evidence that the putative difference-makers indeed makea difference, and the former piece of commonsense is just a manifestationof the latter. Our solution also suggests a strategy for resolving problemswith gruesome antecedents. More broadly, I hope to have provided someevidence that explanatory considerations should play a fundamental role inour understanding of confirmation.20

University of Arkansas

20 Thanks to two anonymous referees for very helpful suggestions for how to properlycontrast Sober’s and Goodman’s accounts with mine.



Explanation and the New Riddle of Induction

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