A Discussion of Hempel and the Paradoxes of Confirmation

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Laying the Raven to Rest: A Discussion of Hempel and the Paradoxes of ConfirmationAuthor(s): John L. PollockSource: The Journal of Philosophy, Vol. 70, No. 20 (Nov. 22, 1973), pp. 747-754Published by: Journal of Philosophy, Inc.Stable URL: http://www.jstor.org/stable/2025089 .

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THE JOURNALOF PHILOSOPHYVOLUME LXX, NO. 20, NOVEMBER 22, I973

LAYING THE RAVEN TO REST: A DISCUSSION OF HEMPELAND THE PARADOXES OF CONFIRMATIONI a landmarkrticle, arl Hempel laid down conditionsfadequacy fora formalanalysis of instance confirmationndthenproduced an analysissatisfyinghose conditions. n thispaper I will show thathis analysisfailstomeetan additional condi-tionwhich is just as obvious as those he laid down. An examina-tionof thereasonsfor thisfailure will lead first o therejectionofall versionsof Hempel's "SatisfactionCriterionof Confirmation"

and thento a new proposal regarding he paradoxes of confirma-tion.What has come tobe called "instanceconfirmation"s confirmationofa generalhypothesis yits"positive nstances."Positive nstancesare those nstanceswhich,upon beingamassed, ead to greater ndgreater onfirmationf thehypothesis. reciselywhatconstitutespositive nstance s a matterof controversy,nd will be discussedbelow, but a necessary ondition is that observation of positiveinstancessupportnot only the general hypothesisbut also newinstances fthehypothesis.or example,everyonegreesthat Aa &Bb) is a positive nstance fa hypothesisf theformx) (Ax D Bx).2Thus, insofar s (Aa & Ba) confirmsx) (Ax D Bx), itmustalso lendsupport to the counterfactual onditional"If we were to find an-otherA, it would also be B." And learningthe truth fmoreandmore instances of this formshould give us greaterand greaterconfirmationf thehypothesis.

I "Studies n theLogicofConfirmation,"ind, LIV, 213,214 (January,pril1945):1-26, 7-121.2Werestricturattentionohypotheseshat re"projectible,"n the ense fNelsonGoodman's act, Fiction and Forecast (Cambridge, ass:Harvard, 955)and my "The Logic of Projectibility," hilosophy of Science,xxxIx, 3 (Septem-ber1972):302-314. etmealso admit t thispointthat omeoftheconclusionsdrawnbeloware at oddswithconclusionsrawn n the atter rticle.747

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748 THE JOURNALOF PHILOSOPHYHempel laid down thefollowing onditionsof adequacyforanysatisfactorynalysisofconfirmation:3Consequence ondition: f p confirms, and q -> r, thenp con-firms.4Conjunctionondition:fp confirmsothq and r,thenp confirms(q&r).Consistencyondition: f p is logically onsistentnd p confirmsq, thenp must e consistent ith .Inverse quivalence ondition:5fp isequivalentoq andp confirmsr,then confirms.

Numerous philosophershave objected to various of these condi-tionson differentrounds,but I findthemacceptable.I will notattempt o defendthemhere.Having laid downhis conditions f adequacy,Hempel proposedan analysis of confirmationwhich satisfies hese conditions.Hisanalysiswas based upon his discovery f the paradox of the ravens.He reasonedas follows.We all agree that (Aa & Ba) is a positiveinstanceof (x) (Ax D Bx). But then ,-Ac & --Bc) mustbe a posi-tiveinstanceof the contrapositivex) (t-'Bx D -,Ax). The latter sequivalentto,and henceentails, x) (Ax D Bx). Thus, bythe conse-quence condition, nything onfirmingx) (,-.Bx D -,Ax) must alsoconfirmx) (Ax D Bx). Therefore, t seemsthat -.Ac & -'Bc) mustalso be countedas a positive nstanceof (x) (Ax D Bx). This seemsparadoxical. For example, if (x) (Ax D Bx) says "All ravens areblack," thenwe should be able to confirm hisby observingnon-blacknonravens. n particular,f we went to a factorymanufactur-ing plastic garbage cans and observedthat all the cans comingoff the assembly line were green, this would confirmthat allravens reblack.This is at leastpeculiar.Rather than reject his formalconclusions,Hempel attemptedto explain awaytheair ofparadox. His explanationwas thatwhat(x) (Ax 2 Bx) says is that everythingn the universeeither s notan A or is a B. Thus observing nything hatsatisfieshisdisjunc-tionshouldconfirmheconditional.This iswhy ,-.-Ac c-,Bc) mustbe counted as a positive instance. However, this proposed ex-planation leads to the resultthat we must count some more sen-tences s positive nstances oo. In particular,,-..Ad& Bd) becomesa positive nstance.Generalizing his ed Hempel tohis Satisfaction

3This listofconditionss notquite thesameas Hempel's riginalist,butitisequivalent o t.4 I use thearrow o standfor ogical ntailment,otthematerial onditional.6 The nameofthis onditions taken rom cheffier,heAnatomyf nquiry(NewYork:Knopf, 963).



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LAYING THE RAVEN TO REST 749Criterion f Confirmation. his result,however, eemseven moreparadoxical than what it was intended to explain. It is simplypreposterous o suppose that we can confirm hat all ravens areblack by finding xamples of nonravensthat are black. Thus, inattempting o explain one paradoxical result,Hempel was led toembrace nother ven more paradoxicalresult.This intuitionof paradox can be given substanceby showingthat, f we add one additional seemingly bviousconditionof ade-quacy to Hempel's previousconditions, o conceptof confirmationcan both satisfy ll these conditionsand embodyHempel's con-clusionsregardingpositive instances.Amassing positive instancesis supposedto giveus more and more confirmationf x) (Ax D Bx).Characteristically,n confirming uch a hypothesis, here will bemanythingswe will know tobe A's withoutknowingwhether heyare B's. For example, if we are confirminghat all ravenshavekidneys,we maycutopen lotsof ravens, ut we aregoing to knowoflots more ravens that we haven't examined. More generally,wemay know that some things re non-A's withoutknowingwhethertheyare B's, that some thingsare B's without knowing whetherthey are A's, and that some things are non-B's withoutknowingwhetherthey are A's. None of this should detractfrom the con-firmationf (x) (Ax D Bx) by its positive nstances. o let us adoptthe additional adequacy condition:Condition f InstanceConjunction:f p is a positivenstance f(x) AxD Bx), and b', 'c', d', and e' are individual onstantsotoccurringn p, then p & Ab & --Ac & Bd & --Be)must onfirm

(x) (Ax Bx).Ifwe nowassumeHempel's conclusions egarding ositive nstancesand assume that all five conditionsof adequacy are satisfied,wecan derivea contradition. ccording o Hempel, (,--Ca& ,--Da) is apositive nstance fboth (x) (Cx D Dx) and (x) (Cx D -Dx). Hence,by the conditionof instanceconjunction, ,--,Ca8& ,Da 8cCb &,,Cc & Dd & ,De) must confirm oth of thesegeneralizations.Then, by the conjunctioncondition, t must confirm heircon-junction.But theirconjunctionentails (x) ,-Cx; so, by the conse-quence condition,thismust also be confirmedy (--Ca & --,Da &Cb & --,Cc& Dd & -De). But thelattercontains counterinstancefor x) -Cx, and so certainly oes not confirmt. Putting this interms f the conditions f adequacy, ,,Ca &c -,Da &cCb 8c -,Cc&Dd &c ..De) is logically nconsistentwith (x) - Cx, and, hence, tsconfirmations ruled out by the consistency ondition. Thus



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750 THE JOURNAL OF PHILOSOPHYHempel's proposals regardingpositiveinstances are incompatiblewiththeconditions fadequacy.

In order to avoid this contradiction, e mustdeny that ,,Ca &,,Da) is a positiveinstanceof both (x) (Cx D Dx) and (x) (Cx D-,Dx). Or, equivalently,we mustagree that one of (,-Ac & .-Bc)and (,,Ad & Bd) is not a positive instance of (x) (Ax D Bx). Al-though both of these purportedpositive instances seem suspect,the latter s the moresuspect.We were led to count it as a positiveinstance only in trying o explain why ,.Ac & -.Bc) is a positiveinstance.And we have a seemingly trong rgumentforcountingthe latteras a positive nstance.Consequently, think t must bedenied that ,,Ad & Bd) is a positive nstance f (x) (Ax D Bx).

IIAlthoughwe have seen thatHempel's analysis s wrong,we havenot reallyseen why t is wrong.The basic idea was verypersuasive.In orderto understand he sourceof the errorwe mustgo back tothe paradox of the ravens.We had an argument o the effecthat(,-Ac & -.Bc) mustbe considered positive nstanceof (x) (Ax DBx), and we were ed to explain thisbymaintaining hat what thegeneralization ays is that everythings either non-A or B, andhence anythingsatisfying his disjunction should confirm thegeneralization.t now seemsto me that theerror ies in supposingthat the generalization hat is confirmed n instance confirmationis properly ymbolizedusing a material conditional. We have al-ready seen that, in instance confirmation,ach instance is sup-posed to support others. More precisely, Aa & Ba) confirms hecounterfactualIf we were to encounter notherA, itwould be B."This has often been put by sayingthatwe confirmaws,not ac-cidentalgeneralizations.fwe wereconvincedthat x) (Ax D Bx), iftrue,would be trueonly coincidentally, henwe would not takethe addition of positive instancesto our evidence as confirmingthe generalization.We take the addition of positive instancestoadd to the confirmation f (x) (Ax D Bx) only insofaras it alsosupportsother nstances, nd hence confirmshegeneralizedcoun-terfactualAnyA would be a B." Let us symbolize his by (x) (Ax> Bx)'. The accumulation of positive instancesconfirms hiscounterfactual,nd it confirmshematerialconditionalonlyderiva-tively, ecause it is entailed by the counterfactual. erein lies theerror n Hempel's argument.The generalization hat is confirmedin instance confirmation oes not simply say that everything seither a non-Aor a B, and hence there s no reason to think t



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LAYING THE RAVEN TO REST 751should be confirmed y everything hat satisfies his disjunction.The generalization hat smost directly onfirmed y ts positive n-stances s counterfactual,nd hence stronger han thisdisjunction.Having brought ounterfactualsnto the picture,we are treadingon notoriously lipperyground. I will make no attempthere toexplain or clarify he counterfactuals nvolved. I can only hopethat our rather fuzzy intuitions regarding counterfactualswillprove equal to the task of understanding t least those featuresofcounterfactuals hicharenecessary orthe presentdiscussion.By maintaining hatwhat s confirmedn instance onfirmationsthe generalized counterfactual x) (Ax Z> Bx), I have explainedwhy (,-,Ad & Bd) should not be regarded as a positive instance.Can we perhaps go all the way and dissolve the paradox of theravens ltogether ymaintaining hat ,,Ac & -. Bc) is nota positiveinstanceeither?To do this t would sufficef contraposition ailedforthe counterfactualsnvolved.Contraposition ertainly oes nothold forcounterfactualsn general.For example,from Even iftheIndians were to hold a rain dance, it would not rain," we cannotconclude "If it were to rain, the Indians would not have held arain dance." However, for the ratherspecial counterfactualsn-volved in stating aws, contraposition oes seem to hold. For ex-ample, fit is truethatanyraven would be black,then t is equallytrue that any nonblack thing would be a nonraven.Thus I amafraid that the appeal to counterfactuals oes not absolve us ofhaving to include (.-'Ac & -.Bc) as a positive nstanceof the gen-eralization.We still have a paradox.

IIIWhy thendoes it seem paradoxical to count (,..-Ac& ~-Bc) as apositive instance of (x) (Ax => Bx)? As Hempel observed, f wechange examples this no longer seems so paradoxical. Consider"All good conductorsof electricityre good conductors f heat."It is not in the least paradoxical to supposewe can confirm hisby finding ubstancesthat are not good conductorsof heat andascertaining hatthey re notgood conductors f electricity ither.Furthermore,f we shift he setting bit, it is not paradoxical tosupposewe can confirm hat all ravensare black byobserving on-black nonravens.Suppose we are somehowprovidedwith a cata-logue of everythingn the universe, istingcertainof their im-portant attributes. f we go throughthe catalogue picking outnonblack things nd thenchecking hatthey re notravens, t cer-tainly eemsthat we could confirmhatall ravens are black.Then whyis it paradoxical to supposewe can confirm hat all



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752 THE JOURNAL OF PHILOSOPHYravens are black by going to a garbage-canfactory nd checkingthatthe cans are all green as they ome out? I thinkthe answer sthatobservation fgreengarbagecans really does not confirmhatall ravens re black.Moreprecisely,lthough"c is a nonblacknon-raven"confirmsAll ravensare black," "c is a green garbagecan"does not, despite thefactthat t entails"c is a nonblacknonraven."Let me explain why.Inductiveconfirmations defeasible.That is, we can have twostatements and q such thatp confirms, but when we conjoinanotherstatement , (p & r) does not confirm . Such an r is adefeater.For example, although (Aa & Ba) confirmsx) (Ax =>Bx), the conjunction [(Aa & Ba) & (Ab & .-iBb)] does not. Theaddition of a counterexample o the evidencedefeatsthe confirma-tion.There ismore thanone kindof defeater or nstance onfirmation.Counterexamples efeat theconfirmationruth-functionally,y en-tailingthefalsity f thegeneralization. ut anotherkindofdefeaterleaves open whether he generalization s true and insteadattacksthe fairness fthesample.6 or example,suppose we have examineda class r of objectsall of whichare A's, and found themall to beB's. This is supposed to confirmx) (Ax -> Bx). But suppose it isthennoticed thateverythingn r is C, althoughnot all A's are C's,and prob Bx/Ax & Cx) > prob Bx/Ax & ~-.Cx) (theprobability-priorto knowingthateverythingn r is C-of an arbitrarybjectbeingB given that t is both A and C isgreater han theprobabilitygiven that it is A and not C).7 This means thatthe inductive am-ple is unfairlyprejudiced in favor of the generalization. f thesample is just a littlebiased, thismay not completelydefeat theconfirmation,ut it at least diminishesthe degree of confirma-tion. We might aythatwhat we have hereare "diminishers" atherthan defeaters. or example, suppose we are examining carswiththe objective of establishingthat no car can go fasterthan onehundred miles an hour. Suppose this is true of every car in our

GIntheterminologyfmy The StructurefEpistemic ustification,"meri-can PhilosophicalQuarterly, onograph (1970):62-78,these re type andtype Idefeaters,espectively.7 To avoid trivializinghis,we mustput somerestrictionsn C. If there ssomeattribute that verythingn r possessesut notall A's possess, henwecould define Cx' to be 'Fx & Bx'. And we can alwaysfind uch an F. Forexample,we might et Fx' be 'x er. Clearly, hisshouldnot defeat hecon-firmationif it did, all confirmationould be defeated).t seems hatthere-striction e needon C is: theway nwhichwe know hat x) xerD Cx)doesnotpresupposeur first nowinghat x) xerD Bx). A moreprecise ormulationfthisrestriction ustawait a satisfactorynalysis fknowing, hich am notnowprepared ogive.



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LAYING THE RAVEN TO REST 753sample. This would confirm he generalization.But if we notice(1) that all cars in the sample have small engines, 2) that notall cars have small engines, nd (3) that theprobability f a car'shavinga top speed less thanone hundredmilesan hour is greaterif the car has a small engine than if it has a large engine, thissignificantly eakens the degreeof confirmation.f the prejudice(i.e., the disparity n probabilities) s sufficientlyreat, this maycompletely estroy he confirmation. ut notice that the generali-zation might still turn out to be true even though it cannotbe confirmed y appealing to this ample.

Analogously, four sample consists f a class of non-B'sthat arenon-A's,but we find 1) thateverythingn r is D, (2) that not allnon-B's are D, and (3) thatprob ,-'Ax/l--.,Bx Dx) > prob Q--Ax/,,Bx & -,Dx), this again constitutes diminisher.n general,oursampler will be the union of a positive ampler1 (a set of A's thatare B's) and a negative ampler2 (a set of non-B'sthat are non-A's).We can diminish he confirmationrovidedbythesample by havinga diminisher or eitherr1 or r2. However, n orderto completelydefeat ucha mixed sample by attacking ts fairness, e mustdefeatbothparts separately. f one part is unfairly rejudiced n favorofthe generalization,but the other is not, the entire sample stillconfirmshe generalization.Although these diminishers o not usually destroy he confirma-tion altogether,nd hence do not constitute efeaters,here s onecase in which theydo constitute efeaters. his is thecase in whichprob Bx/Ax 8cCx) = 1> prob Bx/Ax&g-Cx). For example, sup-pose we are attempting o confirm hat all woodenchairs are madeof oak. To collect our sample we visit a furniture actorymanu-facturingwooden chairs. The sample bears out our hypothesis.But then we discover hatthisfactorymanufacturesnlyoak furni-ture, nd we know that this s not trueof all factoriesmanufactur-ingwoodenchairs.This would makethesamplecompletely seless,and hence defeat the confirmation. he reason the confirmationwould be defeated eems to be the following. f we let C be "wasmanufacturedn a factory roducing only oak furniture,"hen achair'shavingthis ttribute ntailsthat t ismade ofoak,and henceprob (Bx/Ax & Cx) = 1. This means that the chairs we examinedcouldn't have been anything ut oak because of thewaywe choseour sample; hence the generalizationcouldn't have been false ofthe sample. For this reason,therewas no chance of the sample notbearingout the generalization.The generalizationran no riskinthis case,and hence thatthesampledid bear it out in no waycon-



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754 THE JOURNAL OF PHILOSOPHYfirmshegeneralization. o in thiscase we have a genuinedefeaterrather han ust a diminisher.Now let us apply this to the ravens.Suppose our sample con-sists of green plastic garbage cans. The probability of being anonraven given that something s a green plastic garbage can is1. Thus, our sample of nonblack nonravens s totallyprejudicedagainst finding ny ravens in it. There was never any chance offinding ny ravens n it, so we have a fair-sample efeater.This iswhy, ntuitively,we would not regard this sample as confirmingeither that all ravens are black or that all nonblack thingsarenonravens. hus we resolve heparadox oftheravens.On the otherhand, if we pick our sample of nonblack thingsrandomlyfrom a catalogue of all the things n the universe,wewill not automaticallyprejudice our sample against ravens, andfor this reason we feel no reluctance about takingit to confirmthat ll ravens reblack.Also,we are now in a position to explain the often-voicedntui-tionthat thereason t is harder to confirmhat all ravensare blackby looking at nonblack things than it is by looking at ravens, sthat theproportion f nonblackthings n theuniverse s so muchgreater than the proportionof ravens.Because thereare so manymore nonblack things,unlesswe pick our sample veryselectively(e.g.,, choose only birds), it is very ikely that we will findthateverythingn our sample has some attributeD which entails thatit is not a raven.For example,pickingthings t random,we arequite apt to end up withonlyinanimateobjects,becausemost ob-jects are inanimate,and a sample of nonblack inanimateobjectscannotconfirm hat ll ravens reblack.IVIn conclusion, think here rereally wo"paradoxesof theravens."The originalparadox concernedhowobservation fnonblacknon-ravens can confirm hat all ravens are black. This paradox isresolved by appealing to fair-sample efeaters.Hempel, however,compoundedtheparadox bygivingan incorrectxplanationof it,which then led him to maintain that observationof black non-ravensalso confirmshatall ravensare black. The latter s simplywrong,and arises out of supposing that the generalizations hatare confirmed nductivelyare properlysymbolizedby materialconditionals.

JOHN L. POLLOCKUniversityfRochester

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A Discussion of Hempel and the Paradoxes of Confirmation