Explanation Versus Prediction: Which Carries More Evidential

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TheBookofEvidencePeterAchinstein

Printpublicationdate:2001PrintISBN-13:9780195143898PublishedtoOxfordScholarshipOnline:November2003DOI:10.1093/0195143892.001.0001

ExplanationVersusPrediction:WhichCarriesMoreEvidentialPeterAchinstein(ContributorWebpage)

DOI:10.1093/0195143892.003.0010

AbstractandKeywords

Accordingtoonestandardview,apredictionofanewfactalwayscountsasstrongerevidenceforahypothesisthananexplanationofaknownone.Accordingtoanotherview,itisthereverse.Bothviewsareshowntobemistaken.Whatisimportantforevidenceisnotwhetheritwaspredictedorexplained,but(inthesortofcasesusedbyeachside)whatselectionprocedureswereusedtoobtaintheevidence.StephenBrush'sdefenseofexplanationismandPatrickMaher'sdefenseofpredictivismarecriticallyexamined,asisClarkGlymour'sproblemofoldevidence.

Keywords:Brush,evidence,explanationism,Glymour,Maher,oldevidence,prediction,predictivism,scientificexplanation,selectionprocedures

1.TheHistoricalThesisofEvidenceAccordingtoastandardview,predictionsofnewphenomenaprovidestrongerevidenceforatheorythanexplanationsofoldones.Moreprecisely,atheorythatpredicts

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phenomenathatdidnotprompttheinitialformulationofthattheoryisbettersupportedbythosephenomenathanisatheorybyknownphenomenathatgeneratedthetheoryinthefirstplaceandthatthetheorywasusedtoexplain.Sosayvariousphilosophersofscience,includingWilliamWhewellinthenineteenthcenturyandKarlPopperinthetwentieth,tomentionjusttwo.1

StephenBrushtakesissuewiththisonhistoricalgrounds.2Hearguesthat,generallyspeaking,scientistsdonotregardnewphenomenapredictedbyatheory,evenonesofakindtotallydifferentfromthosethatpromptedthetheoryinthefirstplace,asprovidingbetterevidentialsupportforthattheorythanisprovidedbyalreadyknownphenomenaexplainedbythetheory.Bycontrast,Brushclaims,therearecases,includinggeneralrelativityandtheperiodiclawofelements,inwhichscientiststendtoconsiderknownphenomenaexplainedbyatheoryasconstitutingmuchstrongerevidencethannovelpredictions.3

(p.211) Boththepredictionistandtheexplanationistarecommittedtoaninterestinghistoricalthesisaboutputativeevidenceeandahypothesish,viz.

Historicalthesis:Whethereiftrueisevidencethath,orhowstrongthatevidenceis,dependsoncertainhistoricalfactsaboute,h,ortheirrelationship.4

Forexample,whethereiftrueisevidencethath,orhowstrongitis,dependsonwhetherewasknowntobetruebeforeorafterhwasformulated.Varioushistoricalpositionsarepossible,asAlanMusgravenotedyearsagoinaveryinterestingarticle.5Onasimplepredictionistview(whichMusgraveclassifiesaspurelytemporal),eisevidencethathonlyifewasnotknowntobetruewhenhwasfirstproposed.Onanotherview(whichMusgraveattributestoElieZaharandcallsheuristic),eisevidencethathonlyifwhenhwasfirstformulateditwasnotdevisedinordertoexplaine.Onyetathirdhistoricalview(whichMusgravehimselfaccepts),eisevidencethathonlyifecannotbeexplainedbyapredecessortheory,thatis,byacompetingtheorywhichwasdevisedbyscientistspriortotheformulationofh.

Thesearethreeexamplesofhistoricalviewsconcernedwiththetemporalorderinwhicheandhwereformulatedorknowntobetrue.Butotherhistoricalpositionsarepossible.Forexample,itmightbeheldthatwhethereiftrueisevidencethath,andhowstrongthatevidenceis,alwaysdependsonhistoricalfactsconcerninghowtheresultsreportedinewereobtained,forexample,whatsamplingmethodswereinfactusedbythosewhoreportedthate(moreonthisinsection2).

Thehistoricalthesisisnotthateorharethemselvespropositionsaboutparticularhistoricalevents.(Forexample,emightbethatlightcanbediffracted,andhmightbethatlightisawavemotion.)Ratherthethesisisthatevenifneitherenorhdescribesaparticulareventorsetofeventsthatoccurred,whethere,iftrue,isevidencethathdependsupontheoccurrenceofsomeparticulareventorsetofeventspertainingtowhenorhowe,h,ortherelationshipbetweenthemcametobeformulatedorbelieved,orhowtheresultsinewereobtained.

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Isthehistoricalthesistrueorfalse?Itisclearlytrueinthecaseofsubjectiveevidence.Whethereissomeperson'sevidencethathdependsoncertainhistoricalfactsaboute,h,andtheirrelationship,viz.whetherthatpersoninfacthasorhadcertainbeliefsaboute,h,andtheirrelationship.However,defendersofthehistoricalthesisarenotspeakingofsubjectiveevidence.Theyarenotspeakingofsomethingthatsomeonetakestobeevidencebutofsomethingthatis(objective)evidence.Incasesofthelattersort,Iproposetoarguethatthehistoricalthesisissometimestrue,andsometimesfalse,dependingonthetypeofevidenceinquestion.Iwillshowhowthiscomportswithmyowntheoryofpotentialandveridicalevidence.ThenIwillconsiderwhatimplications,ifany,thishasforthe(p.212) debatebetweenBrushandthepredictionistsandfortheproblemofoldevidenceraisedbyClarkGlymour.

Beforebeginning,however,letmementionacuriousbutinterestingfactaboutcertainwellknownphilosophicaltheoriesordefinitionsofobjectiveevidence(otherthanmine),includingCarnap'stheoryofconfirmation,6Hempel'ssatisfactiontheory,7Glymour'sbootstrapaccount,8andtheusualhypotheticodeductiveaccount.Thesetheoriesareincompatiblewiththehistoricalthesis.9Theyholdthatwhether,ortheextenttowhich,eisevidencethathisanapriorifactabouttherelationshipbetweeneandh.Itisinnowayaffectedbyempiricalissuessuchasthetimeatwhichhwasfirstproposed,orewasfirstknown,orbytheintentionswithwhichhwasformulated,orhowinformationreportedinewasobtained.Defendersoftheseviewsmustrejectboththepredictionistandtheexplanationistclaimsaboutevidence.Theymustsaythatwhether,ortheextenttowhich,esupportshhasnothingtodowithwhetherewasfirstformulatedasanovelpredictionfromhorwhetherewasknownbeforehandhwasconstructedtoexplainit.

Accordingly,wehavetwoextremeorabsolutistpositions.Thereistheposition,reflectedinthehistoricalthesis,thatevidenceisalwayshistorical(inthesenseindicated).Andthereisacontrastingposition,reflectedinaprioriviews,thatevidenceisneverhistorical.Doesthetruthlieateitherextreme?Orisitsomewhereinthemiddle?InwhatfollowsIwillpursuethesequestionswithrespecttomyconceptsofpotentialandveridicalevidence,theonesmostcrucialtoscientists(althoughwhatIwillsayisalsoapplicabletoESevidence).

2.SelectionProceduresSupposethataninvestigatordecidestotesttheefficacyofadrugDinrelievingsymptomsS.Thehypothesisunderconsiderationis

h:DrugDrelievessymptomsSinapproximately95%ofthecases.

AsIhaveemphasizedinchapter9,whether(andtheextenttowhich)sometestresultisevidencethatacertainhypothesisistruedependsontheselectionprocedureusedtoobtainthatresult.Herearetwoofthemanypossibleselectionproceduresfortestingh:

SP1:Chooseasampleof2000personsofdifferentages,bothsexes,whohavesymptomsSinvaryingdegrees;dividethemarbitrarilyintotwogroups;giveonegroupdrugDandtheotheraplacebo;determinehowmanyineachgrouphavetheirsymptomsrelieved.

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SP2:Chooseasampleof2000femalesaged5allofwhomhavesymptomsSinaverymildform;proceedasinSP1.

(p.213) Nowsupposethatourinvestigatorobtainsthefollowingresult:e:Inagroupof1000personswithsymptomsStakingdrugD,950personshadreliefofS;inacontrolgroupof1000SsufferersnottakingDbutaplacebononehadsymptomsSrelieved.

IfresultewasobtainedbyfollowingSP2,thene,althoughtrue,wouldnotbeparticularlygoodevidencethath,certainlynotasstrongasthatobtainedbyfollowingSP1.Thereason,ofcourse,isthatSP1,bycontrastwithSP2,givesasamplethatisvariedwithrespecttofactorsthatmaywellberelevant:age,sex,andseverityofsymptoms.(Hypothesishdoesnotrestrictitselfto5yearoldgirlswithmildsymptoms,butassertsacurerateforthegeneralpopulationofsuffererswithvaryingdegreesofthesymptomsinquestion.)

Thismeansthatiftheresultdescribedineisobtained,thenwhetherthatresultsodescribedconstitutesevidencethath,andhowstrongthatevidenceis,dependsonahistoricalfactaboute,viz.howinfactewasobtained.IferesultedfromfollowingSP1,theneisprettystrongevidencethath;ifewasobtainedbyfollowingSP2,theneisprettyweakevidencethath,ifitisevidenceatall.Justbylookingateandh,andevenbyascertainingthateistrue,weareunabletodeterminetowhatextent,ifany,e,iftrue,supportsh.Weneedtoinvokehistory.

Hereisathirdselectionprocedure:

SP3:Chooseasampleof2000personsallofwhomhaveSinvaryingdegrees;dividethemarbitrarilyintotwogroups;giveonegroupdrugsDandD(whereDrelievessymptomsSin95%ofthecasesandblockspossiblecurativeeffectsofDwhentakentogether);givetheothergroupaplacebo.

Ifthiswasusedtogeneratee,thenresultefailstoprovideanysupportforh.And,again,thatthisissocanbeascertainedonlybylearningahistoricalfactaboute,viz.thatwhate(truly)reportswasobtainedbyfollowingSP3.10

Sofarthenweseemtohavesupportforthehistoricalthesisaboutevidence.Canwegeneralizefromexampleslikethistoallcases?Canwesaythatforanye,andanyhypothesish,whether,ortowhatextent,e,iftrue,isevidencethathdependsuponhistoricalfactsabouthowewasobtained?

Consideranothertypeofcase,inwhichtheaimistotestthehypothesis

h:Johnwillwinthelottery.

Atthepresenttimesupposethatthiscanbedoneonlyindirectlybyobtaininginformationaboutwhoboughtticketsandhowmany.Twoinvestigatorsproceedtoobtaininformationofthissort,eachonefollowingadifferentselectionprocedure:(p.214)

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SP4:Determinewhoboughttickets,andhowmany,byaskinglotteryofficials.SP5:Determinethisbyconsultingthelocalnewspaper,whichpublishesthisinformationasaservicetoitsreaders.

Theseresultsareobtained:e1:Investigator1reportsthatinthelottery1000ticketsweresold,ofwhichJohnowns999,andnofurtherticketswillbesold.e2:Investigator2reportsthesamething.

Whethere1ore2orbothareevidencethathdependsonwhatselectionprocedurewasemployedbytheinvestigatorsinobtainingtheinformationintheirreports.Supposethatinvestigator1followedSP4,whileinvestigator2followedSP5.Andsupposethatnewspapers,bycontrasttolotteryofficials,areusuallyunreliableinsuchreports.Then,althoughe1,iftrue,isstrongevidencethath,e2isnot.Inanycase,todeterminewhether,ortheextenttowhich,e1ore2supportshweagainneedtodetermineahistoricalfactconcerninghowtheticketinformationreportedine1ande2wasobtained.

Now,however,letusdistilltheinformationreportedine1ande2intothefollowing:

e:Inthelottery1000ticketsweresold,ofwhichJohnowns999,andnofurtherticketswillbesold.

Inthiscase,whethere,iftrue,isevidencethath,andhowstrongitis,doesnotdependonhistoricalfactsconcerninghowinformationreportedinewasobtained,orconcerningwhenorhowe,h,ortheirrelationshipcametobeformulated,believed,orknown.Tobesure,whether,ortheextenttowhich,e,iftrue,supportshdoesdependuponotherhistoricalfacts,suchaswhetherthelotteryisfair,whethercertainconditionswillinterfere,andsoforth.Butthesearenothistoricalfactsofthekindrelevantforthehistoricalthesisofevidence.Inthiscase,unlikethedrugcase,inorderfore,iftrue,tobeevidencethath,itisirrelevanthoworwhentheinformationinewasobtained,orevenwhetheritwasobtained.Accordingly,wehaveacasethatviolatesthehistoricalthesisofevidence.

Sinceexamplessimilartoeachoftheabovecanbereadilyconstructed,wemayconcludethattherearecasesthatsatisfythehistoricalthesisofevidence,andothersthatfailtosatisfyit.Withrespecttoahypothesish,ifespeaksofobservationsortestsmadethatyieldcertainresults,thenwhethereiftrueisevidencethath,andifso,howstrongthatevidenceis,dependsonwhatselectionprocedurewasemployedinmakingtheseobservationsortests.Thatisahistoricalfactofthekindthatconformstothehistoricalthesisofevidence.However,therearemanycases,suchasthelastonenoted,inwhichedoesnotspeakofobservationsortests,butofcertainfactsthataredescribedindependentlyofobservationsortests.Suchcasesdonotconformtothehistoricalthesis.

Whatimplications,ifany,doesthisholdforwhetherpredictionsorexplanationsprovidestrongerevidence?

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(p.215) 3.PredictionsVersusExplanationsLetusreturntotheoriginalquestionproposedbyBrush.Donovelfactspredictedbyatheoryprovidestrongerevidenceforthattheorythanknownfactsexplainedbythetheory,asWhewellandPopperclaim?Oristhereversetrue?

Apreliminarypointisworthmaking.Asshouldbeobviousfromdiscussionsinearlierchapters,themerefactthatsometheoryorhypothesishwas(orindeedcanbe)usedsuccessfullytopredictsomenoveledoesnotsufficetomakeeevidencethath.Nordoesthefactthatsometheoryorhypothesishwas(orcanbe)usedtoexplainsomeknownfactesufficetomakeeevidencethath.Supposethereisalotterywhose1millionticketsgoonsaleonTuesday;onlyoneticketcanbepurchasedbyanygivenperson;andthedrawingwillbemadeonFriday.Myhypothesisisthatyouwillwinthelottery.SupposeIusethishypothesistopredictthatyouwillbuy1ticket.Thefactthatmypredictionturnedouttobecorrectisnotevidencethatyouwillwinthelottery.SupposethatonSundayitcomestobeknownthatyouareinagoodmood.AndsupposeIusethehypothesisthatyouwonthelotterytoexplainthisknownfact.Thiswouldnotsufficetomakethefactthatyouareinagoodmoodevidencethatyouwonthelottery.Itisreadilyshownthatontheaccountof(potentialandveridical)evidenceIpropose,neitherthefactthatewas(orcanbe)correctlypredictedfromahypothesish,northefactthathwas(orcanbe)usedtoexplainaknownfacte,sufficestomakeeevidencethath.

Accordingly,whatthepredictionistandexplanationistmaywanttosayisthis.Whethereisanovelfactcorrectlypredictedfromahypothesish,orwhethereisaknownfactthathwasinvokedtoexplain,isrelevantforthequestionofwhethereisevidencethath,andparticularlyforthequestionofhowstrongthatevidenceis.Onthepredictionistview,anovelfactcorrectlypredictedbyahypothesisprovidesstrongerevidencethandoesanalreadyknownfactexplainedbythehypothesis.Foranexplanationistitisthereverse.Whichviewiscorrect?

Myansweristhis:Neitherone.Sometimesapredictionprovidesbetterevidenceforahypothesis,sometimesanexplanationdoes,andsometimestheyareequallygood.Whichobtainshasnothingtodowiththefactthatitisapredictionofnovelfactsorthatitisanexplanationofknownones.

Toshowthis,Iwillbeginwithacasethatviolatesthehistoricalthesisofevidence.Hereitshouldbeeasytoshowthatwhethertheputativeevidenceisknownbeforeorafterthehypothesisisformulatedisirrelevantforwhetheritisevidencethathorhowstrongthatevidenceis.Let

h:Thiscoinisfair,thatis,iftossedinrandomwaysundernormalconditionsitwilllandheadsapproximatelyhalfthetimeinthelongrune:Thiscoinisphysicallysymmetrical,andinaseriesof1000randomtossesundernormalconditionsitlandedheadsapproximatelyhalfthetime.

Letussupposethateisempiricallycompletewithrespecttoh(thatis,whethereisevidencethath,andhowstrongthatevidenceis,doesnotdependonempirical(p.216)

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factsotherthane).11Inparticular,whethereisevidencethathdoesnotdependonwhen,how,orevenwhether,ecomestobeknown,oronwhetherewasknownfirstandhthenformulated,oronwhetherhwasconceivedfirstandethenstatedasapredictionfromit.Putativeevidenceesupportshypothesishanddoesso(equallywell)whetherornoteisknownbeforeorafterhwasinitiallyformulated,indeedwhetherornothwaseverformulatedoreiseverknowntobetrueoranyselectionprocedurewasusedtoobtaine.

Thesameholdsinacasethatisanalogousexceptthateisnotempiricallycompletewithrespecttoh.Let

e:Inthelottery1000ticketsweresold,ofwhichJohnowned999atthetimeofthedrawing.h:Johnwon.

Whethereisevidencethath,andifsohowstrongitis,dependsonthetruthofcertainfactsotherthane,suchaswhetherthelotterywasfair.Butitdoesnotdependonwhetherewasknownbeforeorafterhwasformulated,oronwhetherewaseverknown,oronwhatselectionprocedure,ifany,wasemployedtoobtaine.

Soletusfocusinsteadoncasesthatsatisfythehistoricalthesisofevidence.Wemightsupposethatatleastinsuchcasesexplanations(orpredictions)arealwaysbetterforevidence.Returnonceagaintoourdrughypothesis

h:DrugDrelievessymptomsSinapproximately95%ofthecases.

Considertwoevidenceclaims,thefirstapredictionaboutanunknownfutureevent,thesecondareportaboutsomethingalreadyknown:

e1:Inthenextclinicaltrialof1000patientswhosufferfromsymptomsSandwhotakeDapproximately950willgetsomerelief.e2:Inatrialthathasalreadytakenplaceinvolving1000patientswithSwhotookD(weknowthat)approximately950gotsomerelief.

Onthepredictionview,e1(iftrue)isstrongerevidenceforhthanise2.Ontheexplanationview,itisthereverse.Andtosharpenthecase,letussupposethate2,bycontrasttoe1,wasnotonlyknowntobetruepriortotheformulationofh,butthathwasformulatedwiththeintentionofexplaininge2.Whichviewiscorrect?Neitherone.

Letustakethepredictioncasee1first.Whethere1iftrueisevidencethath,andhowstrongitis,dependsontheselectionproceduretobeusedinthenextclinicaltrial.Supposethisselectionprocedurecallsforchoosingjust5yearoldgirlswithverymildsymptomswhoinadditiontoDarealsotakingdrugDwhichamelioratessymptomsSin95%ofthecasesandpotentiallyblocksDfromdoingso.Thene1wouldbeveryweakevidencethathistrue,ifitisevidenceatall.Thisisso,despitethefactthate1isacorrectpredictionfromh,onenotusedingeneratinghinthefirstplace.Bycontrast,supposethattheselectionprocedureusedinthepasttrialmentionedine2ismuch

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betterwithrespecttoh.Forexample,itcallsforchoosinghumansofbothsexes,ofdifferentages,with(p.217) symptomsofvaryingdegrees,whoarenotalsotakingdrugDoranyotherthatcanpreventDfromworking.Thene2wouldbeevidencethath,indeedmuchstrongerevidencethane1.Insuchacase,aknownfactexplainedbyhwouldprovidemoresupportforhthananewlypredictedfactwould.

Obviouslythesituationsherecanbereversed.Wemightsupposethattheselectionprocedureusedtogeneratethepredictionofe1istheonecitedinthepreviousparagraphasbeingusedtogeneratee2(andviceversa).Inthissituationanewlypredictedfactwouldprovidemoresupportforhthananalreadyexplainedone.

Inthesecaseswhethersomefactisevidence,orhowstrongthatevidenceis,hasnothingtodowithwhetheritisbeingexplainedorpredicted.Ithastodowiththeselectionprocedureusedtogeneratethatevidence.Inonesituationwhetheritinvolvessomethingthatisexplainedorpredictedwehaveaputativeevidencestatementgeneratedbyaselectionprocedurethatisagoodonerelativetoh;intheothercasewehaveaflawedselectionprocedure.Thisiswhatmattersforevidence,notwhethertheputativeevidenceisbeingexplainedorpredicted.

4.AResponse:Maher'sAccountAresponseofthepredictionistandexplanationistwillnowbeconsideredinthissectionandthetwothatfollow.Itinvolvesformulatingtheinformationconcerningwhethereisapredictionoranexplanationaspartoftheevidencestatementitselforatleastaspartofthebackgroundinformation.12

Intheexampleabove,let

e:Inaclinicaltrialof1000patientswhosufferfromsymptomsSandwhotakedrugD,approximately95%gotsomereliefb1:eisapredictionfromhb2:hwasdevisedtoexplaineh:DrugDrelievessymptomsSinapproximately95%ofthecases.

Accordingtothepredictionist,e&b1isstrongerevidencethaththanise&b2(oreisstrongerevidencethathgivenb1thanitisgivenb2).

Accordingtotheexplanationist,e&b2isstrongerevidencethaththanise&b1(oreisstrongerevidencethathgivenb2thanitisgivenb1).

PatrickMaheroffersapredictionistresponseofthissort,whichheformulatesasfollows.13Supposethatahypothesishisgeneratedonsomeoccasionby(p.218) amethodM.LetMhbethatMgeneratedahypothesisthatentailsh.LetObethat(asMaherputsit)ewasavailabletoMwhenitgeneratedh.Then,accordingtoMaher,thepredictionistthesisisthis:

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(PT)p(h/Mh&e&O)>p(h/Mh&e&O).14

Theprobabilityontheleftisconditionalontheassumptionthatewasnotusedtogenerateh.(Thisisthepredictioncase.)Theprobabilityontherightisconditionalontheassumptionthatewasusedtogenerateh(whichMahercallsaccommodation).Maherdoesnotclaimthat(PT)holdsuniversally.Buthedoesproveatheoremshowingthatitholdsundercertainconditions,which,heclaims,usuallyobtaininscience.15

Beforediscussinganyoftheseconditions,IwillgiveanexampleMaherhimselfoffers,andthenshowhowhewoulddealwiththedrugcaseabove.Maher'sexampleinvolvescointossinganditpurportstoshowthatpredictionprovidesstrongerevidencethanaccommodation.16Heimaginestwocases,asfollows:

(a)Accommodation.Afaircoinisrandomlytossed99timesbyanexperimenter,andtheoutcomeofeachtossisrecordedinasentencee.Thehypothesishisaconjunctionofewiththeproposition(h)thatthecoinwilllandheadsonthe100thtoss.Letbcontaintheinformationthattheresultsofthe99tosseswerefirstrecordedbytheexperimenter,followingwhichtheexperimenterformulatedtheconjunctivehypothesishtoaccommodatee.(Thisistheexperimenter'smethodMofgeneratinghypothesish.)AccordingtoMaher,

(1)p(h/e&b)=p(h/e&b)=.

Thatis,giventhatoneachofthefirst99tossesthecoinlandedinthemannerdescribedbyeandgiventheaccommodationdescribedinb,theprobabilitythatthecoinwilllandheadsonthe100thtossandthatinthefirst99tossesitlandedinthemannerdescribedbyeisjusttheprobabilitythatitwilllandheadsonthe100thtoss,giventhatitlandedthewayitdidonthefirst99tossesandgivenaccommodation.Assumingthecoinisfairandtossedrandomly,thelatterprobabilityshouldbe.

(b)Prediction.Propositionse,h,andhareasabove,butbischanged.Insteadoffirsttossingthecoin99timesandrecordingtheresults,theexperimenterpredictstheresultsofthefirst99tosses,viz.e,andalsopredictsthatthe100thtosswillyieldheads.(Callthisconjunctivefactb(e).)17Thenthecoinistossed99timesandtheresultseareexactlyaspredicted.Now,accordingtoMaher,

(2)p(h/e&b(e))=p(h/e&b(e))approximately1.

(p.219) LetCPibethattheexperimentercorrectlypredictstheresultoftheithtoss.Wherehisthatthe100thtosswillbeheads,letP(h)bethattheexperimenterpredictsthatthe100thtosswillbeheads.Thene&b(e)isequivalenttoCP1...CP99&P(h).Sofrom(2)wehave

(3)p(h/CP1...CP99&P(h))=approximately1.

Inthiscase,giventhattheexperimentersuccessfullypredictstheresultsofthefirst99tosses,viz.e,andpredictsthatthe100thtosswillbeheads,theprobabilitythathis

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prediction(h)ofheadsonthe100thtosswillbecorrectiscloseto1.Theonlydifferencebetween(1)and(3)isthatin(1)thedatarecordedineareaccommodatedbytheexperimenterusingtheconjunctivehypothesish(=h&e),whilein(3)thedatarecordedinearecorrectlypredictedbytheexperimenterinadvanceoftheexperiment.

Maherattributesthedifferenceinprobabilitiestothefactthatinthesuccessfulpredictioncase,butnottheaccommodationcase,thereisastrongreasontosupposethattheexperimenterhasareliablemethodofpredictingcointosses.Thisisthebasisforhisclaimthatpredictionsprovidestrongerevidencethanaccommodations.

5.MaherandtheDrugCaseReturningnowtomydrugcase,hereishowMaher'saccountwouldproceed.(InwhatfollowsIaltertheexampletoconformtothisaccount.)Supposethereare11clinicaltrials,eachtodeterminetheeffectivenessofdrugDinrelievingsymptomsS.Let

e1:Inthefirst10clinicaltrialsinvolving1000SsuffererswhotookdrugD,95%gotreliefin9outofthe10trials.Inoneofthetentrials,75%gotrelief.e2:Inthe11thtrialinvolving1000SsuffererswhotookdrugD,95%gotrelief.M:Iftherelativefrequencyof95%successintenclinicaltrialsinvolving1000Ssufferersis.9,inferthesame95%successrateinanothertrialinvolving1000SsuffererswhotookD.Me2:Mwasusedtogenerateahypothesisthatentailse2.Mh:Mwasusedtogenerateahypothesisthatentailsh.Oe(i):eiwasavailabletoMwhenMgeneratedei.Oh:eiwasavailabletoMwhenMgeneratedh.h:e1&e2.

Now,accordingtothepredictionistthesis,

(PT)p(h/Mh&e1&Oh)>p(h/Mh&e1&Oh)

Theprobabilityontheleftrepresentsthepredictioncase,theoneontherightrepresentsaccommodation.Thatis,ife1isaprediction,thenhhasahigherprobabilitythanife1isaccommodated(thatis,ife1wasavailabletoMwhenMgeneratedh).

(p.220) Inthepredictioncase,Maherclaims,thefactthate1,whichisentailedbyh,wascorrectlypredicted(indicatedbye1&Oh)booststhereliabilityofthemethodMusedtogenerateh.Thisinturnbooststheprobabilityofh.Bycontrast,intheaccommodationcase,thefactthate1wasaccommodated(indicatedbye1&Oh)doesnotboostthereliabilityofthemethodMusedtogenerateh,ordoesnotboostitasmuchasthepredictioncasedoes.Henceinthiscaseh'sprobabilityisnotashighasitisinthepredictioncase.

OneofthecentralassumptionsofMaher'sgeneraltheoremconcernsthelackofsuchaboostinthecaseofaccommodation.MaherspeaksofthereliabilityRofamethodforgeneratingahypothesis.Thisisthe(objective)probabilitythatthehypothesisgenerated

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bythismethodwillbecorrect.LettheexpectedvalueofthereliabilityofamethodMforgeneratingahypothesishbedenotedbyE(RM(h)).18ConsiderMabove,amethodthatwillgeneratee2.Mahermakesanassumptionforhistheoremwhichhasthefollowingconsequence[p.336,see(22)]:

(1)E(RM(e2)/Me2&e2&Oe2&e1)=E(RM(e2)/e1).

Heassumesthatifeissimplyaccommodated(butnotpredicted),thentheexpectedvalueofthereliabilityofmethodMwithrespecttoe2isnotincreased.

Nowinthedrugcaseabove,thisassumptiondoesnothold.Accordingtoe2,theresultinthe11thtrialsatisfiesM.ThisshouldboosttheexpectedvalueoftheprobabilitythatahypothesisgeneratedbyMiscorrect,whethertheresultsinthe11thtrialwereaccommodatedorpredicted.

Maher'sresponseistoagreethatinthissortofcasethepredictionistthesis(PT)doesnothold.19Insuchacase,hesays,itiscertainwhatthemethodMwouldpredict,thatis,

(2)p(Me2/e2&Oe2&e1=p(Me2/e2&Oe2&e1)=1.

Thisviolatesanassumptionforhistheoremfromwhich(1)abovefollows.Althoughheagreesthat(PT)doesnotholdwhen(2)istrue,heclaimsthatintheusualscientificcases(2)isfalse.Maherwrites:

Onlyinveryspecialcasescanwepredictwithcertaintywhathypothesesscientistswillgenerate.20

Inresponse,notefirstthat(2)isconcernednotwiththeprobabilitythatascientistwillgeneratethehypothesise2butwiththeprobabilitythatmethodMwillgeneratee2.Iagreethattheformerprobabilityisnot1,butitisthelatterprobabilitythatisofconcernin(2).SoisMaherclaimingthatintypicalcasesofprediction,whereascientistusesamethodforgeneratingapredictivehypothesish,theprobabilitythatthemethodgenerateshislessthan1?Theanswerseemstobeyes.TheonlyjustificationMaheroffersforthisansweristhattypicallyinpredictive(p.221) casesthemethodbeingusedisnotunderstoodwellenoughtoallowittobecertainthatthemethodgeneratesthepredictioninquestion.

Asmentionedinnote14,Maheremploysasubjectiveinterpretationofprobabilityinhispredictionistthesis(PT).SohispresentclaimwouldbethatatypicalscientistwhomakesapredictionusingamethodMdoesnotunderstandMwellenoughtobecompletelysurethatMyieldsthatprediction.21ThescopeandinterestofMaher'spredictionistthesis(PT),then,dependsonhowtypicalitisforscientistsusingamethodtopredictthetruthofahypothesishnottounderstandthemethodwellenoughtoknowforsurewhetherhisactuallypredictedbythatmethod.Tosaytheleast,thisisverydifferentfromtheusualpredictionistidea.Moreimportantly,itisonewhosewideapplicabilityseemsquestionable.Frequently,ascientistwillbeinasituationinwhichthemethodofpredictionconsistsinderivingthepredictiondeductivelyfromatheoryorhypothesis,

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wherethescientistknowswhetherornotthederivationiscorrect(withoutnecessarilyknowingwhetherwhatisderivedistrue).

6.Balmer'sFormulaItmaybeusefultociteasimplehistoricalexample,viz.Balmer'sformula,thatissimilarincertainrespectstooneemployedbyMaher.22Whenlightfromhydrogenisanalyzedusingaspectroscope,itisseentoconsistinseriesofsharplinesofdefinitewavelengths.In1885,JohannJakobBalmerintroducedageneralformulathatentailedthewavelengthsofthefourlinesknownbyhimatthattime.Theformulacanberepresentedasfollows:

whereisthewavelengthofagivenline,Risaconstant,andn=3,4,5,6forthefourlines.Balmerdoesnotclaimtobeexplainingwhythelinesoccurorhavethewavelengthstheydo,butsimplytoberepresent[ing]thewavelengthsofthedifferentlinesinasatisfactorymanner.23ThisseemstobeacasesatisfyingMaher'snotionofaccommodation.

(p.222) NowBalmerindicatesthatheusedhisformulatoobtainthewavelengthofafifthline(lettingn=7).Hesaysheknewnothingofsuchalinewhenheperformedthecalculation,andwaslaterinformedthatitexistsandsatisfiestheformula.Sothefifthlinewas,fromhisstandpoint,apredictionthatturnedouttobecorrect.Moreover,hereportsbeinginformedthatmanymorehydrogenlinesareknown,whichhavebeenmeasuredbyVogelandHugginsinthevioletandultravioletpartsofthehydrogenspectrumandthespectrumofthewhitestars(p.362).WhatimpressesBalmer,however,isnotthefactthathehasmadeasuccessfulprediction,butsimplythefactthatallofthelines,whetheraccommodatedorpredicted,satisfyhisformula.Hewrites:

Fromthesecomparisonsitappearsthattheformulaholdsalsoforthefifthhydrogenline....ItfurtherappearsthatVogel'shydrogenlinesandthecorrespondingHugginslinesofthewhitestarscanbepresentedbytheformulaverysatisfactorily.WemayalmostcertainlyassumethattheotherlinesofthewhitestarswhichHugginsfoundfurtheronintheultravioletpartofthespectrumwillbeexpressedbytheformula.(p.362)

AsfarasBalmerisconcerned,itisthefactthatthevariouslines,whetherfirstknownandlateraccommodatedorfirstpredictedandlaterknown,allsatisfyhisformulathatprovidesstrongevidenceforthelastclaimintheabovepassage.

LetB(i)meanthatlineisatisfiesBalmer'sformula.Balmer'sclaimisthat

(3)p(B(5)/B(1)...B(4))isveryhigh.

Thishasnothingtodowithaccommodationorprediction.However,letthemethodforgeneratinghypothesesoftheformB(i)beasfollows:

M:UseBalmer'sformulatoobtainB(i).

1/ = R( 1/ )n 14 n2

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TheprobabilityinwhichMaherisinterestedforaccommodationis(4)p(h/M&e&O),

thatforpredictionis(5)p(h/M&e&O),

wherehisB(5)&e,eisB(1)...B(4),andOsaysthatewasavailabletoMwhenitgeneratedh.

IfMaher'spredictionistthesis(PT)holdsforthiscase,thentheprobabilityin(5)shouldbegreaterthanthatin(4).But,Isubmit,theyareequal.Therelevantconsiderationisthatinbothcaseslines1through4satisfyBalmer'sformula,andthatthehypothesisofinterestconcerningline5wasgeneratedusingtheBalmerformula.ThetruthvalueofclaimsB(1)...B(4)isrelevant,butitisirrelevantwhetherornotthetruthvaluesoftheseclaimswereavailabletomethodMwhenitgeneratedB(5).

Tobesure,insuchacaseMaherwillsaythatBalmer'smethodMissuchthatwhetherB(i)isapredictionusingMisknownforcertain.24ButIsubmit,itisjust(p.223) thesekindsofpredictions,viz.deductionsfromhypothesesand/orapplicationsofquantitativeformulastonewcases,thatarefrequentlymeantbythosewhodefendapredictionistthesis.Moreover,Isubmit,theyarenotatypicalinscience.

7.BrushReduxBrushisclearlydenyingageneralpredictionistthesis.Bycontrast,hecitescasesinwhichscientiststhemselvesregardedknownevidenceexplainedbyatheoryasstrongersupportforthattheorythannewevidencethatwassuccessfullypredicted.Andheseemstoimplythatthiswasreasonable.Heoffersanexplanationforthisclaim,viz.thatwithexplanationsoftheknownphenomena,bycontrastwithsuccessfulpredictionsofthenewones,scientistshadtimetoconsideralternativetheoriesthatwouldgeneratethesephenomena.Now,evenifBrushdoesnotdoso,IwanttoextendthisideaandconsideramoregeneralexplanationistviewthatiscommittedtothefollowingthreethesesthatBrushinvokesforsomecases:

(1)Aselectionprocedurefortestingahypothesishisflawed,oratleastinferiortoanother,otherthingsbeingequal,ifitfailstocallforexplicitconsiderationofcompetitorstoh.(2)Thelongertimescientistshavetoconsiderwhetherthereareplausiblecompetitorstohthemorelikelytheyaretofindsomeiftheyexist.(3)Withputativeevidencealreadyknownbeforetheformulationofhscientistshave(had)moretimetoconsiderwhetherthereareplausiblecompetitorstohthanisthecasewithnovelpredictions.

Iwouldchallengeatleastthefirstandthirdtheses.Inthedrugexampleofsection2,selectionprocedureSP1forthedrughypothesisdoesnotcallforexplicitlyconsideringcompetitorstothathypothesis.Yetitdoesnotseemflawedonthataccount,orinferiortoonethatdoes.However,evensupposingitwereinferior,whetherornotaselection

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procedurecallsforaconsiderationofcompetitorsiscompletelyirrelevanttowhethertheputativeevidenceclaimisapredictionoraknownfactbeingexplained.Inthecaseofaprediction,nolessthanthatofanexplanation,theselectionproceduremaycallforaconsiderationofcompetitors.

Forexample,inourdrugcase,wherehisDrugDrelievessymptomsSinapproximately95%ofthecases,andeisthepredictionInthenextclinicaltrialof1000patientssufferingfromsymptomsSwhotakeD,approximately950will(p.224) getsomerelief,theselectionproceduretobeusedforthenextclinicaltrialmightincludetherule

Inconductingthisnexttrial,determinewhetherthepatientsarealsotakingsomeotherdrugwhichrelievesSinapproximately95%ofthecasesandwhichblocksanyeffectivenessDmighthave.

Suchaselectionprocedurecallsfortheexplicitconsiderationofacompetitortoexplaine,viz.thatitwillbesomeotherdrug,notD,thatwillrelievesymptomsSinthenexttrial.Thisissoeventhougheisaprediction.Moreover,torespondtothethirdthesisabouttimeforconsideringcompetitors,aninvestigatorplanningafuturetrialcanhaveasmuchtimeasshelikestodevelopaselectionprocedurecallingforaconsiderationofacompetinghypothesis.Moregenerally,indesigninganovelexperimenttotestsomehypothesish,asmuchtimemaybespentinprecludingcompetinghypothesesthatwillexplainthetestresultsasisspentinconsideringcompetinghypothesesforolddata.

8.ThomsonVersusHertz:TheWaveTheoryofLightLetmeinvoketwoscientificexamplesemployedearlier.ThefirstinvolvesthedisputebetweenHeinrichHertzandJ.J.Thomsonoverthenatureofcathoderays,discussedinchapter2above.Recallthatin1883Hertzobservedthatthecathoderaysinhisexperimentswerenotdeflectedbyanelectricfield.Hetookthistobestrongevidencethatcathoderaysarenotchargedparticlesbutsometypeofetherwaves.In1897ThomsonrepeatedHertz'sexperimentsbutwithamuchhigherevacuationofgasinthecathodetubethanHertzhadbeenabletoobtain.Thomsonbelievedthatwhencathoderayspassthroughagastheymakeitaconductor,whichscreensofftheelectricforcefromthechargedparticlescomprisingthecathoderays.Thisscreeningoffeffectwillbereducedifthegasinthetubeismorethoroughlyevacuated.InThomson's1897experimentselectricaldeflectionofthecathoderayswasdetected,whichThomsontooktobestrongevidencethatcathoderaysarechargedparticles.

IwanttoconsidertheevidentialreportofHertzin1883,notofThomsonin1897.Let

1. e=InHertz'scathoderayexperimentsof1883,noelectricaldeflectionofcathoderayswasdetected.

2. h=Cathoderaysarenotelectricallycharged.

Hertzconsideredetobestrong(veridical)evidencethath.In1897ThomsonclaimedthatHertz'sresultsasreportedinedidnotprovidestrongevidencethath,sinceHertz'sexperimentalsetupwasflawed:Hewasemployinginsufficientlyevacuated

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tubes.Tousemypreviousterminology,ThomsonwasclaimingthatHertz'sselectionprocedurefortestinghwasinadequate,andhencethateisnot(potentialorveridical)evidencethath.

HerewecanpickuponapointemphasizedbyBrush.Hertz,wemightsay,failedtouseaselectionprocedurecallingforconsideringacompetitortohtoexplainhisresults(viz.thatcathoderaysarechargedparticles,butthatthetubes(p.225) Hertzwasusingwerenotsufficientlyevacuatedtoallowanelectricalforcetoactontheseparticles).ButandthisisthepointIwanttoemphasizeindeterminingwhether,ortowhatextent,eis(potentialorveridical)evidencethath,itisirrelevantwhetherHertz'sewasanovelpredictionfromanalreadyformulatedhypothesishoranalreadyknownfacttobeexplainedbyh.Hertzwritesthatinperformingtherelevantexperimentshewastryingtoanswertwoquestions:

Firstly:Dothecathoderaysgiverisetoelectrostaticforcesintheirneighbourhood?Secondly:Intheircoursearetheyaffectedbyexternalelectrostaticforces?25

Inhispaperhedidnotpredictwhathisexperimentswouldshow.Norweretheresultsofhisexperimentstreatedbyhimasfactsknownbeforehehadformulatedhishypothesish.Onceheobtainedhisexperimentalresultshethenclaimedthattheysupportedhistheory:

Asfarastheaccuracyoftheexperimentallows,wecanconcludewithcertaintythatnoelectrostaticeffectduetothecathoderayscanbeperceived.(p.251)

Tobesure,wemightsaythatHertz'stheoryitselfpredictedsomesuchresults,evenifHertzhimselfdidnot(thatis,evenifHertzdidnothimselfdrawhisconclusionbeforegettinghisexperimentalresults).Butevenifwespeakthisway,Hertzdidnotclaimorimplythathisexperimentalresultsprovidebetter(orweaker)evidenceforhistheorybecausethetheorypredictedthembeforetheywereobtained.NordidThomsoninhiscriticismofHertzalludetooneortheotherpossibility.WhicheveritwaswhetherapredictionoranexplanationorneitherHertz(Thomsonwasclaiming)shouldhaveusedabetterselectionprocedure.ThisiswhatiscriticizableinHertz,notwhetherhewaspredictinganovelfactorexplainingaknownone.

Thesecondexampleinvolvesthenineteenthcenturyargumentforthewavetheoryoflightgiveninchapter7,section7.Theargumentbeganwithtwoclaims:(i)thatlighttravelsfromonepointtoanotherwithafinitevelocity;(ii)thatinotherknowncases,suchassoundwaves,waterwaves,andprojectiles,modesoftransferinvolvingfinitevelocitiesareviaclassicalwavesinamediumorclassicalparticles.From(i)and(ii),itisconcludedthat(h)lightiseitherawavemotioninamediumorastreamofparticles.Inprobabilisticterms,

(1)p(h/(i)&(ii))iscloseto1.

Indeed,itmightbeclaimedthat

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(2)p(thereisanexplanatoryconnectionbetweenhand(i)/(i)&(ii))>,

sothat(since(i)and(ii)weredeemedtrueinthenineteenthcentury),(i)constitutespotentialevidencethath,given(ii).Theclaimthatitdoesconstituteevidencethatlightiseitheraclassicalwaveoraclassicalstreamofparticleswasmadebywavetheoristsaspartoftheireliminativeargumentforthewavetheory.

Now,whatIbelievecanlegitimatelybesaidaboutthiscaseisthatalthough(p.226) (i)constitutesESevidencethath(wheretheepistemicsituationistheoneholdingforwavetheoristsduringthefirstfourdecadesofthenineteenthcentury),itisnotpotentialevidencethath.Thereasonitisnothasnothingtodowithwhetherwavetheoristsemployedhtoexplainthealreadyknownfactthatlighttravelsfromonepointtoanotherinafinitetimeortopredictnewcasesinwhichthisisso.Thereasonisthattheselectionprocedureisbiased,althoughthiswasnotknownbywavetheoristsandcouldnotbeknownuntilthetwentiethcentury.Theitemsselectedtodefendtheclaimthatmodesoftransferinvolveclassicalwavesandparticlessuchaswaterwaves,soundwaves,andprojectilesareallitemsfromthemacroworldofclassicalwavesandparticles;therearenonefromthesubatomicworldsubjecttothelawsofquantummechanicswhichprecludeclassicalwavesandparticles.Accordingly,(1)and(2)arecriticizable,notthefactthatwavetheoristswereexplainingknownfacts(orpredictingnewones).

IendthissectionwithaquotefromJohnMaynardKeynes,whosebookonprobabilitycontainslotsofinsights.Hereisone:

Thepeculiarvirtueofpredictionorpredesignationisaltogetherimaginary.Thenumberofinstancesexaminedandtheanalogybetweenthemaretheessentialpoints,andthequestionastowhetheraparticularhypothesishappenstobepropoundedbeforeoraftertheirexaminationisquiteirrelevant.26

9.TheProblemofOldEvidenceYearsagoClarkGlymourraisedafundamentalobjectiontothepopularpositiverelevancedefinitionofevidence.27Supposethattheprobabilityofeis1.Ifitis,then,assumingthattheprobabilityofhisgreaterthan0,p(e/h)=1.Now,accordingtoBayes'theorem,

(1)p(h/e)=p(h)xp(e/h)/p(e)).

Therefore,ifp(e)=1,thenfrom(1)weobtain(2)p(h/e)=p(h).

Thatis,theprobabilityofh,givene,isthesameasitspriorprobability.Nowonthepositiverelevancedefinition,

(3)eisevidencethathifandonlyifp(h/e)>p(h)

Soifp(e)=1andp(h)=0,thenfrom(2)and(3),ecannotbeevidencethath.

Whyisthisaproblem?Suppose,saysGlymour,thatpriortotheintroductionof

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hypothesish,ewasknownwithcertaintytoobtain.Glymourconcludesthate'sprobabilityisthen1.(Iwillreturntothisclaiminamoment.)Ife'struthwasknownpriortotheintroductionofh,theneisoldevidencewithrespecttoh.Itfollowsfromthepositiverelevancedefinitionofevidencethatoldevidencewithrespecttoahypothesiscannotbeevidencethatthehypothesisistrue,since(p.227) itcannotincreasetheprobabilityofthehypothesis.Thisstrikesmanyasabsurd,sincefrequentlyphenomenaconsideredevidenceinfavorofatheorywereknownwithcertaintytoobtainpriortotheformulationofthetheory.

Insection3ofthischapterweconsideredwhetherpredictionsofnovelfactsprovidestrongerevidenceforatheorythanexplanationsofoldones,orwhetherthereverseistrue.Predictionists(suchasWhewellandPopper)mightwelcomeGlymour'sproblembyrespondingthatifeisalreadyknowntobetruepriortotheformulationofh,thenitcannotbe(verymuch)evidenceforh,evenifitisderivablefromh.Forthem,inorderthatebeevidencethath,oratleastsubstantialevidence,itmustbeanewpredictionandnotafactknownpriortotheformulationofh.

Nowtomakethisclaimistoespousethehistoricalthesisofevidenceformulatedinsection1.Toknowwhethereisanewpredictionandnotafactknownpriortotheformulationofhonemustknowhistoricalfactsabouteandh,viz.whetherandwheneisknownandwhetheritisknownpriortowhenhwasproposed.InmyearlierdiscussionIrejectedthehistoricalthesisasauniversalprinciple.Tobesure,therearecasesinwhichdeterminingwhether,ortowhatextent,esupportshrequiresdeterminingthetruthofsomehistoricalfactaboute,h,ortheirrelationship.Buttherearealsocaseswherethisisnotsoatall.Toclaimthatonlypredictionsprovide(substantial)evidenceforahypothesisistoespouseamistakenhistoricalthesis.

Glymourconsideredtheproblemofoldevidencesufficienttoshowthatoneshouldnotaccept(3),thestandardpositiverelevancedefinitionofevidence.Sinceinchapter4Ihavegivenmanyotherreasonstorejectthisdefinition,whatIwanttofocusonhereisnotthequestionofwhetherGlymour'sobjectionisdevastatingtothepositiverelevanceaccountbutonwhatoneistosayaboutcasesinwhichtheprobabilityoftheputativeevidenceis1.

However,beforedoingsoIreturntoacentralclaimofGlymour'sargument,viz.thatifeisknowntobetrue(whetherornotthiswaspriortotheformulationofh),thenp(e)=1.Isthistrue?Itdoes,indeed,holdforsubjectiveprobability.IfatacertaintimetIknowthateistrue,thenattmysubjectiveprobabilityforeismaximal.28DoesitholdfortheconceptofepistemicprobabilitythatIespouse?

Myansweristhatitdependsonwhatrelativization,ifany,isbeingassumedintheprobabilitystatement.Ifeisknowntobetrue,thenitfollowsthatitistrue.Andiftheprobabilityinquestionisrelativizedtoe,ortothefactthateisknowntobetrue,then,tobesure,theprobabilityofeis1(pe(e)=1).Butthatistrivialanduninteresting.Withotherlesstrivial,moreinteresting,relativizationstheprobabilityisnot1,eventhoughitisknownthateistrue.Forexample,supposeeisasfollows:

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e:Thiscoin,whichwastossed100timesonJanuary1,2000,landedheads95%ofthetime.

(p.228) Supposesomepersonknowsthateistrue.Wemaywanttoconsidertheprobabilityofedisregardingthisknowledge,thatis,pd(Ke)(e),inwhichd(Ke)meansdisregardthefactthateisknowntobetrue.Inthiscasetheprobabilitymaybelessthan1,eventhoughitisknownthateistrue.29Accordingly,withepistemicprobability,fromthefactthateisknowntobetrue,itdoesnotnecessarilyfollowthattheprobabilityofeis1.Itdependsonwhatis,orisnot,beingassumed.

IturnnowtothequestionofwhetherGlymour'sproblemcanbeputinaformwhichdoesnotintroducetheideathateisafact(not)knownpriortotheformulationofh.Usingepistemicprobabilities,supposethatpb(e)=1.Thisisnotahistoricalclaim.Itdoesnotsaythatanyoneinfactknowsorcametoknowthateistrue,orthatanyonedidsobeforeoraftertheformulationofh.Allitsaysisthat,inviewofb,thedegreeofreasonablenessofbelievingeismaximal.Coulditbethecasethat,givenb,eispotentialorveridicalevidencethath?Ordoesthefactthatpb(e)=1precludethispossibility?

Mydefinitionsofpotentialandveridicalevidencedonotprecludethispossibility.Itispossibleforthefollowingconditionsforpotentialevidencetobesatisfied:

1.eandbaretrue2.edoesnotentailh3.pb(h/e)xpb(thereisanexplanatoryconnectionbetweenhande/h&e)>,

evenifpb(e)=1.

Todemonstratethissupposethebackgroundinformationbcontainsthefactthatthereisalotteryconsistingof1000ticketsoneofwhichwillbedrawnatrandom,andthatJohn'swifeMarypurchased950oftheseticketsasapresentforJohn,whodepositedtheminhissafedepositboxwithalegaldocumentsayingthattheticketsbelongtohim.LetebethatJohnowns950ticketsinthislottery.LethbethatJohnwillwin.Ifeandbaretrue,thentheaboveconditionsfore'sbeingpotentialevidencethatharesatisfied.Yetpb(e)=1,sinceinthiscasethebackgroundb(whichwemaysupposecontainsstandardlegalprinciplesofownership)entailse.Givenb,thedegreeofreasonablenessofbelievinge(thatJohnowns950ticketsinthislottery)ismaximal.Yetgivenb,eispotentialevidencethath.And,ifhistrue,itisveridicalevidenceaswell.Thefactthat,inthelightofthebackgroundinformationb,itismaximallyreasonabletobelieveedoesnotatallcountagainste'sbeingevidencethath.Indeed,inthiscase,inthelightofb,eisverystrongevidencethath.

Now,letusconsiderwhat,ifanything,happensifitisunreasonabletobelievee.Ifp(e)=0,thenp(h/e)isundefined.Soletusconsideracaseinwhichehaslow,butnonzero,probability.Predictionistschampionsuchcases,becausethelowertheprobabilityofthepredictione,thehighertheposteriorprobabilityofhone.

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Letthebackgroundinformationbbethatafaircoinwillberandomlytossed100times.Letebethatthecoinwilllandheads100timesinarow.LethbethehypothesisthattheDevilwillinterveneaftereachrandomtoss,causingthecoin(p.229) tolandheadseachtimeduringthefirst100tosses.Inthiscasethehypothesishentails(theprediction)e.Assumingb,theprobabilityofeisverylow: .Yet,givenb,itseemsfarfetchedtosaythate,iftrue,ispotentialevidencethath.Indeed,thedefinitionofpotentialevidenceprecludesthis,evenifeandbarebothtrue,since(givennormalbackgroundinformation)theprobabilityofthedevilhypothesish,evenontheassumptionofe,isextremelylow.Thefactthatthecoinwilllandheads100timesinarow,eveniftrue,isnotpotentialevidencethattheDevilwillintervene.

Canwefollowthepredictionistandsayatleastthis:Wherehentailssomepredictione,thelowertheprobabilityofethestrongertheevidencethateconfersuponh?No,wecannot.Allwecansayisthatthelowertheprobabilityofe,insuchacase,thehighertheprobabilityofhone.Butitdoesnotfollowfromthisthatthelowertheprobabilityofethestrongertheevidencethateconfersuponh,sinceemayconfernoevidenceuponh.Thus,inthepreviousexample,letuschangeeto

e:Thecoinwilllandheads1000timesinarow.

Lethandbbethesameasbefore.Nowwehave ,whichisamuchlowerprobabilitythan .Yetthatdoesnotmakeestrongerevidencethaththaneis,since,onmyconception,eisnotevidencethath.Thethresholdforhighprobabilityrequiredforevidencehasnotbeenreached.

Whatcanbesaidisthis.Ifeisevidencethath,thenthelowertheprobabilityofethestrongeristheevidencethateconfersuponh.Forexample,letbcontaintheinformationthatthiscoinisperfectlysymmetrical.Lethbethehypothesisthatitwilllandheadsapproximatelyhalfthetime.Lete1betheinformationthatwhentossedrandomlythefirst100timesitlandedheadsbetween45and55times.Lete2betheinformationthatwhentossedrandomlythenext1000timesitlandedheadsexactly500times.Wemightsaythat,givenb,bothe1ande2countasevidencethath.Nowinthiscasepb(e2)pb(h/e1).Butinthiscasealso,inthelightofb,e2isstrongerevidencethaththanise1.

Accordingly,whetherehasveryhighorverylowprobabilitydoesnotnecessarilyaffectwhether,ortheextenttowhich,eisevidencethath.Norisitingeneraltruethatifhpredictse,thelowerise'sprobabilitythestrongerise'sevidencethath.

10.Conclusions1.Accordingtothehistoricalthesisofevidence,whethereiftrueisevidencethath,orhowstrongthatevidenceis,dependsoncertainhistoricalfactsaboute,h,ortheirrelationship(forexample,onwhetherewasknownbeforeorafterhwasformulated).Althoughthisthesisholdsforsubjectiveevidence,itdoesnotholduniversallyfortheconceptsofobjectiveevidenceIhaveintroduced.Focusingonpotentialandveridical

(e) = ( )pb 12 100

(e') = ( )pb 12 1000(e) = ( )pb 12 100

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evidence,dependingontheparticularevidenceclaimandtheselectionprocedureemployed,insomecasesthehistoricalthesisholds,inothersitdoesnot.

(p.230) 2.Sometimesanovelfactthatispredictedprovidesbetterevidenceforahypothesisthanaknownfactthatisexplained.Sometimesthereverseistrue.Whichobtainshasnothingtodowithwhetheritisapredictionoranexplanation,butrather,inthecasesinquestion,withtheselectionprocedureusedtogeneratetheevidence.ThisisillustratedinthecaseofHertz'sclaimthattheresultsofhis1883cathoderayexperimentsprovideevidencethatcathoderaysarenotelectricallycharged.Itisalsoillustratedinthewavetheorists'argumentthatlightconsistsofclassicalwavesorclassicalparticles.

3.AresponseofPatrickMaherisexaminedwhichinvolvesformulatingtheinformationconcerningwhethereisapredictionorafactexplainedaspartoftheevidencestatementitselforaspartofthebackgroundinformation.Thisnewformulationwillnotsufficetoestablishthesuperiorpowerofpredictionoverexplanation(oraccommodation),orviceversa,intypicalscientificcasesofinteresttopredictionistsandexplanationists.

4.Glymourraisestheproblemofoldevidenceinordertorejectthepositiverelevancedefinition.Accordingtothatdefinition,ecannotbeevidencethathwheneisoldevidenceknowntobetrue.Predictionistsmaywelcomethisresult,sinceforthem,inorderforetobesubstantialevidencethath,emustbeanewprediction,notafactknownpriortotheformulationofh.Tosaythisistoespousethehistoricalthesisofevidenceasuniversal,whichIdonot.IdiscussGlymour'sprobleminaformthatdoesnotcommitonetothehistoricalthesis.Ifthedegreeofreasonablenessofbelievingtheputativeevidenceeismaximal(correspondingtooldevidence),canebepotential(orveridical)evidencethath?Idemonstratethatitcanbe.Sothereisnocomforthereforthepredictionist.(Nordoesthefactthattheprobabilityofeismaximal,whereeisexplainablebyderivationfromh,guaranteethateispotentialevidencethath;sothereisnocomfortherefortheexplanationisteither.)Asecondquestionisthis:Ifhentailssomepredictione,doesthelowertheprobabilityofemeanthestrongertheevidencethateconfersuponh?Theansweragainisno.Soagainthereisnocomfortforthepredictionist.

Notes:(1.)WilliamWhewell,ThePhilosophyoftheInductiveSciences(NewYork:Johnsonreprint,1967;fromthe1847ed.)KarlPopper,TheLogicofScientificDiscovery(London:Hutchinson,1959).

(2.)StephenBrush,PredictionandTheoryEvaluation:TheCaseofLightBending,Science,246(1989),11241129.

(3.)TowhatextentBrushwantstogeneralizethisexplanationistpositionisaquestionIleaveforhimtoanswer.Therearepassagesthatstronglysuggestamoregeneralposition.Forexample:Thereisevensomereasontosuspectthatasuccessfulexplanationofafactthatothertheorieshavealreadyfailedtoexplainsatisfactorily(forexample,theMercuryperihelion)ismoreconvincingthanthepredictionofanewfact,at

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leastuntilthecompetingtheorieshavehadtheirchance(andfailed)toexplainit(p.1127).Inwhatfollows,Iconsiderageneralizedexplanationistthesis.

(4.)eandhherearepropositions.Asnotedinchapter2,thisisthecustomarypracticeofphilosopherswhospeakofeasbeingevidencethath,eventhoughitisthefactthateistruethatisevidence.Thehistoricalthesis,then,concernsthepropositionseandhortheirrelationship.

(5.)AlanMusgrave,LogicalversusHistoricalTheoriesofConfirmation,BritishJournalforthePhilosophyofScience,25(1974),123.

(6.)RudolfCarnap,LogicalFoundationsofProbability.

(7.)CarlG.Hempel,AspectsofScientificExplanation.

(8.)ClarkGlymour,TheoryandEvidence(Princeton:PrincetonUniversityPress,1980).

(9.)SeeLauraJ.Snyder,IsEvidenceHistorical?,ScientificMethods:ConceptualandHistoricalProblems,PeterAchinsteinandLauraJ.Snyder,eds.(Malabar,Florida:Krieger,1994).

(10.)Itmightbeobjectedthatifthisisso,then,contrarytowhatIhavebeensaying,theconceptofevidenceinvolvedpotentialorveridicalevidencecannotbeobjective.Whethereispotential(orveridical)evidencethathinsuchcaseswilldependonwhatselectionprocedurewasemployed,which,inturn,dependsonwhatsomeperson(s)believedabouteandh,viz.uponwhatwasbelievedabouthowewasgenerated.Myreplyisthatinthecasesinquestionwhethereispotentialevidencethathdependsonwhatselectionprocedurewasinfactemployed,notonwhatbeliefsitsemployer(s)mayhavehadaboutit.

(11.)Microconditionsarebeingdisregarded.Seechapter5.

(12.)SeeEricBarnes,SocialPredictivism,Erkenntnis,45(1996),6989.

(13.)PatrickMaher,Prediction,Accommodation,andtheLogicofDiscovery,PSA,1988,vol.1,273285;HowPredictionEnhancesConfirmation,inJ.M.DunnandA.Gupta,eds.,TruthorConsequences(Netherlands:KluwerAcademicPublishers,1990),327343.

(14.)MaherconstruesphereasthesubjectiveprobabilityforsomerationalpersonwhohasnotyetlearnedthetruthvaluesofMh,e,andO.SeeHowPredictionEnhancesConfirmation,p.327.Withsuitablerelativization,probabilityin(PT)couldalsobeunderstoodinanobjectiveepistemicsenseofthesortintroducedinchapter5.

(15.)HowPredictionEnhancesConfirmation,p.328.

(16.)Prediction,Accommodation,andtheLogicofDiscovery,p.275.Seealsohis

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HowsonandFranklinonPrediction,PhilosophyofScience,60(1993),329340.

(17.)Theexperimenterisusingsomemethod,howeverrandom,forgeneratingpredictions.AlthoughMaherdoesnotsaysoexplicitly,Iassumehewouldclaimthatthemethodbeingusedcanbutneednotbegiveninb(e).

(18.)Ifafunctionhasafinitenumberofvalues,theexpectedvalueisthesumofallproductsconsistingofagivenvalueandtheprobabilityofthatvalue.Ifafunction(suchasprobability)hasinfinitelymanyvalues,theexpectedvalueisdefinedusinganintegral.

(19.)Personalcorrespondence.SeehisHowsonandFranklinonPrediction,PhilosophyofScience,60(June1993),329340;seepp.339340.IamindebtedtoPatrickMaherforhelpintryingtogetmetoexpresshisviewsaccuratelyinthesesections;IhopeIhavedoneso.

(20.)HowsonandFranklinonPrediction,p.340.

(21.)Withanobjectiveconceptofprobability,ofthesortIdefendinchapter5,thisconclusionwouldnotbepermitted,unlessaspecialrelativizationisintroducedthatentailsdisregardinginformationindicatingwhetherMyieldsthepredictioninquestion.

(22.)MahercitespredictionsfromMendeleyev'speriodictableoftheelements.Mendeleyevplacedtheelementsingroupsbasedonatomicweights(ratherthanonotherbases)andshowedthatperiodicrepetitionsofpropertiesoccurinthegroupsandcanbeusedtopredictnewelements.Hedidsoexplicitlyclaimingthatatomicweightsphysicallydetermineandcanexplainpropertiesoftheelementsandcompounds.(SeehisTheRelationsbetweenthePropertiesofElementsandtheirAtomicWeights,reprintedinHenryA.BoorseandLloydMotz,eds.,TheWorldoftheAtom(NewYork:BasicBooks,1966,vol.1),p.306.)Bycontrast,Balmermakesnophysicalorexplanatoryclaimsregardinghisformula.WhileBalmer'sformulaaccommodatesthedata,itismoreclearlynonexplanatorythanMaher'sexample,whilealsobeingpredictive.StephenBrush,bycontrasttoMaher,claimsthattheperiodictablewasregardedbyscientistsasprovidingstrongerexplanatorythanpredictiveevidence.

(23.)JohannJakobBalmer,TheHydrogenSpectralSeries,reprintedinWilliamFrancisMagie,ASourceBookinPhysics(Cambridge:HarvardUniversityPress,1965),360365;quoteonp.360.

(24.)MaherclaimsthathisMendeleyevexampleisoneinwhichhispresentuncertaintythesisholds.Hewrites:NobodywithoutthebenefitofhindsightcouldbecertainthatMendeleyevwouldproposethehypothesishedid(HowsonandFranklinonPrediction,p.340).But,again,thisisnottheprobabilityofconcern.Thequestioniswhether,withrespecttoapredictionMendeleyevmadeusinghisperiodictable,hewasinanydoubtaboutwhetherhistableyieldsthatprediction.Afterconstructinghisperiodictablebasedonatomicweights,Mendeleyevwrites:...itappearstobecertainwhenwelookattheproposedtable,thatinsomerowsthecorrespondingmembersare

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missing;thisappearsespeciallyclearly,e.g.,fortherowofcalcium;inwhichtherearemissingthemembersanalogoustosodiumandlithium(p.310).Also,attheendofhispaperhewrites:Thediscoveryofnumerousunknownelementsisstilltobeexpected,forinstance,ofelementssimilartoAlandSihavingatomicweightsfrom6575(p.312).Mendeleyevseemstohavenodoubtthatthesepredictionsaregeneratedbyhistable.

(25.)HeinrichHertz,MiscellaneousPapers(London:Macmillan,1896).

(26.)J.M.Keynes,ATreatiseonProbability(London:Macmillan,1921).

(27.)ClarkGlymour,TheoryandEvidence,p.86.

(28.)Althoughsubjectivistsacceptthis,somewishtosolveGlymour'sproblembyrefusingtouseone'sactualdegreeofbeliefineindeterminingp(e)butsomecounterfactualdegreeofbelief,suchasone'sdegreeofbeliefineoneverythingoneknowsminuse.SeeColinHowsonandPeterUrbach,ScientificReasoning,p.404.

(29.)ThisepistemicprobabilityisanalogoustoasubjectiveonethatwouldbeusedbysubjectivistsofferingacounterfactualsolutiontoGlymour'sproblemmentionedinthepreviousfootnote.

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