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KnowledgeandConditionals

Knowledgeand Conditionals

EssaysontheStructureofInquiry

RobertC.Stalnaker

GreatClarendonStreet,Oxford,OX26DP, UnitedKingdom

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Acknowledgments

Nineofthepaperscollectedherehavebeenpreviouslypublished,orintwocases,are forthcominginotherpublications.Ithanktheeditorsandpublishersforpermission toreprintthemhere.Thedetailsaboutthesourcesarelistedonpageiv.

Ihavetoomanyintellectualdebtstoacknowledgethemallatthispoint,butletme mentionsomeofthepeoplethathaveaffectedmythinkingonalltheissuesIdiscuss inthechaptersofthisbook.

ManyoftheideasinthesepapersweredevelopedinseminarsIgaveatMITandat ColumbiaUniversityonconditionalsandontopicsinepistemologyoverthepast five orsixyears.Iwasfortunatetohavegroupsofverytalentedphilosophersparticipatinginthoseseminarswhosecriticalandconstructivecontributionstothediscussion helpedmetounderstandtheissues,andinfluencedmyresponsestothem.These includedJessicaCollins,NilanjanDas,KevinDorst,JeremyGoodman,DanGreco, BrianHedden,DanHoek,SophieHorowitz,JensKipper,HarveyLederman,Matt Mandelkern,DamienRochford,BernhardSalow,MiriamShoen field,Jonathan Vogel,andIanWells.

EpistemologyhasbeenalivelyareaofresearchatMIT,andoverawiderrange oftimeIhavebene ﬁtedfromthealmostconstant ﬂowofstimulatingdiscussion, informalandinreadinggroupsaswellasseminars,withgraduatestudentsand colleagues.Inadditiontothosealreadymentioned,Iwanttothankthefollowingwho havehelpedmetounderstandtheseissues,bothduringtheirtimeatMITandlater: RayBriggs,AlexByrne,AndyEgan,AdamElga,NedHall,CasparHare,JustinKhoo, SarahMoss,MiloPhilips-Brown,AgustinRayo,GingerSchultheis,JackSpencer, JasonStanley,EricSwanson,ZoltanSzabo,RogerWhite,SteveYablo,andSeth Yalcin.

MydebtstoTimothyWilliamson,DorothyEdgington,andDavidLewiswillbe evidentthroughoutthesepapers.Witheach,thereistherightmixofagreementand disagreementtomakeforfruitfuldiscussion.Eachhashadaprofoundinﬂuenceon myideas.

Thanks,asalways,toPeterMomtchiloffforhissupportandadvice,andtoDavid Balcarrasforeditorialhelp.

DetailsofFirstPublication

Ithanktheeditorsandpublishersforpermissiontoreprintthefollowingpreviously publishedpapersinthiscollection.

Chapter2 ﬁrstappearedasRobertStalnaker, “LuminosityandtheKKThesis” in Externalism,Self-knowledgeandSkepticism,editedbySanfordC.Goldberg, 17–40,CambridgeUniversityPress.Copyright©2015.Reprintedbypermissionof CambridgeUniversityPress.

Chapter4 ﬁrstappearedasRobertC.Stalnaker, “ModelingaPerspectiveonthe World” in AboutOneself:DeSeThoughtandCommunication,editedbyManuel García-CarpinteroandStephanTorre,121–37.Copyright©2016.Reprintedby permissionofOxfordUniversityPress:https://global.oup.com/academic/product/ about-oneself-9780198713265.

Chapter7alsoappearsasRobertC.Stalnaker, “ExpressivismandPropositions,” forthcomingin UnstructuredContent,editedbyDirkKindermann,AndyEgan,and PetervanElswyk,andisreprintedbypermissionofOxfordUniversityPress.

Chapter10 ﬁrstappearedasRobertC.Stalnaker, “ConditionalPropositions andConditionalAssertions,” in EpistemicModality editedbyBrianWeatherson andAndyEgan,227–48.Copyright©2011,andisreprintedbypermission ofOxfordUniversityPress:https://global.oup.com/academic/product/epistemicmodality-9780199591589.

Chapter11alsoappearsasRobertC.Stalnaker, “CounterfactualsandProbability,” forthcomingin Conditionals,ParadoxandProbability:ThemesfromthePhilosophy ofDorothyEdgington,editedbyLeeWaltersandJohnHawthorne,andisreprinted bypermissionofOxfordUniversityPress.

Chapter12 ﬁrstappearedasRobertStalnaker, “CounterfactualsandHumean Reduction, ” in ACompaniontoDavidLewis,editedbyBarryLoewerandJonathan Schaffer,411–24,Wiley-Blackwell.Copyright©2015.Reprintedbypermissionof theeditors.

Thefollowingchaptersarepublishedhereforthe ﬁrsttime:

“Reﬂection,Endorsement,Calibration ”

“RationalReﬂectionandtheNotoriousUnmarkedClock”

“ContextualismandtheLogicofKnowledge” “DispositionsandChance.”

Introduction

MorethanthirtyyearsagoIwroteabookcalled Inquiry.Thiswasagreattitlefora philosophybook,withitsallusion(orhomage)toclassicworksintheempiricist tradition,anditwasanappropriatetitlefortheaspirationswithwhichthebookwas written:itstopic,Isaidinthepreface,wastheabstractstructureofinquiry.Butitis lessclearthatthiswasanappropriatetitleforwhatwasactuallyaccomplishedinthe booksinceitdidnotgetmuchbeyondpreliminarysettingupoftheissues,andsome expositionofandmotivationfortheformalapparatusthatIplannedtousetotalk aboutthestructureofinquiry.Beforegettingtothemainissues,Ihadtoexplainand motivatemyapproachtotheproblemofintentionality,sketchandmotivatethe formalapparatususedtorepresentthatapproach(possibleworldssemantics),and respondtoproblemsthattheapproachfaced.Thattookupmostofthebook.The restofitfocusedmainlyonanotherpieceofapparatusneededtorepresentthe dynamicsofbelief(aformalsemanticsforconditionals),andIwasabletomakeonly astartonadiscussionoftheroleofthisapparatusinformingandreﬁningbothrules forrevisingbeliefs,andconceptsforgivingatheoreticaldescriptionoftheworld.

Isaidatthetime(againinthepreface)thatIhadbegunthatprojectwithanaïve hopethatIcouldgettothebottomoftheproblemsIwasconcernedwith,butthat IhadlearnedthatthebottomwasfurtherdownthanIthoughtandsowasthen preparedonlytomakeapreliminaryprogressreport.Thepresentcollectionisa furtherprogressreportonthesameproject,butIhavechangedmymindabout gettingtothebottomofthings.I’vedecidedthereisnobottom:thebestwecandoin philosophyistochipawayatbitsandpiecesoftheproblems.Wecanpaint impressionisticbigpicturesthatwehopewillgetonetoseetheissuesinanewand betterway,andwecanconstructmodelsthatachieveprecisiononlyatthecostof idealizationandsimpli ﬁcation,butthatwehopewillthrowsomelightonthe phenomena.Thatmaybeenoughtocountasprogress.

Onegainssomeperspectivefromputtingacollectiontogether,seeingconnections andrecurrentthemesthatonehadnotnoticedwhenworkingontheindividual papers.OnethingthatstoodoutformeasIselectedpapersforthiscollection,and addedtothemto ﬁllingaps,wasthecontinuitywiththeearlierbook,eventhoughall butoneofthepapersinthiscollectionwerewrittenmorethanthirtyyearsafter Inquiry waspublished.Thiscollectionalsohastwoparts,papersonknowledgeand papersonconditionals,andthesepapersdiscussthesamethemesdiscussedinthe twopartsoftheearlierbook.Thefocusofthe ﬁrstparthaschangedfrombeliefto knowledge,butIhavecometoseethattheproblemofintentionality(atleastonmy wayofapproachingit)isessentiallythesameastheproblemofcharacterizing knowledge.Knowledgewhether ϕ,accordingtoasloganIlike,isthecapacityto

makeone’sactionsdependonwhether ϕ.Knowledgeisamatterofcausalsensitivity tofactsthatarethesubjectmatterofone’sknowledge.Myearliergesturesat explainingintentionalitytookasimilarform:Itookbeliefanddesiretobethe basicintentionalstates,butarguedthatbeliefstatesgettheirintentionalcontent fromtheinformationthattheytendtobesensitiveto(undercertainnormal conditions).LookingbackfromthelaterperspectiveofTimothyWilliamson ’ s generalpictureofepistemology,Icametoappreciatethatmyaccountofintentionalityisreallyaversionofhis “knowledge ﬁrst” view:beliefiswhatwouldbe knowledgeiftherelevantnormalconditionsinfactobtained,ortoputittheother wayaround,knowledgeisfullbeliefwhenitisnon-defective.

Thepapersinthesecondhalfofthiscollectiondevelopfurthertheideasabout conditionalsthataresketchedinthelastthreechaptersof Inquiry:theirrolein epistemology,themetaphysicalstatusofthepropositionstheyexpress,andtheir relationtoprobabilisticconcepts,bothcredenceandchance.Intheearlierbook IsketchedanddefendedwhatIcalledtheprojectionstrategyforexplainingobjective modalconceptsasakindofprojectionofepistemicstatesandpoliciesontothe world,arguingthatthisstrategyhelpedtoexplaintherelationbetweenthetwokinds ofconditionals(indicativeandsubjunctive).ThestrategyhasitsrootsinHume,but IcontrasteditwiththekindofreductionistHumeanprojectthatDavidLewis developed.Onmyanti-reductionistaccount,theresultoftheprojectionisconcept formationthatreﬁnesourdescriptiveresourcesfordistinguishingbetweenthe possiblewaysthattheworldmightbe.Inthepapersinthesecondpartofthis collectionIlookinmoredetailatthesesameissues.

Iwillsketchinbroadstrokesthepictureofepistemologythatisguidingme,and thentrytoputtheindividualpapersincontextbysayinghowIseetheirrelationto thisbigpicture.

Themainproblemofepistemologyistoexplainhowwecognitivebeingsareable to findourwayaboutintheworld:howdoweacquireandusetheinformationabout ourenvironmentthatweneedtosucceedinit?Eventhesimplestanimalsacquire anduseinformation,andthey(alongwithsimpleartifacts)provideusefulmodelsof knowledge,butonethingthatdistinguishesthekindofcognitivebeingswearefrom thesesimplecasesisthatwecanreflectonourselvesascognitivebeings;partofthe informationweareabletoacquireanduseisinformationaboutourownplacein theworld informationabouthowweareabletoacquireanduseinformation.The pointisnotjustthatoneoftheinquirieswecanengageinisepistemology.Itisthat anyinquirywillinvolveatleastimplicitconsiderationofthemethodsweareusingto reachtheconclusionswereach,andwhenwearesurprised whensomethingwe tookourselvestoknowisshowntobefalse weareforcedtoreflectonwhatwent wrong:whatassumptionweweremaking,perhapsimplicitly,aboutourepistemic connectionstotheworld,andwhatchangesweneedtomakeinthoseassumptionsto recoverfromourmistakes.Theupshotisthatoneoftheimportantchannelsof informationinvolvedinouracquisitionandrevisionofknowledgeisinformation aboutourselvesandourplaceintheworld.

Aseconddistinctivefeatureofthekindofcomplexcognitivebeingsthatweareis thatwearesocialcreatureswhorelyontheknowledgeofothers.Thatis,knowers otherthanourselvesareinvolvedinthechannelsthroughwhichwereceive

information.Criticalreflectiononthesechannelsofinformationwillbereflection, fromathirdpersonperspective,onhowitisthatcognitivebeingslikeourselvesare ableto findtheirwayaround,what their sourcesofinformationare,andwhatthe worldislikefromtheirperspectives.

So,wedevelopaconceptionoftheworldaprominentpartofwhichisourselves andotherslikeus rationalagentswhoaredevelopingandreﬁningaconceptionof theworldtheyarein.Clarifyingaconceptionofthiskindwillinvolveconsidering differentperspectivesontheworld,andrelationsbetweenthoseperspectives.Ina sense,wearelookingatourselvesfromtheoutside,asagentswhoseinteractionswith natureandwithotheragentsarepartofanobjectiveworldtobedescribedand explained.Butwealsorecognizethatwearen’treallyoutside:Ourthird-personview ofourselvesisdevelopedandreﬁnedwithintheworld,fromperspectiveswithinit. Gettingclearabouttherelationshipsbetweendifferentcognitiveperspectives that ofthetheorist,thatofoneselfatthemoment,thatofoneselfatrememberedand anticipatedtimes,andthatofothers isoneoftheaimsinmanyofthepapersinthis collection.

Thepictureisanaturalisticonethatseescognitivebeingsaspartofthenatural world.TakingapagefromHume,thisnaturalisticpicturegivesnoroletopure reason,beyondtherequirementsofconsistencyandcoherence,initsaccountof inductiveknowledge.ItaketheupshotofHume’sskepticalargumentthatreason cannotjustifyinductivepractice,andhisjudgmentthatallreasoningaboutmatters offactisbasedoncauseandeffecttobesomethinglikethis:wecan’tseparatethe taskofdevelopingandjustifyingrulesfor findingoutabouttheworldfromthe substantivetaskofdevelopingaviewaboutwhattheworldislike.Weapproachboth tasksfromwithin,criticizingandrefiningthemethodsandbeliefsthatwe find ourselveswith.

Todevelopandsharpenthispicture,ithelpstohavesomeformaltools.Thebook begins,inchapter1,withareviewofaformalsemantics,inthepossible-worlds framework,ofknowledgeandbelief.Afeatureofthiswayofmodelingknowledge (pioneeredbyJaakkoHintikka)isthatitprovidesawayofrepresentingpropositions aboutwhatanagentknowsaspropositionsthatarethemselvesthecontentsof knowledge.Intheearlytheoriesofthiskind,justasingleknowerwasmodeled,but theframeworknaturallyextendstoatheorywithmultipleknowerswhohave knowledgeandbeliefsabouttheknowledgeandbeliefsofeachother,sothisisan appropriateframeworkfordevelopingthegeneralpicturesketchedabove.Inthe particularversionofamodeltheoryforknowledgeandbeliefthatIsketchinthis chapter,assumptionsaremadethatpermitbelieftobereducedknowledge,whichis appropriatetothe “knowledge ﬁrst” ideologythatisimplicitintheinformationtheoreticpictureofknowledge.Anditprovidesaframeworkforclarifyingquestions aboutfurtherconstraintsontherelationbetweenknowledgeand(full)belief(where yourfullbeliefsare,roughly,thepropositionsyourightlyorwronglytakeyourselfto know).Belief,inthissense,andknowledgewillcoincidewhenoneisrightabout everything,butwecanconsiderhowmuchwecangeneralizeaboutwhatanagent knowswhensomeofwhatshetakesherselftoknowisfalse.Thatis,whatcanone say,atthislevelofabstraction,abouttheextenttowhicherrorsinsomeofour knowledgeclaimsinfectothersofourknowledgeclaims,andtheextenttowhich

someoftheseknowledgeclaimscanbeisolatedfromothers.Iposethisquestionin themodeltheoreticframework,anddrawsomeconnectionsbetweenanswerstoit andproposalsthataroseintheverydifferentpost-Gettierprojectoftryingtoanalyze knowledgeintermsoftruebelief,plussomefurthercondition.

Thelogicofknowledgesketchedinthe ﬁrstchaptermakessometransparencyor luminosityassumptionsthatarecontroversial.While,asIhavesuggested,mypicture conformsinmanywayswithWilliamson’sexternalistepistemology,itdiverges sharplyfromhisontheissueofluminosity.ButIargueinthesecondchapter,with thehelpofsomesimplemodelsofinformation-carryingdevices,thatthese assumptions particularlytheassumptionthatonewhoknowsthat ϕ isinaposition toknowthatsheknowsit canbereconciledwithathoroughlyexternalistconceptionofknowledge.

Asecondpieceofformalapparatusthatisrelevanttothedynamicdimensionof thepictureisaformaltheoryofbeliefrevision.Themaintaskofthestandardbelief revisiontheoryistospecifyconstraintsonthewayasubjectisdisposedtochangehis overallbeliefstateasaresultofdiscoveringthatsomepriorfullbeliefisfalse.The thirdchapterisacriticaldiscussionofsomeattemptstoextendthestandardtheory togiveanaccountofthewayone’sbeliefrevisionpolicies,aswellasone’sbeliefs, shouldchangeinresponsetothediscoverythatapriorbeliefisfalse.Someelegant theoriesofiteratedbeliefrevisionhavebeenproposed,andtheyhelptoclarifythe terrain,butIarguethattheyallfacecounterexamples.Althoughthemainpoints Imakeinthischapterarenegative,thecounterexamplespointtotheimportanceof meta-information theagent’sknowledgeandbeliefsaboutherownepistemic situation inbeliefrevision.Afullysatisfactorybeliefrevisionwillinvolvean explanationofwhyrevisionwasrequired ofwhatdeviationfromnormalcondition ledonetotakeoneselftoknowsomethingthatonedidnotknow butoftenone learnsonewasmistakenwithoutlearningwhy,andthiscomplicatestheprocessof rationalbeliefrevision.

Thefourthchapterisaboutthewayself-locatingknowledgeandbeliefshouldbe represented.OnthepictureofcognitivebeingsthatIamworkingwith,allknowledge is,inasense,self-locatingsinceallofanagent’srepresentationsgettheircontent fromthatagent’srelationtothethingsthoserepresentationsareabout.Iarguethat thestandardwayofthinkingaboutself-locatingbelief,whichdistinguishessharply betweenknowingwhatpossibleworldyouareinandknowingwhereyouareinthe world,isconfused.Youcanknowwhatcountryyouareinwithoutknowingwhere youareinthecountry,but(Iargue)ignoranceofwhereyouare,orwhattimeitis,is alwaysignoranceaboutwhattheworldislike,whichistosay,aboutwhatpossible worldyouarein.Modelsthatrecognizethiscanhelpgiveaclearerviewofthewaywe thinkabouttherelationsbetweenepistemicperspectives,sincechartingthoserelationsrequirescalibratingtherelationsbetweenthecontentsoftheattitudesof differentagents,andofthesameagentatdifferenttimes.

Fullbelief,onthepictureIamdeveloping,iswhatonetakesoneselftoknow,buta cognitivestatewillalsoincludedegreesofpartialbelief,anda “knowledge ﬁrst” epistemologymustconcernitselfwiththesemore ﬁne-grainedstatesaswell.Questionsabouttherelationshipsbetweendifferentcognitiveperspectiveswillinclude questionsabouttherelationshipsbetweenthecredencefunctionsofdifferentagents,

andofthesameagentatdifferenttimes.Chapters5and6areaboutreﬂectionor deferenceprinciples:principlesthatstateconstraintsonanagent’scredencesabouta credencefunctionotherthanherownatthattime:aboutherownanticipatedor rememberedcredences,aboutthecredencesofanotheragent,oraboutthecredences thatsheideallyoughttohave.Iargueinchapter5thatreﬂectionprinciplesabout oneselfatothertimesoraboutotherscanbedefendedontheconditionthatone endorses thoseothercredencefunctions,whichmeansthatonejudgesthattheyare therightcredencesfortherelevantagenttohave.Inchapter6,Iexploreapuzzle abouttheattitudesthatanagentshouldhaveabouttherationalityofherownpresent attitudes.

Theconceptionthatourcognitiveagentisformingandreﬁninghasanormative dimension.Herinquiriesaskwhattheworldislike,whileatthesametimeasking whatrulesandproceduressheshouldadopttoformbeliefsandpartialbeliefsabout whattheworldislike.Chapter7focusesonthenormativeorpracticaldimension.It sketchesaframework,developedbyAllanGibbard,forrepresentingamixof normativeandfactualbeliefs.WhileIendorseGibbard’sexpressivistframework, Irejecthisowninterpretationofthatframework,arguingthatitblurstheline betweenarealistandanexpressivistconceptionofnorms.Thischapterismostly aboutnormsingeneral,butinthelastpartIlookatsomewaysinwhichthis frameworkhelpstoclarifymorespeciﬁcquestionsaboutepistemicnorms,andthe waystheirapplicationisconstrainedbyfacts.

Theinformation-theoreticconceptionofknowledgeisnecessarilyacontextualist conceptionforthefollowingreason:Knowledgeclaimscanbemadeonlyagainsta backgroundoffactualpresuppositionssince,onthatconception,knowledgeisbased onnaturalisticcausalrelationsbetweenaknowerandtheenvironmentthatisknown. Butthepresuppositionsrelativetowhichknowledgeisdeﬁnedcanthemselvesalways bequestioned,andaddressingthosequestionsrequiresashiftinthecontext. Chapter8developstheinformation-theoreticversionofcontextualismaboutknowledge,comparingandcontrastingitwithacontextualisttheorydevelopedbyDavid Lewisthathasaverydifferentmotivation.

Thesecondpartofthebookcontainspapersthatfocusonconditionalpropositions:theirroleinrepresentingepistemicpolicies,theircontributiontothetheoreticalresourcesfordescribingtheworld,andtheirconnectionswithotherobjective modalnotionssuchasdispositionalpropertiesandchance.

Asnotedabove,allbutoneofthepaperswerewritteninthelasttenortwelve years.Theexceptionischapter9,my ﬁrstpaperonconditionals,whichisnowabout ﬁftyyearsold.Itisincludedheresinceitisthestartingpointofaprojectthatledtoa progressreportsixteenyearslater,andtoanotheronenow.Theformallogicand semanticsforconditionalsdevelopedinthispaperweresimilartothoseinatheory beingdevelopedindependentlyataboutthesametimebyDavidLewis,butthe philosophicalideasguidingourtwotheorieswereverydifferent.Myprojectwasless ambitiousthanLewis’s,disclaiminganyattempttoprovideareductiveanalysis.The aimwasjusttoclarifytheformalstructureofaconceptandtoprovidethesemantic apparatuswithsomeintuitivemotivation.Thetheory’saimwastodoforcounterfactualconditionalswhatKripke’spossibleworldssemanticsdidfortheconceptsof necessityandpossibility,whichwasmanifestlynotareductionofmodalconceptsto

somethingelse.Theprojectionstrategywasnotexplicitinthisearlypaper,butitwas preﬁguredintheappealtoRamsey’sexplanationofindicativeconditionalsto motivateananalysisthathadcounterfactualconditionalsasitsmaintarget.Ramsey’ s suggestionwasabouthowtodecidewhethertoacceptaconditional:addthe antecedent,hypothetically,toyourstockofbeliefs,andaccepttheconditionalif andonlyiftheresultinghypothesizedstockofbeliefsimpliestheconsequent.My question(afterextendingRamsey’ssuggestiontocovercaseswheretheantecedent wasincompatiblewithyourbeliefs)wasthis:Whatshouldthetruth-conditionsfora conditionalpropositionbeifthisisagoodwayofdecidingwhethertoacceptit? Aselectionfunctionfromapossibleworldplusapropositiontoapossibleworldin whichthepropositionistruewasthoughtofasanontologicalanalogueofafunction fromastateofbeliefplusapropositiontoahypotheticalstateofbelief,andsoasa kindofprojectionofarelationbetweencognitivestatesontotheworld.

Whilemyaccountofconditionalspresupposedthattheproblemwastogivetruthconditionsforconditionalpropositions,othersataboutthistimewerearguingthat oneshouldexplainconditionalsentencesassentencesforperformingadistinctive kindofspeechact,orforrepresentingadistinctivekindofconditionalattitude. ErnestAdamsdevelopedaprobabilisticsemanticsthatbeganwiththeideathata conditionalisassertablewhentheprobabilityoftheconsequent,conditionalonthe antecedent,ishigh,andDorothyEdgington,buildingonAdams’swork,developed somepowerfulargumentsforanon-propositionalaccount.Somephilosophers suchasAllanGibbardgaveadividedaccountofconditionals,sidingwithEdgington ingivingaconditionalassertionaccountofindicativeconditionals,butwiththe propositionalistsonsubjunctiveconditionals.BothEdgingtonandIaimedfor uniﬁedaccountsofthetwokindsofconditionals,butaccountsthatallowedfor andexplainedthedifferences.Inchapters10and11Idefendanecumenical approachtothedisputebetweenpropositionalistsandthosewhowanttoexplain conditionalsintermsofconditionalspeechactsandattitudes,arguinginchapter10 thattheconditionalassertionanalysiscanbeformulatedasalimitingcaseofthe propositionalanalysis,andthatitisusefultodososinceithelpstochartthe connectionsandcontinuitiesbetweenindicativeandsubjunctiveconditionals. Chapter11focusesonEdgington ’saccountofcounterfactuals,andontherelations betweencounterfactualsandobjectiveprobability.Iargueinthischapterthatthe propositionalaccountcanallowforindeterminacy,andcanbetterexplainthe phenomenasheusestodefendheraccountoftheroleofcounterfactualsinepistemic reasoning.

Chapter12developsandcriticizesDavidLewis’sreductiveanalysisofcounterfactuals,andtheHumeansuperveniencemetaphysicsthatunderliesit.Lewis’sHumean theorycontrastswiththeusualempiricistdefenseofaHumeanmetaphysics, whichtakesthesuperveniencebasetobeobservationalorphenomenalconcepts. ForLewis,thepropertiestowhichallelseistobereducedarethefundamental propertiesofphysics.HispicturealsocontrastswiththepictureIhavebeendeveloping,whichputscausalnotionsatthecenterbothofthedescriptiveresourcesfor describingtheworld,andoftherulesforourpracticeoflearningabouttheworld. Lewisseparatesmetaphysicalquestionsconcerningwhatthereisafactofthematter

aboutfromepistemologicalquestionsabouttheproperrulesforlearningaboutthose facts,whileItrytodrawconceptualconnectionsbetweenthetwo.

The ﬁnalchapter,aboutdispositionsandchance,isthemostdetaileddiscussionof theprojectionstrategy,andthemostexplicitdevelopmentofanapplicationofit,the applicationtotheconceptofobjectivechance.Iargueintheconclusionofthis chapterthatwhilethegeneralpictureisbasedonconstitutiveconceptualconnections betweenepistemicrulesanddescriptivetheoreticalconcepts(asphenomenalistand veriﬁcationisttheorieswere),itisneverthelessathoroughlyrealistmetaphysical picture.

PARTI Knowledge

1 OntheLogicsofKnowledge andBelief

1.Introduction

Formalepistemology,oratleasttheapproachtoformalepistemologythatdevelopsa logicandformalsemanticsofknowledgeandbeliefinthepossibleworldsframework, beganwithJaakkoHintikka’sbook KnowledgeandBelief (Hintikka1962).Hintikka’ s projectsparkedsomediscussionofissuesaboutiteratedknowledge(doesknowing implyknowingthatoneknows?)andabout “knowingwho,” andquantifyinginto knowledgeattributions.Muchlater,thiskindoftheorywastakenupandappliedby theoreticalcomputerscientistsandgametheorists.¹Theformalsemanticproject gainednewinterestwhenitwasseenthatitcouldbeappliedtocontextswith multipleknowers,andusedtoclarifytherelationbetweenepistemicandother modalconcepts.

EdmundGettier’sclassicrefutationoftheJustifiedTrueBeliefanalysisofknowledge(Gettier1963)waspublishedataboutthesametimeasHintikka’sbook,andit immediatelyspawnedanepistemologicalindustry aprojectofattemptingtorevise therefutedanalysisbyaddingfurtherconditionstomeetthecounterexamples. Revisedanalysesweremetwithfurthercounterexamples,followedbyfurther re fi nements.Thiskindofproject fl ourishedforsomeyears,buteventually becameaninternallydrivengamethat wasthoughttohavelostcontactwith thefundamentalepistemologicalquestionsthatoriginallymotivatedit.Thisway ofapproachingepistemologicalquesti onsnowseemshopelesslyoutofdate,but Ithinktheremaystillbesomeinsightstobegainedbylookingback,ifnotatthe detailsoftheanalyses,atsomeofthegeneralstrategiesofanalysisthatweredeployed.

Therewaslittlecontactbetweenthesetwoverydifferentepistemologicalprojects. The firsthadlittletosayaboutsubstantivequestionsabouttherelationbetween knowledge,belief,andjustificationorepistemicentitlement,orabouttraditional epistemologicalissues,suchasskepticism.Thesecondprojectignoredquestions abouttheabstractstructureofepistemicanddoxasticstates.ButIthinksomeofthe abstractquestionsaboutthelogicofknowledgeconnectwithtraditionalquestionsin epistemology,andwiththeissuesthatmotivatedtheattemptto findadefinitionof knowledge.Theformalsemanticframeworkprovidestheresourcestoconstruct modelsthatmayhelptoclarifytheabstractrelationshipbetweentheconceptof

¹SeeFaginetal.1995andBattigalli&Bonanno1999forexcellentsurveysoftheapplicationoflogicsof knowledgeandbeliefintheoreticalcomputerscienceandgametheory.

knowledgeandsomeoftheotherconcepts(beliefandbeliefrevision,causationand counterfactuals)thatwereinvolvedinthepost-Gettierprojectofdeﬁningknowledge. Andsomeoftheexamplesthatwereoriginallyusedinthepost-Gettierliteratureto refuteaproposedanalysiscanbeusedinadifferentwayinthecontextofformal semantictheories:tobringoutcontrastingfeaturesofsomealternativeconceptions ofknowledge,conceptionsthatmaynotprovideplausibleanalysesofknowledge generally,butthatmayprovideinterestingmodelsofknowledgethatareappropriate forparticularapplications,andthatmayilluminate,inanidealizedway,oneor anotherofthedimensionsofthecomplexepistemologicalterrain.

Myaiminthischapterwillbetobringoutsomeoftheconnectionsbetweenissues thatariseinthedevelopmentandapplicationofformalsemanticsforknowledgeand beliefandmoretraditionalsubstantiveissuesinepistemology.Thechapterwillbe programmatic,pointingtosomehighlyidealizedtheoreticalmodels,somealternativeassumptionsthatmightbemadeaboutthelogicandsemanticsofknowledge, andsomeofthewaysinwhichtheymightconnectwithtraditionalissuesin epistemology,andwithapplicationsoftheconceptofknowledge.Iwillbring togetherandreviewsomeoldresults,andmakesomesuggestionsaboutpossible futuredevelopments.AfterabriefsketchofHintikka’sbasiclogicofknowledge,Iwill discuss,insection2,theS5epistemicmodelsthatweredevelopedandappliedby theoreticalcomputerscientistsandgametheorists,modelsthat,Iwillargue,conflate knowledgeandbelief.Insection3,Iwilldiscussabasictheorythatdistinguishes knowledgefrombeliefandthatremainsrelativelynoncommittalaboutsubstantive questionsaboutknowledge,butthatprovidesadefinitionofbeliefintermsof knowledge.Thistheoryvalidatesalogicofknowledge,S4.2,thatisstrongerthan S4,butweakerthanS5.Intheremainingfoursections,Iwillconsidersome alternativewaysofaddingconstraintsontherelationbetweenknowledgeandbelief thatgobeyondthebasictheory:insection4IwillconsidertheS5partitionmodelsas aspecialcaseofthebasictheory;insection5Iwilldiscusstheupperandlower boundstoanextensionofthesemanticsofbelieftoasemanticsforknowledge;in section6Iwilldiscussaversionofthedefeasibilityanalysisofknowledge,andin section7asimpli fiedversionofacausaltheory.

ThebasicideathatHintikkadeveloped,andthathassincebecomefamiliar,wasto treatknowledgeasamodeloperatorwithasemanticsthatparallelsthepossible worldssemanticsfornecessity.Justasnecessityistruthinallpossibleworlds,so knowledgeistruthinall epistemically possibleworlds.Theassumptionisthattohave knowledgeistohaveacapacitytolocatetheactualworldinlogicalspace,toexclude certainpossibilitiesfromthecandidatesforactuality.Theepistemicpossibilitiesare thosethatremainaftertheexclusion,thosethattheknowercannotdistinguishfrom actuality.Torepresentknowledgeinthiswayisofcoursenottoprovideanykindof reductiveanalysisofknowledge,sincetheabstracttheorygivesnosubstantive accountofthecriteriafordeterminingepistemicpossibility.Theepistemicpossibilitiesaredeﬁnedbyabinaryaccessibilityrelationbetweenpossibleworldsthatisa primitivecomponentofanepistemicmodel.(Where x and y arepossibleworlds,and “R” istheaccessibilityrelation, “ xRy ” saysthat y isepistemicallypossibleforthe agentinworld x.)Theideawastogiveapreciserepresentationofthestructureof anepistemicstatethatwasmoreorlessneutralaboutmoresubstantivequestions

aboutwhatconstitutesknowledge,butthatsharpenedquestionsaboutthelogicof knowledge.Thisformofrepresentationwas,however,farfrominnocent,sinceit required,fromthestart,anextremeidealization:Eveninitsmostneutralform,the frameworkrequiredtheassumptionthatknowersknowalllogicaltruthsandallof theconsequencesoftheirknowledge,sincenomatterhowtheepistemicallypossible worldsareselected,alllogicaltruthswillbetrueinallofthem,andforanysetof propositionstrueinallofthem,alloftheirlogicalconsequenceswillalsobetrueinall ofthem.Therearedifferentwaysofunderstandingthecharacterofthisidealization: ontheonehand,onemightsaythattheconceptofknowledgethatisbeingmodeled isknowledgeintheordinarysense,butthatthetheoryisintendedtoapplyonlyto idealizedknowers thosewithsuperhumanlogicalcapacities.Alternatively,one mightsaythatthetheoryisintendedtomodelanidealizedsenseofknowledge theinformationthatisimplicitinone’sknowledge thatliterallyappliestoordinary knowers.Howevertheidealizationisexplained,thereremainthequestionswhether itisfruitfultodevelopatheorythatrequiresthiskindofdeviationfromreality,andif sowhy.²ButIthinkthesequestionsarebestansweredbylookingatthedetailsofthe waysuchtheorieshavebeen,andcanbedeveloped.

Themostbasictaskindevelopingasemanticsforknowledgeinthepossible worldsframeworkistodecideonthepropertiesoftheepistemicaccessibility relation.Itisclearthattherelationshouldbereﬂexive,whichisnecessarytovalidate theprinciplethatknowledgeimpliestruth,anassumptionthatisjustabouttheonly principleofalogicofknowledgethatisuncontroversial.Hintikkaarguedthatwe shouldalsoassumethattherelationistransitive,validatingthemuchmorecontroversialprinciplethatknowingimpliesknowingthatoneknows.Knowingand knowingthatoneknowsare,Hintikkaclaimed, “virtuallyequivalent.” Hintikka’ s reasonsforthisconclusionwerenotcompletelyclear.Hedidnotwanttobaseitona capacityforintrospection:heemphasizedthathisreasonswerelogicalratherthan psychological.HisproofoftheKKprinciplerestsonthefollowingprinciple:If {Kϕ,~K~ψ}isconsistent,then{Kϕ, ψ}isconsistent,anditisclearthatifonegrants thisprinciple,theKKprincipleimmediatelyfollows.³Thereasonforacceptingthis principleseemstobesomethinglikethis:Knowledgerequiresconclusivereasonsfor belief,reasonsthatwouldnotbedefeatedbyanyinformationcompatiblewithwhatis known.So,ifoneknowsthat ϕ while ψ iscompatiblewithwhatoneknows,thenthe truthof ψ couldnotdefeatone’sclaimtoknowthat ϕ.Thisargument,andother considerationsforandagainsttheKKprincipledeservemorecarefulscrutiny.There isatangleofimportantandinterestingissuesunderlyingthequestionwhetherone shouldaccepttheKKprincipleandthecorrespondingsemantics,andsomechallengingargumentsthatneedtobeansweredifonedoes.⁴ Ithinktheprinciplecanbe

²Iexploredtheproblemoflogicalomniscienceintwopapers,Stalnaker1991and1999b.Idon’t attempttosolvetheproblemineitherpaper,butonlytoclarifyit,andtoarguethatitisagenuineproblem, andnotanartifactofaparticulartheoreticalframework.

³Substituting “~Kϕ” for ψ,andeliminatingadoublenegation,theprinciplesaysthatif{Kϕ,~KKϕ}is consistent,then{Kϕ,~Kϕ}isconsistent.

⁴ Seeespecially,Williamson2000forsomereasonstorejecttheKKprinciple.IrespondtoWilliamson’ s mainargumentinStalnaker2015,reprintedaschapter2ofthisbook.

defended(inthecontextoftheidealizationswearemaking),butIwillnotaddress thisissuehere,provisionallyfollowingHintikkainacceptingtheKKprinciple,anda semanticsthatvalidatesit.

TheS4principles(Knowledgeimpliestruth,andknowingimpliesknowingthat oneknows)wereasfarasHintikkawaswillingtogo.Heunequivocallyrejectsthe characteristicS5principlethatifonelacksknowledge,thenoneknowsthatonelacks it(“unlessyouhappentobeassagaciousasSocrates”⁵),andherehisreasonsseemto beclearanddecisive:

Theconsequencesofthisprinciple,however,areobviouslywrong.Byitsmeans(togetherwith certainintuitivelyacceptableprinciples)wecould,forexample,showthatthefollowing sentenceisselfsustaining:

Thereasonthat(13)isclearlyunacceptable,asHintikkagoesontosay,isthatit impliesthatonecouldcometoknowbyreﬂectionalone,ofanytruth,thatitwas compatiblewithone’sknowledge.Butitseemsthataconsistentknowermight believe,andbejustiﬁedinbelieving,thatsheknewsomethingthatwasinfactfalse. Thatis,itmightbe,forsomeproposition ϕ,that~ϕ,andBKϕ.Insuchacase,ifthe subject’sbeliefsareconsistent,thenshedoesnotbelieve,andsodoesnotknow,that ~ϕ iscompatiblewithherknowledge.Thatis,~K~Kϕ,alongwith~ϕ,willbetrue, falsifying(13).

2.PartitionModels

DespiteHintikka’sapparentlydecisiveargumentagainsttheS5principle,later theoristsapplyingepistemiclogicandsemantics,bothintheoriesofdistributive computersystemsandingametheoryassumedthatS5wastherightlogicfor(an idealizedconceptof)knowledge,andtheydevelopedsemanticmodelsthatseemto supportthatdecision.Butwhilesuchmodels,properlyinterpreted,havetheirplace, Iwillarguethatthetheoristsdefendingthemconflatedknowledgeandbeliefinaway thathasledtosomeconceptualconfusion,andthattheyhaveabstractedawayfrom someinterestingproblemswithintheirintendeddomainsofapplicationthatmore generalmodelsmighthelptoclarify.Butbeforegettingtothisissue,letme firsttake noteofanotherwaythatmorerecenttheoristshavemodi fied,orgeneralized, Hintikka’soriginaltheory.

Hintikka’searlymodelsweremodelsoftheknowledgeofasingleknower,but muchofthelaterinterestinformalepistemicmodelsderivesfromaconcernwith situationsinwhichtherearemultipleknowerswhomayknoworbeignorantabout theknowledgeandignoranceoftheothers.WhileHintikka’searlyworkdidnotgive explicitattentiontotheinteractionofdifferentknowers,thepotentialtodosois implicitinhistheory.Boththelogicandthesemanticsoftheknowledgeofasingle

⁵ Hintikka1962,106.

⁶ Ibid.,54.InHintikka’snotation, “Pa” isthedualoftheknowledgeoperator, “Ka” : “ ~Ka~ ”.Iwilluse “M” for~K~).

knowergeneralizeinastraightforwardwaytoamodelformultipleknowers.One needsonlyaseparateknowledgeoperatorforeachknower,andinthesemantics,a separaterelationofepistemicaccessibilityforeachknowerthatinterpretsthe operator.Onecanalsointroduce,foranygroupofknowers,anoperatorforthe common knowledgesharedbythememberofthegroup,whereagrouphascommon knowledgethat ϕ ifandonlyifallknowthat ϕ,allknowthatallknowthat ϕ,allknow thatallknowthatallknow,etc.allthewayup.Thesemanticsforthecommon knowledgeoperatorisinterpretedintermsofanaccessibilityrelationthatisdeﬁnableintermsoftheaccessibilityrelationsfortheindividualknowers:thecommonknowledgeaccessibilityrelationforagroupGisthetransitiveclosureofthesetof epistemicaccessibilityrelationsforthemembersofthatgroup.⁷ IfRG isthisrelation, thentheknowerswhoaremembersofGhavecommonknowledgethat ϕ (inpossible world x)iff ϕ istrueinallpossibleworldsthatareRG relatedtoworld x.The generalizationtomultipleknowersandtocommonknowledgeworksthesameway, whateverassumptionsonemakesabouttheaccessibilityrelation,andonecandeﬁne notionsofcommonbeliefinanexactlyanalogousway.Thepropertiesofthe accessibilityrelationsforcommonknowledgeandcommonbeliefwillderivefrom thepropertiesoftheindividualaccessibilityrelations,buttheywon’tnecessarilybe thesameasthepropertiesoftheindividualaccessibilityrelations.(Thoughifthe logicofknowledgeisS4orS5,thenthelogicofcommonknowledgewillalsobeS4or S5,respectively).

Theoreticalcomputerscientistshaveusedthelogicandsemanticsforknowledge togiveabstractdescriptionsofdistributedcomputersystems(suchasofficenetworks oremailsystems)thatrepresentthedistributionand flowofinformationamongthe componentsofthesystem.Forthepurposeofunderstandinghowsuchsystemswork andhowtodesignprotocolsthatpermitthemtoaccomplishthepurposesforwhich theyweredesigned,itisusefultothinkofthemascommunitiesofinteracting rationalagentswhousewhatinformationtheyhaveaboutthesystemasawholeto servetheirowninterests,ortoplaytheirpartinajointproject.Anditisusefulinturn forthoseinterestedinunderstandingtheepistemicstatesofrationalagentstothink ofthemasanaloguesofthekindofsimpli fiedmodelsthattheoreticalcomputer scientistshaveconstructed.

Adistributedsystemconsistsofasetofinterconnectedcomponents,eachcapable ofbeinginarangeoflocalstates.Thewaythecomponentsareconnected,andthe rulesbywhichthewholesystemworks,constraintheconﬁgurationsofstatesofthe individualcomponentsthatarepossible.Onemightspecifysuchasystembypositing asetofncomponentsandpossiblelocalstatesforeach.Onemightalsoincludea componentlabeled “nature” whoselocalstatesrepresentinformationfromoutside thesystemproper. Global stateswillben-tuplesoflocalstates,oneforeach component,andthemodelwillalsospecifythesetofglobalstatesthatare admissible Admissibleglobalstatesarethosethatarecompatiblewiththerulesgoverningthe

⁷ Moreprecisely,ifRi istheaccessibilityrelationforknower i,thenthecommon-knowledgeaccessibility relationforagroupGisdeﬁnedasfollows: xRGy iffthereisasequenceofworlds, z₁,... zn suchthat z₁ = x and zn = y andforall j between1and n–1,thereisaknower i 2 G,suchthat zjRi zj+1

waythecomponentsofthesysteminteract.Theadmissibleglobalstatesarethe possibleworldsofthemodel.Thiskindofspeciﬁcationwilldetermine,foreachlocal statethatanycomponentmightbein,asetofglobalstates(possibleworlds)thatare compatiblewiththecomponentbeinginthatlocalstate.Thissetwillbethesetof epistemicallypossibleworldsthatdetermineswhatthecomponentinthatstate knowsaboutthesystemasawhole.⁸ Speciﬁcally,if “ a ” and “b” denoteadmissible globalstates,and “ ai ” and “bi” denotetheithelementsof a and b,respectively(the localstatesofcomponent i),thenglobalworld-state b isepistemicallyaccessible(for i) toglobalworld-state b ifandonlyif ai = bi.So,applyingthestandardsemanticrule fortheknowledgeoperator,component(orknower) i willknowthat ϕ,inpossible world a,ifandonlyif ϕ istrueinallpossibleworldsinwhich i hasthesamelocal statethatithasinworld-state a.Oneknowsthat ϕ ifone’slocalstatecarriesthe informationthat ϕ. ⁹

Nowitisobviousthatthisepistemicaccessibilityrelationisanequivalence relation,andsothelogicforknowledgeinamodelofthiskindisS5.Eachofthe epistemicaccessibilityrelationspartitionsthespaceofpossibleworlds,andthecrosscuttingpartitionsgiverisetoasimpleandelegantmodelofcommonknowledge,also withanS5logic.Gametheoristsindependentlydevelopedthiskindofpartition modelofknowledgeandhaveusedsuchmodelstobringouttheconsequencesof assumptionsaboutcommonknowledge.Forexample,itcanbeshownthatincertain games,playerswillalwaysmakecertainstrategychoiceswhentheyhavecommon knowledgethatallplayersarerational.Butaswehaveseen,Hintikkagavereasonsfor rejectingtheS5logicforknowledge,andthereasonsseemedtobedecisive.Itisclear thataconsistentandepistemicallyresponsibleagentmighttakeherselftoknowthat ϕ inasituationinwhich ϕ wasinfactfalse.Becauseknowledgeimpliestruth,it wouldbefalse,insuchacase,thattheagentknewthat ϕ,buttheagentcouldnot knowthatshedidnotknowthat ϕ withouthavinginconsistentbeliefs.Ifsuchacase ispossible,thentherewillbecounterexamplestotheS5principle(~Kϕ ! K~Kϕ). Thatis,theS5principlesrequirethatrationalagentsbeimmunetoerror.Itishardto seehowanytheorythatabstractsawayfromthepossibilityoferrorcouldberelevant toepistemology,anenterprisethatbeginswithskepticalargumentsusingscenarios inwhichagentsaresystematicallymistakenandthatseekstoexplaintherelation betweenknowledgeandbelief,presupposingthatthesenotionsdonotcoincide.

⁸ Amorecomplexkindofmodelwouldspecifyasetofadmissible initial globalstates,andasetof transitionrulestakingglobalstatestoglobalstates.Thepossibleworldsinthiskindofmodelarethe admissibleglobal histories thepossiblewaysthatthesystemmightevolve.Inthiskindofmodel,onecan representthedistributionofinformation,notonlyaboutthecurrentstateofthesystem,butalsoabouthow itevolved,andwhereitisgoing.Inthemoregeneralmodel,knowledgestatesaretime-dependent,andthe componentsmayhaveorlackinformation,notonlyaboutwhichpossibleworldisactual,butalsoabout where(temporally)itisinagivenworld.Thedynamicdimension,andtheparallelswithissuesabout indexicalknowledgeandbelief,arepartoftheinterestofthedistributedsystemsmodels,butIwillignore theseissueshere.

⁹ Possibleworlds,onthiswayofformulatingthetheory,arenotprimitivepoints,astheyareintheusual abstractsemantics,butcomplexobjects sequencesoflocalstates.Butanequivalentformulationmight beginwithagivensetofprimitive(global)states,togetherwithasetofequivalencerelations,oneforeach knower,andonefor “nature.” Thelocalstatescouldthenbedeﬁnedastheequivalenceclasses.

Differenttheoristshavedifferentpurposes,anditisnotimmediatelyobviousthat themodelsofknowledgethatareappropriatetotheconcernsoftheoreticalcomputer scientistsandgametheoristsneedberelevanttoissuesinepistemology.ButIthink thatthepossibilityoferror,andthedifferencesbetweenknowledgeandbeliefare relevanttotheintendeddomainsofapplicationofthosemodels,andthatsomeofthe puzzlesandproblemsthatcharacterizeepistemologyarereﬂectedinproblemsthat mayariseinapplyingthosetheories.

Asweallknowtoowell,computersystemssometimesbreakdownorfailtobehave astheyweredesignedtobehave.Insuchcases,thecomponentsofadistributed systemwillbesubjecttosomethinganalogoustoerrorandillusion.Justasthe epistemologistwantstoexplainhowandwhenanagentknowssomethingseven whenheisinerroraboutothers,andisinterestedinmethodsofdetectingand avoidingerror,sothetheoreticalcomputerscientistisinterestedinthewaythatthe componentsofasystemcanavoidanddetectfaults,andcancontinuetofunction appropriatelyevenwhenconditionsarenotcompletelynormal.Toclarifysuch problems,itisusefultodistinguishknowledgefromsomethinglikebelief.

Thegametheorist,oranytheoristconcernedwithrationalaction,hasaspecial reasontotakeaccountofthepossibilityoffalsebelief,evenundertheidealizing assumptionthatintheactualcourseofevents,everyone ’sbeliefsarecorrect.The reasonisthatdecisiontheoristsandgametheoristsneedtobeconcernedwithcausal orcounterfactualpossibilities,andtodistinguishthemfromepistemicpossibilities. WhenIdeliberate,orwhenIreasonaboutwhyitisrationaltodowhatIknowthat Iamgoingtodo,IneedtoconsiderpossiblesituationsinwhichImakealternative choices.Iknow,forexample,thatitwouldbeirrationaltocooperateinaone-shot prisoners ’ dilemmabecauseIknowthatinthecounterfactualsituationinwhich Icooperate,mypayoffislessthanitwouldbeifIdefected.AndwhileIhavethe capacitytoinﬂuencemypayoff(negatively)bymakingthisalternativechoice,Icould not,bymakingthischoice,inﬂuenceyourpriorbeliefsaboutwhatIwilldo;thatis, yourpriorbeliefswillbethesame,inthecounterfactualsituationinwhichImakethe alternativechoice,astheyareintheactualsituation.Sinceyoutakeyourself(correctly,intheactualsituation)toknowthatIamrational,andsothatIwillnot cooperate,youthereforealsotakeyourselftoknow,inthecounterfactualsituation Iamconsidering,thatIamrational,andsowillnotcooperate.Butinthatcounterfactualsituation,youarewrong youhaveafalsebeliefthatyoutaketobeknowledge.Therehasbeenacertainamountofconfusionintheliteratureaboutthe relationbetweencounterfactualandepistemicpossibilities,andthisconfusionisfed, inpart,byafailuretomakeroominthetheoryforfalsebelief.¹

Eveninacontextinwhichoneabstractsawayfromerror,itisimportanttobe clearaboutthenatureoftheidealization,andtherearedifferentwaysofunderstandingitthataresometimesconfused.Butbeforeconsideringthealternativewaysof makingtheS5idealization,letmedevelopthecontrastbetweenknowledgeand belief,andtherelationbetweenthem,inamoregeneralsetting.

¹⁰ TheseissuesarediscussedinStalnaker1996.

3.BeliefandKnowledge

SetasidetheS5partitionmodelsforthemoment,andconsider,fromamoreneutral perspective,thelogicalpropertiesofbelief,andtherelationbetweenbeliefand knowledge.Itseemsreasonabletoassume,atleastinthekindofidealizedcontext wearein,thatagentshaveintrospectiveaccesstotheirbeliefs:iftheybelievethat ϕ, thentheyknowthattheydo,andiftheydonot,thentheyknowthattheydonot. (TheS5, “negativeintrospection” principle,(~Kϕ!K~Kϕ),wasproblematicfor knowledgebecauseitisintensionwiththefactthatknowledgeimpliestruth,but thecorrespondingprincipleforbeliefdoesnotfacethisproblem.)Italsoseems reasonabletoassumethatknowledgeimpliesbelief.Giventhefactthatouridealized believersarelogicallyomniscient,wecanassume,inaddition,thattheirbeliefswillbe consistent.Finally,tocapturethefactthatourintendedconceptofbeliefisastrong one subjectivecertainty weassumethatbelievingimpliesbelievingthatone knows.So,ourlogicofknowledgeandbeliefshouldincludethefollowingprinciples inadditiontothoseofthelogicS4:

(PI) ‘ Bϕ ! KBϕ positiveintrospection (NI) ‘ ~Bϕ ! K~Bϕ negativeintrospection (KB) ‘ Kϕ ! Bϕ knowledgeimpliesbelief (CB) ‘ Bϕ ! ~B~ϕ consistencyofbelief (SB) ‘ Bϕ ! BKϕ strongbelief

Theresultingcombinedlogicforknowledgeandbeliefyieldsapurebelieflogic, KD45,whichisvalidatedbyadoxasticaccessibilityrelationthatisserial,transitive, andEuclidean.¹¹Moreinterestingly,onecanprovethefollowingequivalencetheorem: ‘ Bϕ $ MKϕ (using “M” astheepistemicpossibilityoperator, “~K~”).This equivalencepermitsamoreeconomicalformulationofthecombinedbeliefknowledgelogicinwhichthebeliefoperatorisdeﬁnedintermsoftheknowledge operator.Ifwesubstitute “MK” for “B” inourprinciple(CB),wegetMKϕ ! KMϕ, which,ifaddedtoS4yieldsthelogicofknowledge,S4.2.Alloftheotherprinciples listedabove(with “MK” substitutedfor “B”)aretheoremsofS4.2,sothislogicof knowledgebyitselfyieldsacombinedlogicofknowledgeandbeliefwiththe appropriateproperties.¹²

Theassumptionsthataresufficienttoshowtheequivalenceofbeliefwiththe epistemicpossibilityofknowledge(onebelievesthat ϕ,inthestrongsense,ifand onlyifitiscompatiblewithone’sknowledgethatoneknowsthat ϕ)mightalsobe madeforaconceptof justified belief,althoughthecorrespondingassumptionswillbe morecontroversial.Suppose(1)oneassumesthatjustifiedbeliefisanecessary

¹¹KD45addstothebasicmodalsystemKtheaxioms(D),whichisour(CB),(4)Bϕ!BBϕ,whichfollows immediatelyfromour(PI)and(KB),and(5)~Bϕ ! B~Bϕ,whichfollowsimmediatelyfrom(NI)and(KB). ThenecessitationruleforB(If ‘ϕ,then ‘Bϕ)andthedistributionprinciple(B(ϕ ! ψ) ! (Bϕ ! Bψ))can bothbederivedfromourprinciples.

¹²Thedeﬁnabilityofbeliefintermsofknowledge,andthepointthattheassumptionsabouttherelation betweenknowledgeandbeliefimplythatthelogicofknowledgeshouldbeS4.2,ratherthanS4,were ﬁrst shownbyWolfgangLenzen.Seehisclassicmonograph,Lenzen1978.

conditionforknowledge,and(2)oneadoptsan internalist conceptionofjustification thatsupportsthepositiveandnegativeintrospectionconditions(ifonehasjustified beliefthat ϕ,oneknowsthatonedoes,andifonedoesnot,oneknowsthatonedoes not),and(3)oneassumesthatsincetherelevantconceptofbeliefisastrongone,one isjustifiedinbelievingthat ϕ ifandonlyifoneisjustifiedinbelievingthatoneknows that ϕ.Giventheseassumptions,justifiedbeliefwillalsocoincidewiththeepistemic possibilitythatoneknows,andsobeliefandjustifiedbeliefwillcoincide.Theupshot isthatforaninternalist,adivergencebetweenbelief(inthestrongsense)and justifiedbeliefwouldbeakindofinternalinconsistency.Ifoneisnotfullyjustified inbelieving ϕ,oneknowsthis,andsooneknowsthatanecessaryconditionfor knowledgethat ϕ islacking.Butifonebelievesthat ϕ,inthestrongsense,thenone believesthatoneknowsit.So,onebothknowsthatonelacksknowledgethat ϕ, andbelievesthatonehasknowledgethat ϕ.

TheusualconstraintontheaccessibilityrelationthatvalidatesS4.2isthefollowing convergenceprinciple(addedtothetransitivityandreflexivityconditions):if xRy and xRz,thenthereisawsuchthat yRw and zRw.ButS4.2isalsosoundand completerelativetothefollowingstrongerconvergenceprinciple:forall x,thereisa y suchthatforall z,if xRz,then zRy.Theweakconvergenceprinciple(addedto reflexivityandtransitivity)impliesthatforany finite setofworldsaccessibleto x, thereisasingleworldaccessiblewithrespecttoallofthem.Thestrongconvergence principleimpliesthatthereisaworldthatisaccessibleto all worldsthatare accessibleto x.Thesemanticsforourlogicofknowledgerequiresthestronger convergenceprinciple.¹³

Justas,withinthelogic,onecandefinebeliefintermsofknowledge,sowithinthe semantics,onecandefineadoxasticaccessibilityrelationforthederivedbelief operatorintermsoftheepistemicaccessibilityrelation.If “R” denotestheepistemic accessibilityrelationand “D” denotesthedoxasticrelation,thenthedefinitionisas follows: xDy =df (z)(xRz ! zRy).AssumingthatRistransitive,reflexive,andstrongly convergent,itcanbeshownthatDwillbeserial,transitive,andEuclidean the constraintsontheaccessibilityrelationthatcharacterizethelogicKD45.

Onecanalsodefine,intermsofD,andsointermsofR,athirdbinaryrelationon possibleworldsthatisrelevanttodescribingtheepistemicsituationofourideal knower:Saythattwopossibleworlds x and y are epistemicallyindistinguishable toan agent(xEy)ifandonlyifshehasexactlythesamebeliefsinworld x asshehasin world y.Thatis, xEy =df (z)(xDz $ yDz).Eisobviouslyanequivalencerelation,and soanymodaloperatorinterpretedintheusualwayintermsofEwouldbeanS5 operator.Butwhilethisrelationisdefinableinthesemanticsintermsofthe epistemicaccessibilityrelation,wecannotdefine,intheobjectlanguagewithjust theknowledgeoperator,amodaloperatorwhosesemanticsisgivenbythisaccessibilityrelation.

¹³Thedifferencebetweenstrongandweakconvergencedoesnotaffectthe propositional modallogic, butitwillmakeadifferencetothequantiﬁedmodallogic.Thefollowingisanexampleofasentencethatis validinmodelssatisfyingstrongconvergence(alongwithtransitivityandreﬂexivity)butnotvalidinall modelssatisfyingweakconvergence:MK((x)(MKϕ ! ϕ)).