conditionals: a unifying ranking-theoretic perspective€¦ · i will proceed as follows: first, in...

volume15,no.1 january2015

Conditionals:

A Unifying Ranking-

Theoretic Perspective

Wolfgang SpohnUniversity of Konstanz

© 2015 WolfgangSpohnThis work is licensed under a Creative Commons

Attribution-NonCommercial-NoDerivatives 3.0 License. <www.philosophersimprint.org/015001/>

1. Introduction1

The topic of conditionals is an extremely important one. It lies atthebottomof somanyphilosophical issues (causation,dispositions,lawlikeness,etc.),andcurrenttheoriesofconditionalsseemtofairlyground these issues.On theotherhand, the topichasbecomeevermessier. Philosophical opinions grossly diverge, not only about de-tails,butalsoaboutsuchfundamentalquestionsastowhetherornotconditionalshavetruth-values.Andthelinguisticphenomenologyisso rich, the interactionwith tense,mood,negation, quantifiers, etc.,socomplicated,and thepragmaticsso imperspicuous thatplausibleexamplescanbefoundforandagainstnearlyeveryclaimoraccount.Thesituationseemsquitedesperate.

Onemight say that the situation is inevitable; there isno reduc-tionofcomplexrealitytosimpletheories.However,Iamconvincedthatthepresentconfusionisalsobroughtaboutbythefactthatthediscussionsofthelast45yearshavefocusedonsuboptimalparadigms,propositionallogic,probabilitytheory,objectivesimilarityspheres,orsubjectiveentrenchmentorderings:allofthemarenotoptimallysuit-edforlayingfoundationstoanykindofconditionals.

This conviction grounds inmy further conviction that the philo-sophicalapplicationsofconditionallogicarebetterdealtwithbyrank-ingtheory,asIhaveextensivelydisplayedinSpohn(2012,ch.12–15).If this should reallybe so, thedeeper reasoncanonlybe that rank-ingtheoryisbettersuitedfortreatingconditionalsthemselves.Inthispaper,Icannotgiveafullcomparativeargument,butmyambitionisatleasttodisplaythatrankingtheoryiswellsuitedforacomprehen-siveandunifiedaccountofconditionals.Thebasicreasonforbeingsosuitedcanbesummarizedinonesentence:conditionalsofallkindsexpress our conditional beliefs or something about them, and ourconditionalbeliefsaremostadequatelyrepresentedbyrankingtheory.Ihaveextensivelyargued for the secondclaim fromSpohn (1983b)till Spohn (2012); I can’t repeat this here andwill restrictmyself to

1. Iamdeeplyindebtedtoseveralrefereesformanyhelpfulcommentsofvari-ouskinds.

ImprintPhilosophers’

wolfgangspohn Conditionals: A Unifying Ranking-Theoretic Perspective

philosophers’imprint –2– vol.15,no.1(january2015)

“circumstancesaresuchthat”readingofconditionals,whichwillbein-troducedinsection6.Thiswillleadus,insection7,toanexplicationofthewideclassofcausalconditionals,i. e.,conditionalsrepresentingacausalrelationwithinourexpressivisticframework.ThiswillconcludemytransgressionbeyondtheRamseytest.Dependingonthewaywecount,wewillthusendupwithmorethantwentyexpressiveoptions.

Section 8will finally turn to the crucial issuewhether and howmy determinately expressivistic perspective can be reconciledwithour deeply entrenched intuition that conditional discourse is aboutmattersof fact, i. e., truth-evaluable.Such reconciliationmight seemimpossible;but,infact,weshallfindthatitgoesquitefar.Section9wrapsupthepaperbyoncemoreemphasizingtheunifyingperspec-tiveofferedhere.

Inaway, this is chapter 18ofmybookSpohn (2012)—or ratherchapter16(sothatthelasttwochapterswouldhavetobedeferred).Indeed,Ihadmentionedthisasapainfuldesideratum.However,Ihadneither space, norpower, nor certitude enough to include thepres-enttopicthere.Hencethispaperwilloftenrefertothatbook.Still,itshouldbeself-contained.

2. Expressivism

Inordertostartfromscratch, letmeintroducethesymbol> fortheschematic conditional, i. e., foranyconjunctionsomehowrepresentinga conditional relation; for indicative and subjunctive, past, present,andfuture,open,semi-factual,andcounterfactual,backtrackingandnon-backtracking, material, strict, variably strict, and suppositional,epistemicandcausal,inferentialandcontentconditionals.Theseandevenmorequalificationscanbefoundintheliterature,whichtrytoclassify conditionals according to different criteria. The schematicconditional> istoevencoverconjunctionslike“evenif,”“although,”or“because,”whichalsorepresentconditionalrelations.“yevenifj”roughlyexpressesthatyistobeexpected(even)givenorconditional on j.“yalthoughj”roughlyexpressesthatywasnottobeexpectedgivenj.“ybecausej”atleastrepresentsthatywasboundtoobtain

indicatingthebasicpoints.Themaintaskofthispaperwillbetoun-foldthefirstclaim.

Wewillseethattheexpressionofconditionalbeliefsisnotrestrict-edtotheRamseytest.Therearemanymorethingsaboutthemthatcanbeexpressedaswell.Indeed,theexpressivisticstrategyadoptedherewillextendtosubjunctiveandcounterfactualconditionals.More-over,wewillseethatthisstrategyisnotcommittedtodenyingtruth-values toconditionals; tosomeextent theycanberecovered.Somesuchmiddlecourseseemsexactlyright;neitherflatlydenyingtruth-valuesnorsanguinelydistributingtruth-valuesforall(nested)condi-tionalswilldo.

Iwillproceedasfollows:first,insection2,Iwantto,unoriginally,suggest thatourvariegatedconditional idiombasically serves toex-pressourconditionalbeliefs.Insection3,Iwillsuggest,andcandonomorethansuggest,thatrankingtheoryisthebesttoolforanalysisbecause it is thebestaccountofconditionalbelief.Thiswillbeoneprincipalshiftproposedinthispaper.

Thenextsectionswillcarryouttheotherprincipalshift:whenweaskhowconditionals express conditionalbeliefs,we shouldnotbeoverwhelmedbytheexcessivelycomplexlinguisticmaterialwithallits syntactic and pragmatic interactions.We should rather focus onwhatmightbeexpressed.Thisstudycanbeasclear,systematic,andpossiblycompleteastheunderlyingaccountofconditionalbelief; itisthisstudythatwillbecarriedouthere.Thehopethenisthat,oncewehaveaclearandsystematicoverviewoftheinterpretativeoptions,wecanapplyittothelinguisticmaterialandhavegoodguidelinesforstudyingallthoseinteractions.However,thispaperwillnotredeemthishopeindetail.

Theinterpretativeorexpressiveoptionswillberich.Ofcourse,theRamsey test is thefirst thing that comes tomind; section 4will bedevoted to it.Mycrucialobservationwillbe, though, that therearemanymoreexpressiveoptions;theexclusivefocusontheRamseytestisfatal.Thus,insection5,Iwilldiscussrelevance,whichisbasicallyamatterofconditionalbeliefs.EvenmoreinterestingiswhatIcallthe



Thisideashouldclearlyfavorexpressivism,i. e.,theapproachofdo-ingsemanticsbydescribingthementalstatesexpressedbylinguisticmeans.Expressivismisindeedtakenseriously,mainlyasametaethi-calpositionconcerningthemeaningofmoralsentences,butalsoasageneralsemanticstrategy(cf.,e. g.,Gibbard[1990]).Thelabel“expres-sivism”shouldnotinvoketoonarrowassociations.IdonotpursueanyspecificexpressivisticprogramsuchasMerin(2003)andSchroeder(2008).Also,whenyousearchforassertibilityoracceptabilitycondi-tionsinsteadoftruthconditions,Ialreadytakeyouasengagingintheexpressivisticbusiness.

Eventalkoftruthconditionsmaybecompatiblewithexpressivism.Atthispointitisusefultonotethatwemayspeakoftruthandtruthconditionsinanemphaticorshallowsense.WhenIreferredabovetotruthconditionsofconditionals,Iintendedtheemphaticsenseaccord-ingtowhichatruthconditionisanobjectivematteroffactnotrelativetosubjectsorperspectives.And,inthissense,itisatleastproblematicwhetherconditionalshavetruthconditions,justasitisatleastprob-lematic toassumemoral facts.However,onemayaswelldeclareaconditionalor,say,anevaluativeassertiontobetruenotobjectively,butonlyrelativetothespeaker’sbeliefsorpreferences.Letuscallthisatruthconditionintheshallowsense.ThenIfindnorelevantdiffer-encetoexpressivism.Assigningsuchashallowtruthconditiontotheassertionisthesame—Itakeit—assayingthatitexpressesthosebe-liefsorpreferencesor—morecautiously—somethingaboutthosebe-liefsorpreferences(acautionthatwillbeappropriatethroughoutthepaper).Thisequationmaybetoosimpleintheend(seeKölbel[2002]formoresophisticatedviews),butitwilldoforourpurposes.

So, why not prefer expressivism? Why is truth-conditional se-mantics still the favored approach, even in philosophy? Certainly,thedeepestandmostdifficultreasonismarkedbytheso-calledlin-guisticturn,thetransitionof18thand19thto20thcenturyphiloso-phy,anditsinsightthatmentalstatesandtheircontentsareidentifi-ableonlywithreferencetoexternalstatesofaffairs.Hence,itseems,we must first study what utterances mediately represent, namely

givenj. The list can easily be extended. Such conditional relationstotallypervadeordinary language.Consideringtheschematiccondi-tionalisjustifiedbymyaimtoexplainaspaceofpossibilitiesofwhatconditionalscouldmeanandnot toexplain thespecificmeaningofanyspecificconditional.Therefore,Itakeanewsymbol,whichis—asfarasIknow—notyetputtospecificuseintherelevantliterature.

Wewillthusbeconsideringthesentenceschema“j > y.”Iwillrightawayrestrictourinvestigationtoconditionalassertionsandwillnotlookatotherillocutionaryrolesthatmaybeconditionalizedaswell;assertionsare largeenoughafield.j and ystandforunconditionalorcategoricalsentences;IwillexplainwhyIdonotconsidernestedconditionals.Moreover,Iwillnotdistinguishbetweensentencesandutterancesbecausethisdistinctionwillnotbecomerelevantinthispa-per.Iwillsaythatsentencesrepresentpropositions(=truthconditions),insofaras theydo,andexpressmental states, inparticularbeliefsoranyotherfeaturesofepistemicstates.Throughoutthepaper,thesen-tencesjandyare,respectively,takentorepresentthepropositionsA andB.So,jandyexpressthebeliefsinAandinB.(ThisentailsthatItakebelieftobeapropositionalattitude;Icannotburdenthepaperwithissuesofhyperintensionality.)Whether“j > y”alsorepresentsapropositionisanopenissuewhichwillbeconsideredonlyinsection8.Before,Iwillonlydiscusswhat“j > y”mightexpress.

Somuch for terminologicalpreliminaries.What then could start-ingfromscratchmean?Forme,itmeansstartingwithsemantics,withthemeaningofconditionals.Howarewetodescribemeanings?Whatislanguagegoodfor?Primarilyforexpressingourmentalstatesandattitudes.Atleast,ourmentalstatesandattitudesaretheimmediatecausalpredecessorsofourspeech;so,whateverelseitsignifiesisme-diatedbythem.Ofcourse,theprimarypurposeneednotbethemostimportant or most interesting. (This emphasizes the speaker’s sidewhilethehearerhasthecomplementarytaskofunderstandingwhatisexpressed.)Letmetakethisforgrantedhere;thisisnottheplaceforfoundationaldisputesaboutphilosophyoflanguage.



truthconditionsmaybeawaytoavoidtheproblem.Butifweinsistonemphatictruthconditions,itisnotclearwhoseproblemthisis.Usu-ally,itistakenasachallengetotheexpressivist.However,itmayaswellbeseenasaproblemfortruth-conditionalsemanticstointegratesentencesthathaveapparentlynoemphatictruthconditions.

I shall not attempt to resolve these intricate issues here. Also, Ihappily concede that truth-conditional semantics isfine, as far as itgoes.Myonlypointwillbethat,inanemphaticsense,itisnotgoodenoughforallmentalstateswemightwishtoexpress—noteveninthederivativewayjustmentionedforpropositionalattitudesbesidesbelief.Iwillimplementanexpressivisticstrategyfordealingwiththeexceptionsthatareinthefocusofthispaper,andIwillarguethatthisstrategyissuperior,howeverwesolvetheensuingproblems.

The exceptions first coming to mind are utterances like “ouch,”whichexpressespainandhasnotruthcondition(onlyasinceritycon-dition;namely,actuallybeing inpain). If thiswere theonlykindofexception,onemightaswellneglectit.Butitisnot.Iamverysurethatconditionalbeliefisamentalstatethatescapesthetruth-conditionalapproachaswell;conditionalbeliefshavenotruthconditions!Thisissoimportantwithinourpresentcontextthat itdeservesa label:CB-noTC.(Here, truthconditionsareagaintobeunderstoodintheem-phaticsense.ThiswillbemydefaultunderstandingintherestofthepaperunlessIsayotherwise.)

This claim is often taken to have been shown by Lewis (1976),calledthe‘bombshell’byEdgington(1995,p.271),wherethefollowingisproved:assumethatforalljandy, “j > y”representsapropositionor truthconditiontheprobabilityofwhichis identicalwiththecon-ditionalprobabilityP(B | A)(recallmyconventionaboutj, y, AandB). ThenP canonlybeaverytrivialprobabilitymeasure(inaspecificsense).Bennett(2003,ch.7)takesthistobeoneofseveralroutestoNTV,theclaimthatindicativeconditionalshavenotruth-value.How-ever, thedialectic situation isnotquite clear, asBennett’s chapter 7thoroughlydisplays.Maybethereisnotreallyaproblemabouttruthconditions,butonlyoneaboutembeddingsofconditionals?

(emphatic)truthconditions,beforewecanknowwhattheyimmedi-atelyexpress.Surely,Frege’sso-calledantipsychologism is thehall-markofthatextremelyhealthytransition,whateveritspresentstatus(see,e. g.,Burge[1979]).

Anotherpoint is that thestates linguisticallyexpressedaremain-ly propositional attitudes. Propositions are truth conditions and be-lief—the paradigmatic propositional attitude—is truth-evaluable.Here,truthmaywellbetakenintheemphaticsense.InsteadoftakingjtoexpressthebeliefinA,wemaythereforestraightawayconsiderj asrepresentingA. Thus,truth-conditionalsemantics,evenintheem-phaticsense,maycarryusmostoftheway,evenifitdoesnotliterallyapplytodeonticlanguage,etc.,expressingotherattitudesthanbelief.Thesamepointisreflectedinspeechacttheorythatdistinguishesil-locutionaryroleandpropositionalcontent.

Thebasicreasonforpreferringtruth-conditionalsemantics,oftenconsideredtobedecisiveeveninphilosophy,isthatsemanticsmustproceedcompositionallyandthatweknowhowtostaterecursivese-manticrulesintermsoftruthandreference(inpossibleworlds).How-ever, thedialectic situation isnotsoclear.Wemaybecontentwithusingashallownotionoftruthfortruth-conditionalsemantics.Then,asstatedabove,mynotionofexpressivismisbroadenoughtoencom-passthisprocedure.Orwemayinsistthattruth-conditionalsemanticsrefers toemphatic truth.Thismay induce thechallenge thatexpres-sivismshouldnotbestatedintermsofshallowtruthconditions,butshouldprovideasemanticrecursiondirectlyintermsofmentalstatesexpressed.This ishowKölbel (2002, ch. 4–5)distinguishes ‘soft’ (=shallow)truthfromexpressivismandhowSchroeder(2008,sec.I.2)setsupthebasicproblemofexpressivism;here,bothintendamoreambitiousnotionofexpressivism.Merin(1999,2003,2006)hasmadeproposalshowtomeetthosedemands.Ileaveitopenherewhetherornotambitiousexpressivismcansolvethisproblem.

Atthispoint,wealsoslipintothenotoriousFrege-Geachproblemof how to treat complex sentencesparts ofwhich are tobe treatedtruth-conditionallyandotherparts inanexpressivisticway.Shallow



importantforourcognitivelife.Itgovernsthedynamicsofbelief,orrather itsrationaldynamics.Thebasicrule is thatconditionalbeliefturns intounconditionalbeliefupon learning that the conditionob-tains.Infact,thisistoocrudearuleofconditionalization;butallmoresophisticatedandmoreadequaterulesforchangingbeliefsbuildonthenotionofconditionalbelief(seeSpohn[2012,sec.5.4;ch.9]).Wemightalsosaythatallofourlearningorinductivestrategies,allofournon-deductiveinferencesdependonourconditionalbeliefs.Onecan-notoveremphasizetheirimportanceforepistemology.Theverysameremarkswouldapplytoconditionalprobabilities.

In any case, onemustnot assume the above equation: that con-ditional belief is unconditional belief in conditionals. This idea hasgeneratedconsiderableconfusionandis,atbest,aplausiblehypoth-esisforsomekindofconditionals.Weshoulddispensewiththisidea.Rather, conditional belief—just like conditional probability—is apurelyepistemologicalnotionwellcharacterizedbyitscentralroleforthedynamicsofbelief,andassuchindependentofanysemanticcon-siderationsconcerningparticularlinguisticmeans.Itdoesnotderivefromsemantics,butcanreverselygroundexpressivisticsemantics.

Finally,itseemsclearthat,ifourconditionalidiomexpressesany-thing,itexpressesconditionalbelieforsomethingaboutit;nootherprominentattitudeisinsightthatcouldfillthisexpressiverole.Andreversely,ifconditionalbeliefissofundamentalforourcognitivelife,asjustclaimed,thenitcannothideintheunderground;itshouldfindsomelinguisticexpression.Butwhichcouldthatbe,ifnotthecondi-tionalidiom?Again,onemustnotsaythatitisstillunconditionalbeliefthatisexpressed;namely,beliefinaspecialkindofconditionalpropo-sitions.Thiswouldpresupposewhatweare trying to analyze.Anditwouldleaveconditionalbeliefitselfwithoutexpression,which — as stated — isnottobeequatedwithunconditionalbeliefinconditionals.

Wemighteventuallybeabletoreturntotheclaimthatcondition-alsexpressunconditionalbeliefinconditionals,justasanyassertionexpressesthebeliefintheasserted.Butifso,thenonlyaftercarryingouttheprojectionstrategywithwhichStalnaker(1984,ch.6–8)has

Thereisnopointnowinunfoldingthatdialecticsituation.FortherelevanceofthedebateconcerningNTVforCBnoTCisnotfullycleareither;itverymuchdependsontherelationbetweenconditionalsandconditionalbelief.Therelevancewouldbeimmediateifconditionalswere assumed to represent conditional propositions, and if condi-tionalbeliefwouldthenbeequatedwithunconditionalbeliefinsuchconditionalpropositions.Themostpertinent impossibilityresult forthisset-upisthatofGärdenfors(1986),whichassumesthisequationbystrictlyadheringtotheRamseytest.Thisresulttransfersthe‘bomb-shell’tobeliefrevisiontheoryandshowsthatthereisnopropositionrepresentedby“j > y,”whichisacceptedifandonlyifBisacceptedafterrevisionbyA.However,thisequationisdoubtful;hence,difficul-tieswithassigningtruthconditionstoconditionalsdonotautomati-callyspeakinfavorofCBnoTC.

Therefore,Iprefertoomitthedetourviaconditionalsandavoidtherelianceonsuchanequation.Idonotacceptit,anyway,notbecauseconditionalbeliefwouldbesohardtograsp,butbecauseconditionalsaresovariedandambiguous.Ialsoneednotrefertoembeddingsofconditionals.Rather,theissuecanbetackleddirectly,andissointheproofinSpohn(2012,sec.15.3)thatconditionalbeliefisnotobjectiv-izable,i. e.,cannotgenerallybeassignedtruthconditions.However,Iamrunningahead,sincethisproofpresupposestheranking-theoreticrepresentationofconditionalbeliefandtheappertainingobjectiviza-tiontheory,whichIcannotrepeathere.Still,thisismydecisivereasonforCBnoTC.

Thus,conditionalbelief isnot,and isnot reducible to,aproposi-tional attitude. It is rather abi-propositional attitude, as itwere.Eachof the two propositions it relates, the condition and the condition-allybelieved,isatruthcondition;theirrelation,however,cannotbegraspedintruth-conditional(letalonetruth-functional)terms.Ifso,anexpressivistic semantics of conditionals cannot be reduced to truth-conditionalsemantics.

Ihavenotyetdiscussedwhatconditionalbeliefisatall;thisisthetopicofthenextsection.Sofar,itshouldonlybeclearthatitismost



insteadof sentential terms).Asmentioned, I shall considerproposi-tionsasobjectsofbeliefandignoreproblemsofhyperintensionality.

So,howshouldweaccountforconditionalbelief?Intheliterature,thisquestionisdistractinglyintertwinedwiththeissueofaccountingforconditionals.Asmallminority(e. g.,Lewis[1976],Jackson[1987])defendstheviewthatanindicativeconditional“j > y”maybasicallybe interpreted as thematerial implication “j → y,” thus expressingBel(A → B)—thebeliefinthematerialimplication.Forthehalf-truthofthisview,seebelow.Therefore,onemightbetemptedtoidentifytheconditionalbeliefBel(B | A)withtheunconditionalbeliefBel(A → B).However,nobodyhasproposedanythinglikethis;itwouldbecrazy.For,ifwetakeAtobefalse,wetakeA → BaswellasA → tobetrue,andthen,accordingtothisproposal,wewouldbelieveBaswellasconditionalonA.However,evenconditionalbeliefisrationallyboundtobeconsistent(atleastunderallconditionsnotconsideredtobeim-possible).Hence,thisproposalwouldbeinadequate.

Perhapsthemostpopularviewtodayisthattreatingconditionalityinepistemictermsmeanstreatingitbyconditional(subjective)prob-abilities(see,e. g.,Adams[1965,1975],Edgington[1995,2003,2008]);thisispartofthesuccessstoryofBayesianismincontemporaryformalepistemology. Iamverysympathetic to thisapproach;but Iamnotfullysatisfied.Conditionalprobabilitiesindeedprovidethebestofthereceivedmodelsofepistemicconditionality.Theproblem,however,isthatbeliefisnotprobability,andconditionalbeliefisnotconditionalprobability.

The most plausible connection between belief and degrees ofbelief is thatbelief issufficientdegreeofbelief,which iscalledtheLockean thesis (byFoley [1992]).The literaturealways interprets itin terms of probabilities. However, thus interpreted, the Lockeanthesisisnottenable.Thebasicpointisthis:itisafundamentallawofrationalbeliefthat,ifyoubelieveAandbelieveB,or—what’sthesame—ifyou takeA andB tobe true, thenyoualso takeA ∩ B tobetrue,orbelieveit.(Well,onemaycontestthislaw,but,withoutit,hardlyanytheoryofrationalbeliefisleft.Wemuststartsomewhere.)

paradigmatically struggled. In section 8, I shall indicate a clear andrigorousversionoftheprojectionstrategyforintroducingtruth-evalu-ableconditionalpropositions.

Theupshotisasfollows:ifCBnoTCisrightand,ifexpressivismisthereforethesemanticstrategytobeemployed,thenanyinvestigationof conditionalsmust startwith studyingconditionalbeliefs and theexpressiverelationbetweenconditionalsandconditionalbeliefs.ThisiswhatIshalldointherestofthepaper.

Theupshotisnotnew,ofcourse.ItisembodiedintheRamseytest,whichderivesfromRamsey(1929,p.142ff.)anddirectlytakescondi-tionalstoexpressconditionalorsuppositionalbeliefs.Wewillseethattherearemanymoreexpressiveoptions.Moreover,theRamseytestisratheronlyaguidingideathathasfoundvariousexplicationsintheliterature.So,disagreementstartswhenwegettothedetails.

3. Conditional Belief

Howshouldweaccountforconditionalbelief?Let’satleastintroducesymbols:Bel(A)representsunconditionalbelief inA,andBel(B | A)representsconditionalbeliefinBgivenorconditionalonA.Thesub-jectandthetimeofbeliefmaybeleftimplicit;allmytermsforepis-temicstatesrefertothepresentattitudesofthespeaker.Again,AandBstandforpropositions.Apropositionisasetofpossibilitiesandthusatruthconditionofasentence,i. e.,thesetofpossibilitiesinwhichthesentenceistrue.

To bemore explicit, letW be the set of all possibilities in a giv-encase(youmay,butneednotthinkofpossibilitiesasfullpossibleworlds;theymaybesmallworlds,centeredworlds,oranyothermutu-allyincompatibleandjointlyexhaustiveitems);andletAbeaBooleanalgebraofsubsetsofW,whichisclosedundernegation,conjunction,anddisjunction.Ishallnotassumeanyotherclosureproperties.Aisthesetofpropositionsathand,andA andBaretakenfromA. isthenegationofA, A ∩ BtheconjunctionofAandB, A ∪ Btheirdisjunc-tion,andA → B= ∪ B thematerial implication(inset-theoretical



thisaprobabilisticversionofRamsey’s test. In itself, itdoesnotyetprovideasemanticsforA > B;itonlysayshowcredibleoracceptableaconditionalis,whateveritmeans.However,Adamsingeniouslyturnsthisintoacriterionofvalidityoflogicalinferenceswithconditionals:“ifaninferenceistruth-conditionallysoundthentheuncertaintyofitsconclusion cannot exceed the sum of the uncertainties of its prem-ises”(Adams[1975,p.3]).Theformalversionisthis:aninferenceissound iff foreache > 0 there isad > 0 such that, forallprobabilitymeasures,theconclusionhasprobability≥ 1–e,ifallofthepremiseshaveprobability≥ 1–d(cf.Adams[1975,p.57]).Thereby,Adamsisabletoaccountforthelogicalbehaviorof indicativeconditionals inthestandardform.

All this is very nice, and in away I have no quarrelwith it. Ad-ams’approachhasbeendeeplyandextensivelydeveloped;see,e. g.,Bamber(2000)forameticulousinvestigation,whichaddstoAdams’above probabilistic definition of ‘entailment with surety’ a rich ac-countof‘entailmentwithnearsurety.’AdamsandBamberapproach(conditional)beliefbyapproaching(conditional)probability1.Theydonotequatethetwo,becausetheymovewithinstandardprobabilitytheorywhereinwhichprobabilitiesconditionalonsomethinghavingprobability0areundefinedand,hence,beliefsconditionalonsome-thingdisbelievedcouldnotbeexplainedonthebasisofsuchanequa-tion.Onemight,however,fullyendorsethisequationifoneresortstoPoppermeasuresinsteadofstandardprobabilities.Thisideahasbeenexecuted,e. g.,byHawthorne(1996).Again,oneendsupwiththestan-dardlogic,whichincludesrationalmonotony.

Mymainreservationaboutall this is the following:myproposalbelowwillcaptureconditionalbeliefdirectlyandinmuchsimpleraway.There is noneed at all for theseprobabilistic detours and sur-rogates, no need for Popper measures or those quite involved e,d-Limitconstructions.Weneednotmaintainthefalseandsuperfluouspretense that we could capture belief in probabilistic terms or ap-proachbeliefbyapproachingprobability1. Inparticular,weshouldnotequatebeliefwithprobability1viaPoppermeasures.Thisisnot

However,ifyourprobabilityofAishigh(abovetherelevantthresh-old)andthatofBisalsohigh,thatofA ∩ Bneednotbe.Thus,thisfundamental lawofrationalbeliefrefutestheprobabilisticLockeanthesis.Thispointishighlightedbythewell-knownlotteryparadox(cf.Kyburg[1961,p.197]).

Theissuehasprovokedavigorousdiscussionwithquiteafewep-icycles. Forperhaps themost advancedprobabilistic accountof theLockeanthesis,seeLeitgeb(2014);buteventhishas itshitches, forinstance,bymakingbeliefascriptionsinmyviewexcessivelycontext-orpartition-sensitive(hediscusses thisobjectiononp.152–159).So,withoutengagingintodetaileddiscussion,myconclusionisasfollows:thereisnogoodwaytosavetheprobabilisticLockeanthesis.Beliefandprobabilityareincongruentphenomena.AndIamconvincedthatdispensingwithbeliefandturningtoJeffrey’s(1992)radicalprobabi-lismisnosolutioneither.(ForallthisseemyextensivediscussioninSpohn[2012,sect.3.3,andch.10].)

Thepointextendstoconditionalbeliefandprobability.Everyoneacceptsthefollowinglogicallawforindicative(andsubjunctive)con-ditionals:ifj > yandj > c,thenj > y & c.Thiswellfitstothegen-erallyacceptedlawofrationalconditionalbeliefsaying:ifBel(B | A)andBel(C | A),thenBel(B ∩ C | A).However,wegetnoneofthisifweidentifyconditionalbeliefwithhighconditionalprobability.Hence,itseemsinadequatetotreatconditionalsandconditionalbeliefinprob-abilisticterms.

This seems tocontradictAdams (1965, 1975)whohasdevelopedthestandardlogicofconditionalsinprobabilistictermsinamostat-tractiveway,includingtheabovelawaboutconjunction.(ItakethisstandardlogictobethebasicsystemVofLewis[1973,p.132]forun-nestedconditionals,possiblywithadditionalaxioms.)Butthereisnocontradiction.AdamsstartswithwhatisnowcalledAdams’ thesis:that“theprobabilityofanindicativeconditionaloftheform‘ifAisthecase,thenBis’isaconditionalprobability”(Adams[1975,p.2]),i. e.,P(A >

B)=P(B | A),providedAandBdonotcontainaconditionalinturn(i. e.,belong to factual language,asAdamssays).Onemaywell call



unconvincinginmyview(butseemyextensivediscussionsinSpohn[2012,sec.5.6;ch.8]).

IhaveproposedasolutionthatfullysolvestheproblemofiteratedbeliefchangeinSpohn(1983b,sec.5.3;1988,sec.4)bywhatisnowcalledrankingtheory;thetheoryisfullydevelopedanddefendedinSpohn(2012)andpartiallyinmanyearlierpapers.Thisrichtheorizingismyultimate justification formaintaining that ranking theory pro-videsthemostadequateaccountofconditionalbelief.IadmitthatIhardlyarguedhereforthisclaim;Ihaveonlyindicatedsomecentralproblemswithsomeofthemainalternatives.However,Ishouldnotfurtherexpandthecomparativebusiness;itmustsufficeherethatthisclaimhasatleastsomeinitialplausibility.

Letmeintroducethebasicconceptsfortheyarecrucialfortherestofthepaper:

Definition:kisanegative ranking functionforA,theBooleanalgebraofpropositionsoverW,iffkisafunctionfromAintoN ∪ {∞},thesetofnaturalnumbersplusinfinity,suchthatforallA, B, ∈ A:

(1) k(W)=0andk(∅)=∞,

(2) k(A ∪ B)=min{k(A),k(B)}.

k(A)iscalledthe(negative)rankofA.Ifk(A)<∞,thentheconditional rankofB given Aisdefinedas

(3) k(B | A)=k(A ∩ B)–k(A).

Negativeranksrepresentdegreesofdisbelief(thisiswhyIcallthemnegative).Thatis,k(A)=0saysthatAisnotdisbelieved,andk(A)=n > 0saysthatAisdisbelieved(todegreen).Accordingto(1)and(2)wehavemin{k(A),k( )}=k(W)=0;thatis,atleastoneofk(A)and k( )mustbe0.Thismeansthatyoucannottakeboth—Aand —tobefalse;thisisabasicconsistencyrequirement.Butwemayhavek(A)=k( )=0,inwhichcasekhasnoopinionaboutA.BeliefinA, Bel(A),isthesameasdisbeliefin andthusrepresentedbyk( )>0.

onlyphenomenologically,butalsotheoreticallyinadequate.AsHaw-thorne (1996) shows, Gärdenfors’ (1979, 1981) belief revision theo-reticaccountofconditionalsgetstherebyprobabilisticallyreproduced,sincethe0–1-structureofPoppermeasuresisequivalenttoepistemicentrenchmentorderings(asalreadyprovedinSpohn[1986]).Inturn,this entails that my reservations below concerning belief revisiontheoryextendtotheuseofPoppermeasures inthepresentcontext.Finally,sincesubjectiveprobabilitiescan’tbetrueorfalse(intheem-phaticsense),Adams’approachcannotpointawayforconditionalstobetrueorfalse.Intuitively,however,atleastsomeconditionalscanbetrueorfalse.Hence,theprobabilistmusteitherrejectthisintuitionorgoforheterogeneousaccountsofconditionals.However,ifweap-proachconditionalsintermsofbeliefinsteadofprobability,thisawk-wardalternativewillnotarise(seesections6–8below).

So,letusnotrejecttheprobabilisticapproach,butletusputittoone side inorder tomake room forother considerations.Howelsecouldwe grasp conditional belief?Curiously, belief revision theoryoriginatesdirectlyfromtheRamseytest.Gärdenfors(1979,1981)wasmotivatedbythistesttodirectlyinquireintotherationalbehaviorofbeliefrevision,ofwhattobelieveaftersupposingoracceptingapossi-blybelief-contraveningproposition.Thishasdevelopedintoso-calledAGMbeliefrevisiontheory(accordingtoAlchourrónetal.[1985]).ItiscanonizedinGärdenfors(1988);ithas,however,foundmanyhotlydebatedvariants(cf.,e. g.,Rott[2001]).

The importance of this field cannot be overemphasized; it wasabout thefirstgenuinephilosophicalemancipationof formalepiste-mology fromtheprobabilisticparadigm.However, inSpohn(1983b,sec.5.2;1988,sec.3),Iraisedtheproblemofiteratedbeliefchange:be-liefrevisiontheoryisunabletoprovideacompletedynamicsofbelief;itcanaccountonlyforthefirst,butnotforfurtherchanges.Thisalsoentailsthatitdoesnotprovideanadequatenotionofconditionalbe-lief.Theproblemhasbeenthoroughlyattended;see,e. g.,Rott(2009).Letmejustsay,againwithoutengagingintodetailedargument,thatallproposalswithintheconfinesofbeliefrevisiontheoryhaveremained



probabilities.Inwhatfollows,IshallkeepthingssimpleandsticktothefirstinterpretationofBel(A)ask( )>0.

Despitethesharpinterpretationalcontrast,thereisalsoastrikingsimilaritybetweentherankingaxioms(1)–(3)andtheaxiomsofprob-ability including the definition of conditional probability; by takingthelogarithmofprobabilitiesrelativetoasmall(orinfinitesimal)basethe latter roughly (oralmostexactly) translate into the former.Thisgeneratesalotofsimilaritiesofamathematicalnature.(ForarigoroustranslationseeSpohn[2012,sec.10.2].)

OnemightthinkthatAdamsattemptstoprobabilisticallyapproxi-matethebehaviorofconditionalrankswithhise,d-limitconstructions.However, as indicated above, he rather approximates the 0–1-struc-tureofPoppermeasuresorepistemicentrenchmentorderings.Inanycase,weshouldnottrytoapproachranksinprobabilisticterms;itissomuchmorestraightforwardtosimplyreplaceprobabilitiesbyranks.Thisis,inanutshell,whatIshallpropose.

Itmay seem awkward toworkwith negative ranks representingdisbelief,becauseofthedoublenegationsinvolved(thiswasanotherpointhamperingthereceptionofrankingtheory).However, it isnoproblemtodirectlyrepresentbelief.IfkisanegativerankingfunctionforA,wemaydefinethepositive ranking function bforAby:

(5) b(A)=k( ).

bthusrepresentsdegreesofbelief;b(A)>0saysthatAisbelieved(af-terwehaveputthethresholdztooneside)andb(A)=0saysthatA isnotbelieved.Wemayalsodirectlyaxiomatizepositivefunctionsbytranslating(1)and(2)bymeansof(5).Thus:

(6) b(W)=∞andb(∅)=0,

(7) b(A ∩ B)=min{b(A),b(B)}.

(7)saysthatyourdegreeofbeliefinaconjunctionequalsyourweak-estdegreeofbeliefintheconjuncts — andthusentailsthefundamental

Similarlyforconditionalranks;theyrepresentconditional degrees of disbelief.Definition(3) is intuitivelyplausible: itsaysthatyouarriveat your degree of disbelief inA ∩ B,when you add your degree ofdisbeliefinA andyourdegreeofdisbeliefinB,giventhatAshouldbetrue.Conditionalranksalsorepresentconditionalbelief:k(B | A)=0 saysthatBisnotdisbelievedgivenA;k(B | A)>0representsdisbeliefinBgivenA;andk( | A)representsbeliefinB given A, i. e.,Bel(B | A).Againwehave:

(4) eitherk(B | A)=0ork( | A)=0orboth.

Thatis,youcannothavecontradictorybeliefsunderanyconditionA whichyoudonottaketobeimpossible,i. e.,forwhichk(A)<∞.Giv-endefinition (3), (4) is indeedequivalentwith (2).Thismeans thatrankingtheoryessentiallyassumesnothingbutconditionalconsisten-cy—andthushasextremelystrongnormativefoundations.

Thecrucialpointisthatbeliefsmaybeweakerorfirmerandtheyare still beliefs. This is our everydaynotion, and it is respectedbyranking theory; ranksare intended tomeasure thosedegreesofbe-liefs.However,initiallyitwasunclearhowtheydoso;ranksmayhaveappearedtobearbitrary.Thishascertainlyhamperedtheacceptanceofrankingtheory.Thesituationhaschanged,though,withHildandSpohn(2008),wherearigorousmeasurementtheoryforranksisof-feredintermsofiteratedcontractions.Itcorrespondstothemeasure-mentofprobabilities,thedifferencebeingthatranksaremeasuredonaratioscaleandprobabilitiesonanabsolutescale—cf.alsoSpohn(2012, ch.8).Hence,wearedealingwith twodifferentkindsofde-grees of belief, ranks and probabilities, and only one of them alsorepresentsbelief.

Infact,thereisnoneedtosaythatbeliefinA—Bel(A)—isrepre-sentedbyk( )>0;wemightaswellrepresentitbyk( )>z,forsomefixedthresholdz > 0.Thelawsofbeliefcomeouttheverysame;it’sonly that belief is vague and can be takenmore or less strictly, de-pendingonthethresholdz.Thus,theLockeanthesisisabsolutelycor-rect,ifthedegreesofbeliefitreferstoaretakentoberanksandnot



ourspecificframework,thetesttellsthattheschematicconditional“j

> y”expressesthefollowing:

(I.1) t(B | A)>0,

i. e.,Bel(B | A), theconditionalbelief inBgivenAorunderthesup-positionofA (asbefore,jrepresentsAandyrepresentsB).Wemightalsosaythat(I.1)providesassertibilityconditionsfor“j > y,” inthesensethat“j > y”isassertibleonlyforsomeonewhosedoxasticstatesatisfies(I.1).Orwemightsaythat(I.1)providesacceptabilitycondi-tionsfor“j > y,”inthesensethat“j > y”isacceptableonlyforsome-onesatisfying(I.1);puttingitthisway,however,tendstorefertothehearer’sandnottothespeaker’sside.(Theremaybesubtledifferencesbetween assertibility and acceptability; see Douven and Verbrugge[2010].) I prefer to continue speakingofwhatwemight express in-steadofassertibilityoracceptabilityconditions.

There isnoneedhere torehearse the tremendousplausibilityofthethesisthat indicativeconditionalsareoftencharacterizedbytheRamseytest,i. e.,express(I.1).

Interlude 1: The Logic of the Ramsey Test According to (I.1)Letmedisplaytheensuinglogic:rankingfunctionsentailasemanticsforanon-iteratedfragmentL1ofconditionallogicimplementingtheRamseytest.Thesyntaxissimple:letL0bethelanguageofproposi-tionallogic,andlet>standfortheconditional.Then,ifjandyaresentencesofL0, j > yisasentenceofL1,andifjandyaresentencesofL0orL1,propositionalcombinationsofjandyaresentencesofL1, too.Thus,nonestingsof>canoccurinL1.

Thesemanticsrunsthus:letVbethesetofvaluations(ofthesen-tenceletters)ofL0.Forj ∈ L0andv ∈ V v ⊨0 jsaysthatjistrueinv.DefineV(j)={v | v ⊨0 j}tobethesetofvaluationsinwhichjistrue. Moreover,foranyrankingfunctionkforV,letB(k)={j | t(V(j))>0} bethesetofsentencesexpressingbeliefsheldink(ortheassociatedtwo-sidedrankingfunctiont),andletCB(k)={j > y | t(V(y)|V(j))>

lawthatyoubelieveaconjunctioniffyoubelievebothconjuncts.Thedefinition(3)ofconditionalnegativerankstranslatesinto:

(8) b(B | A)=b(A → B)–b( ).

Atfirstanunlikelytranslation,butitscontentishighlyintuitive,sayingthatyourdegreeofbeliefinamaterialimplicationisyourcorrespond-ingconditionaldegreeofbeliefplusyourdegreeofbeliefinthevacu-oustruthoftheimplication,i. e.,inthefalsityofitsantecedent.Iwillunfoldtheimportanceof(8)below.However,with(8),itisparticular-lyclearthatpositiverankshavenoformalanalogywithprobabilities.Thisexplainsmydeterminatepreferencefornegativeranks.

Wemayevenintegratepositiveandnegativeranksintoonenotion,whichIcallatwo-sided ranking function tdefinedby:

(9) t(A)=b(A)–k(A)=k( )–k(A).

Conditionaltwo-sidedranksaredefinedanalogously:

(10)t(B | A)=b(B | A)–k(B | A)=k( | A)–k(B | A).

Two sided-ranks are intuitivelymost intelligible, because they repre-sentbeliefanddisbeliefatonce:Aisbelievedordisbelievedorneitheriff,respectively,t(A)>0,<0,or=0—similarlyforconditionaltwo-sidedranks.ThisiswhyIshalloftenrefertotwo-sidedranksbelow.Howevertheformalbehavioroftwo-sidedranksisclumsy;itisbestaccessibleviadefinition(9)andthebehaviorofthecomponentconcepts.

Thelongandtheshortofallthis:ifwewanttotheoreticallycapture(rational)conditionalbelief,webestdoitby(3)and(8).Positiveandnegative conditional ranks are, respectively, conditional degrees ofbeliefanddisbelief;andifthesedegreesarenon-zero,theyrepresentconditionalbeliefanddisbelief.Weshouldproceedwithourexpres-sivisticexplorationofconditionalsintermsofthisrepresentation.

4. The Ramsey Test

TheexpressivisticviewofourtopicwasintroducedbyRamsey(1929);thustheRamsey testisthenaturalstartingpointofourinvestigation.In



thoughnot completely inevery relevantaspect; thebusinessmightbecometricky.However,thisbusinessisnotmyinteresthere.

TherestrictiontothefragmentL1isimportant.TheRamseytest(I.1)cannotmakeimmediatesenseofnestingsofconditionals.Thesameholdsinprobabilisticterms.WehaveAdams’thesis,whichobeysthisrestrictiontoL1 (or ‘factual language’)andis justaprobabilisticver-sionoftheRamseytest(whenceitisdoubtfulwhetherP(j > y)canbeunderstoodasaprobability).AndthereisStalnaker’s thesis(cf.Stal-naker[1970,sec.3]),whichextendsAdams’thesistonestedcondition-als,andthustreats“j > y”asrepresentingapropositionandattemptstoassigntoitaproperunconditionalprobability.Alas,itfoundersatLewis’trivializationtheorem.So,webettersticktotherestriction.

Interlude 2: The Equivalence ThesisWiththehelpof(I.1),wecanclarifythestatusoftheso-calledequiva-lencethesis,whichattractedalotof interest.Itsaysthatat leasttheindicativeconditional>issimplythetruth-functionalmaterialimpli-cation→,perhapsamendedbysomesuitablepragmaticbackground.Fornestedconditionals,itisknowntoleadtononsensicalresults.(Seetheproof ofGod’s existence inEdgington (1995, p. 281),with truth-functional “if”: “IfGoddoesnotexist, then it’snot thecase that if Ipraymyprayerswillbeanswered. Idonotpray.ThereforeGodex-ists.”).However,restrictedtothefragmentL1 ,ithasalotofplausibil-ity(whenceitsprominentdefenderssuchasGrice[1975],Lewis[1976,p.305ff.],andJackson[1987,ch.1–2]).Thehalf-truthofthispositioniswellexplainedbytheRamseytest.Withthedefinitions(9)and(10),(8)immediatelyentails:

(11) ift(A)≥ 0,thent(B | A)>0ifandonlyift(A → B)>0.

(Becauseof(9)and(10),(11)isequivalentto:ifk(A)=0,thenk( | A)>0iffk(A ∩ )>0.Becauseof(8),thisisequivalentto:ifk(A)=0,thenk(A ∩ )–k(A)>0iffk(A ∩ )>0.Andthisisobviouslytrue.)

Thatis,ifAisnottakentobefalse,BisbelievedconditionalonA ifandonlyifthematerialimplicationA → Bisbelieved.Instillother

0}bethesetofconditionalsentencescorrespondingtotheconditionalbeliefsink.

NowwemayrecursivelydefinetruthforallsentencesinL1relativetoavaluationv ∈ VandarankingfunctionkforVbyspecifyingthefollowingrecursivebase:〈v, k〉 ⊨ p iffv ⊨0 p foranysentenceletterp ofL0,and〈v, k〉 ⊨ j > yiffj > y ∈ CB(k).NotethatwetherebyprovidewhatIabovecalledshallowtruthconditionsforthesentencesinL1.

Thenwehaveachoice:wemaycallc ∈ L1 semi-epistemically logically true, ⊨se c,iff〈v, k〉 ⊨ cforallvaluationsv ∈ V andallrankingfunctionskforV.Orwemayepistemicallyrestrictthatnotionbyrequiringthatall(unconditional)beliefsinkmustbetrueinthevaluationv.Thatis,wemaydefineasentencec ofL1tobeepistemically logically true, ⊨e c,iff〈v, k〉 ⊨ c forallrankingfunctionskforVandallvaluationsv ∈ V suchthatv ⊨0 jforallj ∈ B(k).

Itiseasilychecked,then,thattherestrictionofLewis’logicV(cf.Lewis[1973a,p.132])tothefragmentL1iscorrectandcompletewithrespect to⊨se. In particular, neither Centering norWeak Centeringholdwithrespectto⊨sesimplybecausethereisnorelationbetweenthefactsaccordingtovandtheconditionalbeliefsaccordingtoarank-ingfunctionkforV.Bycontrast,itisLewis’logicVC,restrictedtothefragmentL1(whichalsoresultsfromAdams’probabilisticsemantics),thatiscorrectandcompletewithrespectto⊨e,since⊨especifiessucharelation.Inparticular,WeakCenteringholdssincet(B | A)>0entailst(A → B)>0(seealso(11)below)andCenteringholdssincet(A ∩ B)> 0entailst(B | A)>0.However,accordingto⊨etheseaxiomsonlyin-dicatearelationbetweenconditionalandunconditionalbeliefs.ThisagreeswithhowGärdenfors (1988,p. 148ff.)setsup things;healsoacceptstheseaxioms.

This interlude showed that the ranking-theoretic route leads togenerallyacceptedlogics,asitshouldbe.Inprinciple,thelogicfortheotherexpressiveoptionstobediscussedbelowcouldbeworkedoutinthesameway,sincethebehaviorofrankingfunctionsiswellknown,



implicationswithinthesettingofdynamicorupdatesemantics.How-ever, I sense a certain confusionof perspectives in his attempt.Up-date semantics describes how the common ground of interlocutorschanges in response to certain utterances. Thus, leaving subtletiesaside,itbasicallydescribeshowthebeliefsofahearerchangethroughutterances.However,Gillies (2004, sec. 4) describes the update po-tentialofanopenconditionalinanauto-epistemicway,accordingtowhich theopenconditionalmakesaclaimaboutwhat isalready inthe common ground. Either, the conditional agreeswithwhat is al-readycontainedinthecommonground(oracceptancebase)andthenconfirmsitwithoutchangingit,oritdoesnotagreeandthusmakesthecommongroundcollapse.Thisdoesnotseemtobeanadequatedescriptionofwhatgoesonon thehearer’s side.And the speaker’ssideismoredirectlydescribedby(I.1)and(11)withoutengagingintodynamicsemantics.

Beforeproceedingtothenextmajortopic,letmementionvariousotherexpressiveoptionsthatgowithoutsaying,butshouldbeexplic-itlylistedhere.First,ifwecanexpressconditionalbeliefaccordingto(I.1),wecanalsoexpressotherconditionalepistemicattitudes;thatis,wemayuse“j > y”alsoforexpressing

(I.2)t(B | A)=0,or

(I.3)t(B | A)<0,

orcombinations thereof.For instance, indicativemight-conditionalsusuallyexpress(I.1or2),i. e.,t(B | A)≥ 0;“ifitrains,hemaycomelate”expressesthatIdonotbelievehimtobeintimegiventhatitrains.

Andletusnotforgetthatinuttering“j > y”wemightalsoexpressourattitudestowardsAbyitselfandtowardsBbyitself,i. e.,whether

(II.1)t(A)>0,(II.2)t(A)=0,or(II.3)t(A)<0,

andwhether

(III.1)t(B)>0,(III.2)t(B)=0,or(III.3)t(B)<0,

terms,ifnotBel( ),“j > y”maybetakentoexpresseitherbelief,sincetheyamounttothesamething.

Thus, (11)refutes the fullequivalencethesis thatequates the in-dicativeconditional>andthetruth-functionalmaterialimplication→ without any restriction. However, onemight say that it is trueunder the assumption that the antecedent isnot taken tobe false.This assumption is thus an adequate epistemic characterizationofthe pragmatic background required by the equivalence thesis. Oronemightsay that theequivalence thesisholds forso-calledopenconditionals (see (II) below),which are indicative conditionals ad-ditionally characterized by the speaker being indeterminate abouttheantecedent(i. e.,t(A)=0).Accordingto(11),beingopeninthissenseisnotrequired;t(A)≥ 0isenoughofanassumption.Butnotethatopenconditionalsaretherebyonlyepistemicallyandnotinanywaylinguisticallydefined.

The equivalence thesis derives its plausibility from the innocentprincipleoften called thedirect argument: “j ory” entails “ifnotj, theny.”Thisappearsmostconvincing,andindeedhasmanycorrectinstances.Jackson(1987,sec.1.1)provestheequivalencethesiswiththedirectargument(andtwofurther,stillmoreinnocentprinciples).However,(11)explainswhat iswrongwiththedirectargument; theentailmentholdsonly if thedisjunctivepremise “jory” isopenaswellinthesensethat“jory”isnotassumedbecausejitselfisalreadytaken to be true. That is, the direct argument itself holds only con-ditionally, andnounconditional equivalence thesismaybederivedfromit.ThisagreeswithStalnaker’s(1975,sec.IV)accountofthedirectargument.AndEvansandOver(2004,p.114)pointtothesamefact,whentheysaythatitmakesadifferencewhetheronehasaconstruc-tiveoranon-constructivejustificationforthedisjunctivebelief“jory.”Ifhavinganon-constructivejustificationfor“jory”onlymeansbelievingthedisjunctionwithoutbelievinganyofthedisjuncts,thentheirexplanationcomestothesame.

Gillies(2004)makesanalternativeattempttocapturesimilaritiesaswellasdifferencesofopenorepistemicconditionalsandmaterial



hadbeenthecase”usuallyexpresses,amongotherthings,(II.3)and(III.1)andisthusalsocalledasemi-factual.Withalittleingenuityonecanpresumablyfindinstantiationsforallninecombinationsof(II.1–3)and(III.1–3).

IshouldfinallyaddthatwecannotonlyexpressbeliefinA,etc.,butalsostrengthofbeliefinA.Therearemanymodifiersinnaturallan-guage indicatingstrengthofbelief,at leastroughlyandvaguely.(In-trospectiondoesnotrevealprecisedegreesofbelief,sothatnomoreprecisioncanbeexpectedfromtheexpressivemeans.)Becausethesemodifiersaresowidespread,itmayseemthatexpressivismmusttakeaprobabilisticroute,at leastas farasassertivespeech isconcerned.However, it can seem so only as long as probabilities are the onlymodelofdegreesofbelief.Therefore,I insistthatthesedegreescanalsobeinterpretedasranks.AndIinsistthatsensitivitytodegreesofbeliefmustnotblindusforthefactthatthebasicphenomenontobeexpressedisbeliefitself.Bethisasitmay,Ishallnotponderabouttheexpressionofstrengthofbelief,becauseit isnotspecifictothecon-ditionalidiom;thosemodifiersareequallycommoninunconditionalassertivespeech.

5. Relevance

IntherestofthepaperIwanttoexplorehowwecangobeyondtheRamseytest.There ismuchmore to theepistemicrelationbetweenthepropositionsrepresentedby theantecedentandtheconsequentoftheschematicconditional“j > y”;it’snotjusttheconditionalbelief.Anotherrelation—amostimportantoneindeed—isepistemic relevance.Fivedecadesagoorso,relevancewasaresiduelefttothepragmaticwastebasket,butonlybecausetherewasnowaytocapturerelevancewiththemeansofextensionallogic.Aniceexampleforthisinabilityis found inFrege’sclaimthat “but”has thesamesenseas “and,” i. e.,is truth-functionally equivalent to it, and differs only in tone.How-ever,itseemsplainlywrongtorelegateepistemicrelevancetomattersoftone.(Foramonographonhowtodobetter,seeMerin[1996].)AmoreseriousexampleisthecelebratedHempel-Oppenheimtheoryof

i. e.,whetherwetaketheantecedentandtheconsequentofthecondi-tionaltobetrueortobefalse.Ofcourse,wecanexpress(II)and(III)alsobyunconditionalsentences,simplybyassertingjory,etc.,andweusuallydo.However,usingtheconditionalidiomisalmostalwaysaccompaniedbyexpressingsomeversionof(II)and(III).

Thisclaimbecomes intelligiblewhenwenote that theexpressiv-isticapproachissofarneutralastohow(conditional)beliefsareex-pressed,i. e.,whethertheyareexpressedasanassertion,apresupposi-tion,oranimplicatureoftheutterance.Fromalinguisticpointofviewthese are, of course, important semantic andpragmaticdistinctions.Fromanexpressivisticpointofview,however,thesedistinctionscomelaterandmaybeinitiallyneglected,asIwilldohere.Therefore,itiscorrecttosaythatconditionalsmightbeusedforexpressing(II)and(III),evenifonlyasapresuppositionorimplicature.

Herearesomeexamples: Ialreadymentionedopenconditionalsthatbydefinitionexpress(II.2)(andusuallyexpress(III.2)aswell).If counterfactuals deserve their name,we thereby express, by pre-suppositionorimplicature(orevenpresuppositionalimplicature,assuggestedbyLeahy[2011]),thatwetaketheirantecedentandtheirconsequent to be false, i. e., as counterfactual. However, no rulewithoutexception.Bysaying“ifhehadtakenarsenic,hewouldhaveshownexactlythosesymptomswhichhedoesinfactshow”(Ander-son1951),Imakeaninferencetoapossibleexplanation,i. e.fromthebeliefintheconsequent(III.1)atleasttothepossibilityoftheante-cedent,thusexpressing(II.1or2).(Seethediscussionofoption(IV.1)belowforthepotentialcorrectnessofsuchaninference.)So,Iwillcontinuespeakingofconditionalsusuallyexpressingthisandthat.Itisextremelydifficultforlinguiststostatestricterrules,andIwillnotengageintheirbusiness,thoughIhopetofacilitateitbyextendingtheexpressiveoptions.

Otherexamplesfor(II)and(III)are“evenif”and“because,”whichIhavealsosubsumedundertheschematicconditional>.“y becausej”isfactiveandthusexpresses,amongotherthings,(II.1)and(III.1),i. e.,beliefinAandinBwhile“ywouldhavebeenthecase,evenifj



(Formyreasonsforstating,e. g.,(IV.1),asaboveandnotast(B | A)>t(B),seeSpohn[2012,p.106f].)

Notethatexplicatingrelevanceasraisingorloweringdegreeofbe-lief—asisdonein(IV)—requiresthefullresourcesofrankingtheory.TheywerenotyetrequiredfortheRamseytest(I).Thatis,forthisex-plicationofrelevance,oneneedscardinalconditionaldegreesofbeliefas ranking-theoreticallydefined in(3)and(8) (orasofferedbycon-ditionalprobabilities).Onlytherebyoneisabletocomparedegreesofbeliefundervaryingconditions,asisrequiredby(IV).Bycontrast,purely ordinal conceptions such as the entrenchment orderings ofbeliefrevisiontheory(cf.Gärdenfors1988,ch.4),thoughsufficientlypowerfulfordealingwiththeRamseytest(I),isunabletodeliversuchcomparisonsinanadequateway.Thisremarkalsoappliestothesimi-larityspheresofLewis(1973a),whichareonlyorderedwithoutquan-titativedistances.Thus,adequatelyrepresentingrelevanceisacrucialpointinmyviewwhererankingtheoryissuperiortothosealternativetheories(asIhavepointedoutalreadyinSpohn[1983a,note18]).

So,mysuggestion is that theschematicconditional “j > y”maybeused to express somekindof relevance (IV.1–3)—and indeed ismostlysoused.Admittedly,thisiscommonlydonebywayofimplica-ture.However,Isaidalreadythattheexpressivistcan,atleastinitially,proceedwithoutdistinguishingassertion,presupposition,and impli-cature. So, the indicative conditional “if” usually expresses positiverelevance.WhenIsay“ifitrains,hewillcomelate,”ItherebyexpressthatIdonotbelievehimtobelate,ifitdoesnotrain.Thiseffectisusu-allyexplained in termsofGriceanconversationalmaxims,butnotethatthosemaximsthemselvesgroundinthenotionofrelevance.“IfOswalddidnotshootKennedy,someoneelsedid”clearlyexpressespositiverelevance:giventhatOswalddidshootKennedy,Idonotbe-lieve—orbelievelessfirmly—thatsomeoneelsedidaswell.(Moreonthisfamousexamplebelow.)“Because”alsoexpressespositiverel-evance;itdiffersfrom“if”onlywithrespectto(II)and(III).“Hecamelate,becauseitrained”doesexpressthesameconditionalrelationas

deductive-nomologicalexplanation,whichessentiallyfounderedatitsinabilitytoincorporaterelevanceconsiderations.(ThisstoryisnicelytoldinSalmon[1989],particularlyinch.3.)

Rott(1986)noticesthesignificanceofrelevancetoconditionallog-ic,andMerin(2007)elaboratesonit.Philosopherstriedvariousideastounderstandrelevance(oneideabeingrelevancelogic;cf.AndersonandBelnap[1975]).SperberandWilson(1986)isverywellreceivedinlinguisticpragmaticsandpsychology.However,asilluminatingastheirobservationsontheroleofrelevanceinhumancommunicationare,theirgeneralcharacterizationisemptyofwhatrelevancebasicallyis:“Anassumptionisrelevantifandonlyifithassomecontextualef-fectinthatcontext”(p.122).Thereisnopointhereintryingtosurveythemanyattemptsatcapturing relevance. Inmyview, theepistemi-callybasicsenseofrelevanceisexplicatedin(subjective)probabilitytheorybyitsnotionof(in-)dependence:AisrelevanttoBiffBproba-bilisticallydependsonA, i. e.,iffP(B | A)≠ P(B | ),i. e.,iffAmakesadifferencetotheepistemicassessmentofB.Clearly,wecanalsodistin-guishpositiveandnegativerelevance.Thisisindeedthebasicnotionofinductivelogicandconfirmationtheory(cf.,e. g.,Carnap1971).Anditisclearlythedominantparadigmforaccountingforrelevance.

This probabilistic notion is fine, and ifwewere concerned onlywithrelevance,wemightperhapsreturntotheprobabilisticparadigm(althoughthedistinctions(IV.1a–d) introducedbeloware importantanduseful,theycannotbeduplicatedinprobabilisticterms).However,wehavechangedtheparadigmforgoodreasons,and,therefore,itisnoteworthythatepistemicrelevanceiscapturedinrankingtheoryatleastaswell.Aispositively relevant, irrelevant,ornegatively relevanttoB, if,respectively,Araises,doesnotchange,orlowersthedegreeofbelief,i. e.,thetwo-sidedrankofB, i. e.,iffrespectively:

(IV.1)t(B | A)>t(B | ),

(IV.2)t(B | A)=t(B | ),

(IV.3)t(B | A)<t(B | ).



althoughSpohn(2012,theorem6.7)coversalotofground.(Thesameholds,bytheway,forprobabilisticpositiverelevance.)

Imightpointout,though,thatrankingtheoreticpositiverelevanceissymmetric:ifAispositivelyrelevanttoB, BispositivelyrelevanttoA.Anditholdsforthenegationsaswell;thatis,ifAispositivelyrelevanttoB, ispositivelyrelevantto .Thesameappliestonegativerele-vance.Itiswellknownthattheverysameclaimsholdforprobabilisticrelevanceordependence.Thereisadeeplyentrenchedtendencynotonlytocorrectlyapplymodus ponensandmodus tollens,butalsotocom-mittheallegedfallaciesofaffirmingtheconsequent(from“ifj,theny”andyinferj)anddenyingtheantecedent(from“ifj,theny”andnon-jinfernon-y).Apparently,thereisatendencytoread“if”as“iff”(indeed,thereisnophoneticdifference).If“if”expressespositiverele-vance(IV.1),thistendencymaybeexplainedbythefactsjustobserved.Theprobabilisticversionofthisexplanationseemspresentlytobefa-voredbypsychologists(cf.OaksfordandChater[2007,p.118ff.]);butitmayaswellbegiveninranking-theoreticterms(seeOlsen[2014,ch.III–IV]).Onemightthinkthatthegeneralvalidityofthesymmetryofpositiverelevancegoestoofar;manyconditionalsdonotdisplaythissymmetry.However,thisneednotmeanthattheydonotexpressposi-tiverelevanceatall.Itmayalsomeanthattheyexpressaspecifickindofpositiverelevance.

Indeed,animportantobservationisthattheranking-theoreticop-tion(IV.1)maybefurtherdifferentiated.Foralongtime,IfeltjustifiedincallingAareason for B,ifAspeaksforB,ifAsupportsorconfirmsB, ifAstrengthensthebeliefinB—thatis,ifAispositivelyrelevantforB,if(IV.1)obtains.(Forfurtherjustification,seeSpohn[2012,ch.6].)Also,Ichosethislabelinordertoindicatethelargephilosophicalreso-nancespaceofthenotionofpositiverelevance.Thepointnowisthattherearevariouskindsofreasonsorpositiverelevance.IfAisareasonforB,itraisesthedegreeofbeliefinB.Butfromwheretowhere?Inprobabilistic terms,nospecific raisingsstandout.However, in rank-ing-theoretictermswecandistinguishfourcases(withself-explaining

before.(Moreon“because”below.)Therearemanymorewaystoex-presspositiverelevance(IV.1).

“Hewill come latewhether ornot it rains” expresses irrelevance(IV.2).Andnegativerelevance(IV.3)mayhaveevenmoreexpressivemeansthanpositiverelevance.“Despite”basicallyindicatesnegativerelevanceandsodoes“but.”Itisgenerallydevianttosay“Fa,butFb,”forinstance:“Annsings,butBobsings”(whereas“FaandFb,”or“AnnsingsandBobsings”isperfectlyokay).AninterestingexplanationliesinCarnap’sprincipleofpositiveinstantialrelevance(cf.Carnap[1971,sec.13]),whichsaysthat,intheabsenceoffurtherbackgroundinfor-mation,oneinstanceofafeatureispositivelyandnotnegativelyrel-evantfornextencounteringafurtherinstanceofthatfeature.(Forthisobservation,seeMerin[1996,1999].)“Hecamelate,althoughitrained”expressesthat,givenitrained,itcameasasurprisethathecamelate;apparently,onewouldratherhaveexpectedhimtocomelatewithoutrain.Fromtheexpressivisticpointofview,relevanceisacentralepis-temicaspecttobeexpressedandnotmerelysomepragmaticadd-on.

WhetherconversationalrelevanceasgenerallyrequiredbyGrice’smaximsofconversationcanbefullycapturedwiththeabovenotionofrelevanceisadifferentquestionthatneednotconcernushere.Butitmightbeworthtrying.For instance, inso-calledbiscuitcondition-als(“therearebiscuitsonthesideboardifyouwantsome”),theirrel-evanceoftheantecedentfortheconsequentissalient,butnotclaimed,e. g.,bya“whetherornot”construction.Still,theconversationalrele-vanceofsuchaconditionalcanbewellexplainedinthepresentterms(namelybythepositiverelevanceoftheantecedentaswellasthecon-sequentoftheexampleforthegoalofeatingsomething—seeMerin[2007]fordetails).

Notethattheexpressiveoptions(II),(III),and(IV)arelogicallyin-dependent;unconditionaldegreesofbeliefinAandBarecompatiblewithanydirectionof relevancebetweenA andB. Itwouldbe inter-estingtodevelopthelogicofpositiverelevanceconditionals, i. e.,ofconditionalsexpressing(IV.1).Letmeonlyremarkthatthebehaviorof positive relevance is not straightforward nor completely known,



Ramsey test (I.1), then it entails positive relevance (IV.1); indeed,A mustthenbeanecessaryandsufficientreasonforB, i. e.,(IV.1b+c)ap-ply.Bycontrast,if“j > y”standsfor“ybecausej”expressingbeliefinAandB,(II.1),and(III.1),andpositiverelevance(IV.1),thenAmustbeasufficientorsupererogatoryreasonforB,andsoforth.Thereisnoplaceheretostudyalltheseinteractions.

6. Circumstances

Itmightappear that (I)–(IV)exhaustourexpressiveoptions for theschematicconditional“j > y.”WecanexpressourattitudestowardsAandBbythemselvesandhowweepistemicallyrelateAandB.Andsincewehave refrained fromattending to specificdegreesofbelief,nothingseemsleftout.Nothing?No,thereisatleastonefurthermostimportantclassofbeliefsthatwemightexpresswithconditionals.Theideaisindicatedatmanyplacesintheliterature;itisperhapsobvious.However,asmyremarksattheendofthissectionwillshow,Icannotfindthattheideahasfoundacleardescription,letaloneapropertheo-reticaltreatment.LetmeexplainwhatIhaveinmind:

Wemightstartwith the infamoussamplepairofQuine(1960,p.222)concerningtheKoreanwar:

(12) IfCaesarwereincommand,hewouldusetheatomicbomb.

(13) IfCaesarwereincommand,hewouldusecatapults.

The pair was designed to demonstrate the hopeless context-depen-denceandindeterminacyofcounterfactualdiscourse.Ifindthecasenotsohopeless,though.(12)directsourattentiontoacertainissueorquestionunderdiscussion:whatkindofpoliticalleaderwasCaesar?Violent,audacious,prudent,compromising,etc.?(13)raisesadifferentquestion:whatkindofwarfaretechnologywasavailableatCaesar’stimes?(SeeRoberts[1996]forageneralaccountofthepragmaticroleofthe‘questionunderdiscussion.’)

Formally, aquestion is representedbyapartitionof thepossibil-ityspaceW;forinstance,apsychologicalpartitioneachcellofwhich

labels),andwemighthaveaninterestinexpressinganyofthembyusingaconditional“j > y”:

(IV.1a)t(B | A)>t(B | )>0, i. e.,A isasupererogatory reason forB.

(IV.1b)t(B | A)>0 ≥ t(B | ),i. e.,Aisasufficient reasonforB.

(IV.1c)t(B | A)≥ 0 > t(B | ),i. e.,Aisanecessary reasonforB.

(IV.1d)0 > t(B | A)>t(B | ),i. e.,Aisaninsufficient reasonforB.

Thus,e. g.,AisasufficientreasonforB,ifBisbelievedgivenA,butnotgiven .Onlykinds(b)and(c)arenotdisjoint;Amaybeanecessaryand sufficient reason forB. The samedistinctionsmaybemade fornegativerelevance.Inthisway,theexpressiveoptionsfor“j > y”dif-ferentiatefurther.Itiscertainlyapointinfavorofrankingtheorythatitisabletorepresentthesedistinctionsanddefinitelyapointcountingagainstprobabilitytheorythatitcannotcapturesuchafamiliarnotionasthatofasufficientreason,whichmustnotbeshortenedtotheno-tionofalogicallysufficientreason.

Again, thepropertiesof thesekindsof reasonarenotstraightfor-wardandnotcompletelyknown.Someexpectations (trainedordis-tortedbydeductive logic)maybedisappointed,e. g., the relationofbeing a sufficient reason isnot transitive. And ifA andA’ are suffi-cientreasonsforB, A ∩ A’neednotbe!Moreover,inordertocontinueontheaboveremark,thefourkindsofreasonsarenotsymmetricbythemselves;onlytheirdisjunction(IV.1)is.(Formoreonthatbehavior,seeSpohn[2012,sec.6.2].)

Finally,thelogicalindependenceof(IV)from(II)and(III)nolon-gerholdsforthesubtypes(a)–(d).Therearemanyinterestinginterac-tionsof(II)andIII)withthesesubtypes.Forinstance,ifAisasuper-erogatoryreasonforB, Bmustalsobebelievedunconditionally.Andif“j > y”isacounterfactual“ifjhadobtained,ywouldhaveobtained”andthusexpresses,viapresuppositionorimplicature,thebeliefinthefalsityofAandB, (II.3)and(III.3),and if itmoreoverexpresses the



conditionsorpropositions.Thenwemayusetheconditional“j > y”forexpressingthebeliefthatoneofallthoseconditionsinPobtainsgivenwhichwebelieveBconditionalonA—formally:

(V.1) t(C*)>0,i. e.Bel(C*),whereC*= {C ∈ P | t(B | A ∩ C)> 0}.

Iwillcall(V.1)the“circumstances are such that” readingofconditionals,becauseitexpressesthebeliefthatcircumstancesC aresuchthattheconditionalbeliefinBgivenAcanbemaintained.

Whatistherelationbetween(I.1)and(V.1)?Nottrivial,butquiteclose.Thisiswhyonemaygeteasilyconfusedaboutwhatpreciselyisexpressed.First,wemightwonderwhetherBisbelievedgivenAandC*,ifBisbelievedgivenA andC foreachcellC ⊆ C*ofthepartitionP.Theanswerisyes:

(14) ForC*asdefinedin(V.1)wehavet(B | A ∩ C*)>0.

(Thisistheorem14.14ofSpohn2012.)Intuitively,(14)saysthatwhatisconditionallybelievedgiveneachdisjunctC ⊆ C*isso,too,giventheir disjunction C*. Then it is also unconditionally believed, onemightthink,whenthatconditionC*isitselfbelieved,asstatedin(V.1).This,however,wouldbeafallacy;thisfurtherconclusionholdsonlyundercertainassumptions:

(15) ForC*asdefinedin(V.1),ift(B | A ∩ C*)>0,thent(B | A)> 0ifft(C* | A)>0.Moreover,t(C* | A)>0isentailedbyt(C*)>0andt(A)≥0.Hence,t(A)≥0issufficient,andt(C* | A)>0isnecessaryandsufficient,forinferring(I.1)from(V.1).

(Thisispartoftheorem14.81ofSpohn2012.)So,perhaps,theRamseytest (I.1) does not fully capture, but is only entailed by what is ex-pressed;thisisso,ifitis(V.1)thatisexpressed.Ofcourse,thealterna-tiveadditionalpremises,t(A)≥0ort(C* | A)>0,mayormaynotbe

representsacompletepossiblepsychologicalconditionofCaesar,oratechnologicalpartition,etc.Thequestionthenis:whichcellofthepartitionisthetrueone?Theremustbeexactlyonethatistrue.Butaninformativeanswerneednotidentifythetruecell;itneedonlysaythatthetruecellliesinsomesubsetofthepartition—forinstance,inoneofthemany“compromising”cellsoftheveryfine-grainedpsycho-logicalpartition.

Whatdo(12)and(13)assert?Clearly,theygiveananswertotheirrespectivequestion.(12)says:Caesarisakindofpersonsuchthathewouldusetheatomicbomb,ifincommand.And(13)says:thekindoftechnologyavailabletoCaesarwassuchthathewouldusecatapults,ifincommand.Andbyuttering(12)and(13),Iexpressthecorrespond-ingbeliefs.

Abitmoreformally:letj=“Caesarisincommand”representingthepropositionA andy=“Caesarusestheatomicbomb”representingthepropositionB;so,(12)isabbreviatedas“j > y.”With(12),IclaimthatCaesarisofacertainpsychologicalcharacteristicC*.So,IbelieveC*, i. e.,t(C*)>0.Somehow,C*isdeterminedwiththehelpof“j > y”.Buthow?WejustsaidthatC*isthecharacteristicsuchthatgivenitandCaesar’sbeingincommand,hewouldusetheatomicbomb,i. e.,suchthatt(B | A ∩ C*)>0.IsC*therebyuniquelydetermined?No.Wehavetobeabitmorecareful.WhatIreallyexpressismybeliefthatCaesarbelongstooneofthosecellsCofthepsychologicalpartitionforwhichIbelieveBgivenA ∩ C.Thatis,IbelieveinthedisjunctionC*ofthosecellsCforwhicht(B | A ∩ C)>0.Andthisdisjunctionisindeedunique.

Wecouldgo through the sameexercisewith (13) and theapper-tainingtechnologicalpartition.Thisamountstoacleardescriptionoftherelevantcontext-dependenceof(12)and(13):thecontextcreatedbytheutterancesthemselvesinthesecasesconsistsinacertainissueorpartition,andwithinthatcontextitisdeterminatewhatthecondi-tionalassertionclaimsorexpresses.

The abstract representation of the situation is this: let P be apartitionofW, i. e., a setofmutuallydisjointand jointlyexhaustive



(16) ForC*asdefinedin(VI.1b)wehavet(B | A ∩ C*)>0≥t(B | ∩ C*),i. e.,thatAisasufficientreasonforB givenC*.

And the assumption of (VI.1b) thatC* itself is believed oncemorehelpsustofurtherconclusions,inanalogyto(15):

(17) ForC*asdefinedin(VI.1b),givent(C* | )≥0andgiveneithert(A)≥0ort(C* | A)>0,(VI.1b)entails(IV.1b),i. e.,thatA is(unconditionally)asufficientreasonforB.

(Forproofs,seetheorems14.14and14.81inSpohn[2012].)Again,onemust be clear aboutwhat the additional premisesmean. But let usnolongerdwellontechnicalities.Forthemoralshouldbeobviousbynow:byallmeans,wemustcloselyobservethemanyformalrelationsbetween the various expressive options as determined by rankingtheory;withoutbeingclearabouttheserelationswenevergainclarityaboutthevariousconditionalsandtheirrelations.

Youmayhavealreadynoticedthemostinterestingfeatureof(V.1)and(VI.1);namely, that,according to them,conditionalsexpressanunconditionalbelieft(C*)>0,which is truth-evaluableandhenceei-thertrueorfalse.Thiswellconformstoourintuition.Lookat(12)and(13)again.Wemaywellhaveadisputeaboutthem,andweallthinkthatthisisafactualdispute.WhatkindofcharacterorpoliticalleaderwasCaesar?Washereallysorecklessandaggressiveas(12)claims?Whichevidencedowefind inhisbiographyandhiswritings fororagainst(12)?Andsoforth.

Thisobservationismostimportant.Insection2,Istronglyempha-sizedCBnoTC, the claim that conditional belief has no truth condi-tions,andtherebymotivatedtheexpressivisticapproach.Eveniftheargumentsweregood,theyappearedcounter-intuitive.Nowwehaveapartialexplanationforthisintuition.MaintainingCBnoTCdoesnotentailthatconditionalsdonothavetruthconditionsatall;theyfailtohavethemonlyinsofarastheyexpressonlyconditionalbelieforrel-evance,astheydoaccordingto(I)and(IV).However,theymayalsoexpressmorecomplexfeaturesdefinableintermsofconditionalbelief,

plausiblysatisfiedinagivencase.(Foranexampleoftheirviolation,seetheendofsection7.)

TheforegoingdiscussionwasguidedbytheRamseytest;thatis,itfocusedontheadditionalconditionsCunderwhichtheconditionalbeliefinBgivenAismaintained.However,thediscussionof(12)and(13)mighthavebeenevenmoreplausible in termsof positive rele-vance.Theconditional“j > y”mightaswellbeusedforexpressingthebeliefthatoneofallthosecellsinPobtainsgivenwhichAistakentobepositivelyrelevanttoB—formally:

(VI.1)t(C*)>0, i. e.,Bel(C*),whereC*= {C ∈ P | t(B | A ∩ C)>t(B | ∩ C)}.

Now, expressive options proliferate. Just as (V.1) builds on theRamseytest(I.1),onemightintroduceoptions(V.2)and(V.3)buildingon(I.2)and(I.3).Similarly,onemightdefineoptions(VI.2)and(VI.3),paralleling(IV.2)and(IV.3).Anditwouldcertainlybeworthwhiletodifferentiate(VI.1)inthesamewayaswedifferentiated(IV.1),accord-ingtothevariouskindsofreasonorpositiverelevance.Letmepickoutjustoneinstance:

(VI.1b)t(C*)>0,whereC*= {C ∈ P | t(B | A ∩ C)>0≥t(B | ∩ C)}.

According to (VI.1b), “j > y” expresses thebelief that somecircum-stancesobtainunderwhichAisasufficientreasonfor B.

Is(VI.1)relatedto(IV.1)inthesamewayas(V.1)to(I.1)?No,asser-tionsparallelto(14)and(15)donotobtain.Thereasonisthattheposi-tiverelevanceofA forB givenvariousCaccordingto(VI.1)maybeanyofthekinds(a)–(d),andthenit isunpredictablehowtheserelevan-ciesmix.(InaprobabilisticsettingthisiscalledSimpson’sparadox.)

However,ifwefocusonaspecifickindofpositiverelevance,aswedoin(VI.1b),thepicturechanges.Thenwehaveinanalogyto(14):



holds, I think, also vis à vis the variants which Edgington (1995, p.293ff.)andBennett(2003,p.85ff.)bringupinordertoperfectthesym-metrybetweenthetwocontradictoryopinions.Eitherthereissomehidden fact of thematter thatbreaks the symmetry—and thenoneofthetwoopposingindicativeconditionalsiscorrect—orthereisnosuchfactofthematter(ofwhatwillhappen,e. g.,inBennett’sexample,whenall threegatesareopen);andthennoneof the twoopposingindicativeconditionalsiscorrectunderreading(V.1).

Letmepointtosomeprecedentsof(V)and(VI)intheliterature.First, I am struckby the similarityof (V.1) to the formula (8)ofAd-ams(1975,p.131),whichissupposedtotreatcounterfactualswithinhisprobabilisticframeworkandwhichhasbeenfurtherdevelopedbySkyrms(1981);(V.1)indeedlooksjustlikearanking-theoretictransla-tion of that formula. I amnot aware thatAdams’ two factormodel,ashecallsit,hasmadeadeepimpressionontheongoingdiscussionperhapsbecausehehimselfwasnotsoconfidentofitcallingitad hoc onp. 132.A furtherpointmayhavebeen that that formuladeliversonlyadubiousexpectedassertibilityvalueforcounterfactualsandnotaprobability,asAdamshimselfnotesonp.132andasSkyrms(1981)makesmoreexplicit.Whateverthereasonsforthisneglect,theydonotapplyto(V.1),whichwasintroducedheredefinitelynotinanad hoc wayandclearlyspecifiesthebeliefsexpressedandnotsomeexpectedassertibilityvalue—thetheoreticaluseofwhichisveryobscure.Thisisafurtherpointinfavoroftheranking-theoreticversustheprobabi-listicpointofview.

TheotherprecedentIhaveinmindisthatithasbecomequitecus-tomarytospeakofthebasisofaconditional(cf.,e. g.,Edgington[1995,p.283]),whichissupposedtoconsistinthefactsthatmakethecondi-tionaltrueorassertible.Speakingofthecategoricalbaseofadisposi-tionmaybearnotonlyaterminologicalresemblance.WhenBennett(2003,ch.22)givesacentralroletotheevidenceorexplanatorybasesofconditionals,hehasagainsomethingsimilarinmind.Theseusagesof“base”willfitstillbettertothespecialcaseofcausalconditionals

somethingthatisindeedtruth-evaluable,and(V)and(VI)showhowtheymightdoso.Ishalldeepenthisobservationinsection8.

Interlude 3: Sly PeteLetmeexemplify(V)and(VI)withanothermuchdiscussedexample,theSlyPetestoryofGibbard(1981,p.231).Itisaboutapokergame.Zack saw thehandofPete’sopponent and signalled it toPete. Jackneednotknowthis,buthesawbothhandsandthusthatPete’swasthe losingone. JackandZackhave to leave the roomandcanonlyspeculateabouttheoutcome.So,Zackbelievesandsincerelyasserts“ifPetecalled,hewon,”whereasJackbelievesandsincerelyasserts“ifPetecalled,helost.”

Gibbardarguesthatbotharerightintheirwaysandthatnothingbreaksthesymmetrybetweenthem.Hence,heconcludesthatthetwoconditionalscannotbeassignedtruthvaluesandonlybeinterpretedinanepistemicallyrelativizedway,asexplainedbytheRamseytest.Manyhaveacceptedthisargument,as faras indicativeconditionalsareconcerned.Itisalsogristformymill.

However,itisnotclearthatindicativeconditionalsmustbeinter-pretedaccording to theRamsey test (I.1), (neglectingrelevancecon-siderations). The “circumstances are such that” reading (V.1) mightbeequallyormoreappropriate.WemaywellsupposethatJackandZackhavenodeepinductivedisagreementandwillagreeoncetheysharetheirinformation.AccordingtowhatLycan(2001,p.168)callstheHardLine(which iscommonlyacceptedaccording tohispolls),theywill thenagreethatJackisrightbecausehehastheoverridinginformation.However,eventhisisnotsoclear.AnothercircumstancemightbethatSlyPeteisadeftorevenperfectcheat;hewillsucceedifhecallsanddecidestocheat.Inthatcase,Zackwouldberight.

Thepointisthis:onceweengageinthiskindofdiscussion,weap-plytothoseindicativeconditionalsthe“circumstancesaresuchthat”reading(V.1),accordingtowhichitisafactualquestionoverwhoisright(unlessthereisaninductivedisagreementwhichpersistsevenafterallfactsarecommonlyknown—moreonthisbelow).Thispoint



[2000],Woodward[2003])alsoclaimstobeavariantorspecificationofthecounterfactualapproach.

Causal conditionals are very common, andmost counterfactualsaretobeinterpretedinacausalway.TheRamseytestisconsideredtobeinadequateforthem,sinceitestablishesonlyanepistemicandnotacausalrelation.Therefore,theopinionprevailsthattheyconstituteadifferenttypeofconditionalsthatrequiresadifferentaccount,say,intermsoftheStalnaker/Lewissemanticsorintermsofstructuralmod-elsorequations.Contrarytothisopinion,Iclaimthatthereisnoneedtochange the framework.Causalconditionals indeeddonot followtheRamseytest(I.1),buttheydofollowthe“circumstancesaresuchthat”reading(VI.1).Letmeexplain.

First,wemayassumethattheBooleanalgebraAoverthesetWofpossibilities isgeneratedbyasetA*ofsimple propositions, i. e., eachpropositioninA isapossiblyverycomplexBooleancombinationofpropositionsinA*.AndwemayfurtherassumethateachofthesimplepropositionsinA*referstoafixedtemporallocation.“It’sfreezinginKonstanzonMarch19,2014,”“IdonotsleepwellonMarch20,2014,”“myflightstartsat6am,March21,2014”:eachoftheserepresentsasimple proposition referring to a (possibly coarse-grained) specifictime.Complexpropositionsbycontrastneednothaveadeterminatetemporallocation.

Wemay imagine then that the possibilities inW, as far as theycanbecharacterizedbyA*,aremaximalconsistentconjunctionsofsimplepropositionsinAortheirnegations;eachpossibilityisanen-tirepossiblehistoryoftheform“firstA’1andthenA’2andthenA’3…”,whereeachA’iisasimplepropositionoritsnegation.Thosehistoriesneednotbecompletehistoriesinanyabsolutesense;theywouldbeso only if the possibilities inW were full Lewisian, or ratherWitt-gensteinian,possibleworlds.Hence,thehistoriesinWaremoreorless fine-grained depending on the richness ofA*. These informaldescriptionsaregoodenoughforourpresentpurposes.Ofcourse,aformaltreatmentwouldhavetobefullyexplicitaboutthesealgebraic

discussed in thenextsection.Finally,whenLewissays thatcounter-factuals,andthusthesimilarityorderingonwhichtheyground,super-veneonthecharacterofactualworld(cf.Lewis[1986,p.22]),hereferstosuchabasisonagranderscaleinamoremetaphysicalmoodandmuchmoredebatableway.

However,at the threeplacescited, thisbasis remainsratherdimandwithoutfurthertheoreticaltreatment;Iamnotawarethatithasbeenelaboratedelsewhere.(V)–(VI)offersubjectivecounterpartsofthatbasis,namelythe(context-orpartition-relative)propositionC*of(V)and(VI).Thispropositionmaywellbecalledthebasisonwhichtherelevantconditionalbelief isheld,and itsexplicationopens theway toa rigorousanddetailedstudyof thebasisof conditionals sounderstood.

7. Causal Conditionals

(V)and(VI)areimportantgeneralschemes,referringtoasomehowcontextuallygivenpartitionP.Stillmore important isaspecialcase.ThenextbigclaimIwanttodefendinthispaperisthatallcausal con-ditionalsinstantiatescheme(VI)inaspecificcontext-independentway.

Causal conditionalsare thoseconditionalswhichare interpretedassomehowrepresentingcausalrelations.However,thiswayoftalk-ingmaypresupposethatcausalrelationsaresomethingthatobtainsobjectively. Within our expressivistic approach, we better say thatcausal conditionals are those expressing causal beliefs, i. e., beliefsabout causal relations. Causal conditionals are strongly correlatedwiththesubjunctivemoodandthecounterfactualphrasing(althoughthese correlations are never completely reliable). Indeed, themostprominenttheoryofdeterministiccausationisthecounterfactualone,whichstartedoutbydefiningcausalrelationsintermsofcounterfac-tuals(cf.Lewis[1973b]).Mattershavebecomemoreandmorecom-plicated,duetonumerouscausalpuzzleslikeoverdeterminationandvariouskindsofpreemption(cf.Collinsetal.[2004]).Thenowadaysevenmorefashionableinterventionistaccountofcausation(cf.Pearl



mustbeunderstoodinanon-backtrackingway—thatis,asnotaffect-ing the past, as leaving the past unchanged and thus as letting theactualhistorybefixedandgiven.Theverysamepointisemphasizedintheinterventionistapproach:theideaofaninterventionispreciselytokeephistoryfixedandmiraculously,asitwere,towiggleonlywiththecause.So,conditioningonthehistoryisaverycommonidea.

Whatisunusualaboutmyshort-circuit is itsappealtotheepiste-mologicalnotionofa(conditional)reason,thusturningcausationintosomething relative toourepistemicstate.However,even thismovehasgoodprecedent.It’sHume’smove,andithasbeenoneofthemostbewilderingmovesintheentirehistoryofphilosophy.Ontheotherhand,Icannotfindthat ithasbeenconvincinglyrefuted; itremainsathorninthefleshofphilosophy.MyreasonforfollowingHumeissimpleandpowerful:namely,thatallobjectivistconceptionsofcausa-tionhavebeenunable tocomeupwithanadequatenotionofposi-tiverelevance.Allinall,allthosecausalpuzzles(overdetermination,preemptionbytrumping,etc.)canbebestsolvedwiththeepistemicnotionofpositive relevance (see inSpohn[2012, ch. 14]).Asupple-mentary reason is that the epistemic relativizationof causation canbeundonetoa largeextent;weneednot forswearourobjectivisticintuitions(seesection8below).

Weshouldnotfurtherdigressintothephilosophyofcausation;letmereturntomyexplanationofcausalconditionals.Sofar,IsaidthatAt isacauseof Bt’ iff,giventheactualhistory H, At ispositivelyrelevanttoBt’ (accordingtotheepistemicstatekort).Butwehavetobeabitmoreprecise.WhatistheactualhistoryH?AsIhavearguedseveraltimes(cf.Spohn[2012,sec.14.4]),weshouldfocusonAtbeingadirect causeofBt’ andthentaketheentirehistoryuptotheeffectatt’exceptthecauseattastherelevanthistoryH. Moreover,weshouldnotmere-lyrefertotheactualhistory;weshouldmakeexplicit thattherearemanypossiblehistoriesandthateachpossibilityorworldhasitsownhistory(uptot’)anditsowncausalrelations.So,let denotethehistoryorthecourseofeventsinthepossibilitywuptotime t’ withtheexceptionoft. Ofcourse,howrich isdependsontherichness

matters,andthenissuesofgranularitywouldloomlarge(cf.Spohn[2012,sec14.9]).

Next,wemay observe that the antecedent and the consequentofaconditionaloftenrefertosimplepropositions.Thegeneralrea-sonmay be logical simplicity, but in the case of causal condition-als the reason is that antecedent and consequent refer to singularcauseandeffect,whichhavetohaveafixedtemporallocation.So,acausalconditionalhastheform“j > y,” wherej nowrepresentsAt referringtot,andy representsBt’ referring to t’; forinstance,ifIhaddroppedtheglass(rightnow),itwouldhavebroken(asecondlater).Wemayassumethatt’ islaterthantandneglectherephilosophicalproblemsaboutwhetheracausemightbesimultaneouswithorevenlaterthanitseffect.

Now,whatdoesitmeanthatAtisacauseofBt ’?Thereisnottheslightesthopeofadequatelydealingherewiththisissue.(SeeSpohn[2012,ch.14]foracomprehensiveexpositionanddefenseofmyview.)LetmeonlybrieflysketchtheanswerIendorseformorethan30years:Atisacause ofBt ’ifAtispositivelyrelevanttoBt ’insomesense,orifAt

makesacontributiontoBt ’;thatis,ifwithinthegivencourseofeventsoronthebasisof theactualhistoryH AtwassomehowrequiredtobringaboutBt ’.Thus,Atisacause of Bt’ ifgiventheactualhistory H At

ispositivelyrelevantto Bt ’, i. e., if t(Bt ’ | At ∩ H) > t(Bt ’ | t ∩ H).Thus,veryroughly,causesarereasonsgiventheactualhistory.The

topic“reasonsandcauses”isanimportantoneinepistemology,andformanycenturiesithasproducedprofoundconfusionsundervary-inglabels.WhenIamshort-circuitingthisissuehere,thismaybetak-enasilluminatingorascontinuingconfusion.Inanycase,myshort-circuithasgoodprecedent.Tobeginwith,conditioningonthehistoryHhasbeenfirstexplicitlyproposedbyGood(1961–3)withinstatisticalattemptsatcausation,anditiswidelyaccepted,e. g.,ineconometrics(seeGranger1969). Theideaisalsopresentinthecounterfactualap-proach.Inaway,thecrucialissueaboutcounterfactualsiswhatisco-tenable—tobekeptfixed—withthecounterfactualassumption.Andthenitisalwayssaidthat,whenthecounterfactualisacausalone,it



(VII.1)t(H*)>0,whereH*={w ∈ W | t(Bt’ | At ∩ )>0}, or

(VIII.1)t(H*)>0,whereH*={w ∈ W | t(Bt’ | At ∩ )>t(Bt’ | t ∩ )}.

Letmecall(VII)and(VIII)the“history is such that” readingofcon-ditionals. It isclear that (VII)and(VIII) ramify in thesameway(V)and(VI).Itisalsoclearthatmyexplanationsconcerningcausalcon-ditionalsarecapturedby(VIII.1)thatfocusesonconditionalpositiverelevance;itsramificationswouldincludeconditionalsexpressingsuf-ficientand/ornecessarycausation.Perhaps,though,wedonotwanttoincludethisandonlyexpressthebeliefthathistoryissuchthatBt’ mustobtaingivenAt;andthenoption(VII.1)ispertinent.Finally,itisclearthat,iftheconditional“j > y,”istoexpress(VIII.1),thenitistobereadas “y,becausej.”Wemayalso take(VIII.1)asexpressingaconditionaloftheform“non-j >non-y”;then,indeed,itisthecoun-terfactual“ifjhadnotbeenthecase,ywouldnothavebeenthecase.”

This concludesmy list of expressive options for conditionals. ItgoesfarbeyondtheRamseytestand,asmymanyexampleshavedis-played,mostofthoseoptionsarerequiredforaccountingfortherichlinguisticphenomena.

Interlude 4: The Oswald/Kennedy CaseLetmedemonstratethepowerofmyapproachwiththefamouspairintroducedbyAdams(1970):

(19) IfOswalddidn’tkillKennedy,someoneelsedid.

(20)IfOswaldhadn’tkilledKennedy,nooneelsewouldhave(andKennedywouldhavebeenalive,forawhileatleast).

LetO=“OswaldkilledKennedy”andS=“someoneelsekilledKenne-dy.”Forthesakeofsimplicity,letOandSstandbothforthesentencesandthepropositionsrepresentedbythem.Then(19)maybeabbrevi-atedas >1 and(20)as >2 .Thethrustofthepairisobvious

ofthesetA* ofsimplepropositionsoriginallyassumed.So,myfinalexplicationforthepresentpurposesisthis:

(18) At is a direct cause of Bt’ in the possible worldw (relative to the ranking function t) iffAt ispositively relevant to Bt’ given , i. e., ifft(Bt’ |

At ∩ )>t(Bt’ | t ∩ ).

Extendingthistoanaccountof indirectcausationisamostdelicateissue(cf.Spohn[2012,sec.14.11–13]),whichneednotconcernushere.

Now,atlonglast,wearepreparedtoaddresscausalconditionals.WhenIutterthecounterfactual“ifjhadnotbeenthecase,ywouldnothavebeenthecase,”withjandyasaboveandintendingacausalinterpretation,IexpressmybeliefthatAtisa(direct)causeofBt ’.Howmight thisbeliefbeunderstood inviewof (18)?Well, it’s thebeliefthattheworldorhistoryissuchthatAtisa(direct)causeofBt ’;it’sthebeliefinthetruthconditionof“Atisa(direct)causeBt ’,”inthesetofworldswinwhichthedefiniendumordefiniensof(18)issatisfied.

Takeanexample: “If ithadnot rained,hewouldnothavecomelate.”Asacausalconditionalthissaysthesameas:“Hecamelate,be-causeitrained.”Thereareplentyofpossiblehistoriesinwhichthisisnottrue,plentyofotherpossiblecausesthatcouldhavedelayedhimwithouttherainplayinganyrole.Inassertingoneofthetwosentenc-es,Iexpressmybeliefthatnoneofthesealternativehistoriesobtains.

Now, we can finally see how causal conditionals fall under theschemes(V)and(VI).Eachschematicconditionaloftheform“j > y,”againwithjandyasabove,canbeunderstoodwithoutanycontex-tualcluesasreferringtothesetorpartition ofpossiblehistories

uptot’withtheexceptionoft;thequestionunderdiscussionis—so to speak—howwas history?And thenwe can take the con-ditional“j > y”asexpressing(V.1)or(VI.1)referringtothatspecificpartition,e. g.:



Myfurthersuggestionwasthatcounterfactualsareinterpretedac-cordingtoscheme(VII.1)(againneglectingrelevanceconsiderations).Ifweapplythisto(20),or>2,(20)saysthathistory(=H*)issuchthat,if andH*,then .So,H*agreeswiththeWarrenreportconfirmingthatOswaldwasasingleassassin,and wouldbeadisjunctionofalternativehistorieswithotherormultipleassassinsincluding,e. g.,aconspiracy.Thus,(20)expresses:

(23)t(H*)>0andt( | ∩ H*)>0.

Alltheseconditionalandunconditionalbeliefs(21)–(23)perfectlygotogetherinoneconsistentdoxasticstate—infactours—andthusarewellexpressedinoneandthesamecontextby(19)and(20).Forinstance,thenegativerankingfunctionmightbethis:

Sincethemiddleleftupperboxcontainstheonly0, kbelievesO, , andH*;thisaccountsfor(21)andonehalfof(23).Moreover,wehavek( )=min{6, 4, 3, 4}=3, k( ∩ )=4,andthusk( | )=1;i. e.,S isbelievedgiven ,asrequiredby(22).Finally,thetwoupperrightboxessaythatk(S | ∩ H*)=6 – 4 = 2;i. e.,Sisdisbelieved,andbelieved,underthiscondition,asytheotherhalfof(23)requires.

Isn’tthereamistake?Accordingto(15),(23)seemstoentailt( | )>0andthustocontradict(22).However,noneofthealternative

additionalpremisesin(15)neededforthisinferenceholds.t( )≥0 woulddoasadditionalpremise,butitisdeniedby(21);wedobelievethatOswaldkilledKennedy.t(H* | )>0woulddoaswell.However,theabovefiguresentailthatk(H* | )=k( ∩ H*)–k( )=4 – 3=1.GiventhatOswalddidnotkillKennedy,wedonotsticktoourhistori-calbeliefH*;historymusthavebeendifferenttheninoneoftheways

andpowerful:both(19)and(20)aretrue,oratleastclearlyacceptable.Hence,>1 and>2mustbetwodifferentkindsofconditionals.Sincethen, it seemed that theories of conditionals have to bifurcatewithregardtowhatisusuallyclassifiedasindicativeandsubjunctivecon-ditionals—amostdramaticeffectofthepair(19)and(20).

Stalnaker(1975)triedtopreserveunitybyproposingthat(19)and(20)involveacontextshiftsothat>1and>2aretobeinterpretedintwodifferentcontexts.(20)canbeutteredonlyinacontextwhereOisassumedtobetrue,whereas(19)makessenseonlyinacontextwhereOistreatedasopen.AndthenStalnakergoesontoexplainhoweachconditionalmaybeacceptableinitsowncontext.However,Idonotfindtheclaimabout(19)convincing.It’sperfectlyacceptabletosay:“IdobelievethatOswaldkilledKennedy;wealldo.Butifitwasn’thim,someoneelsemusthavekilledKennedy.” Idon’tseehereacontextshiftinitiatedby“but.”Whyshouldoneassumehereapretenseofbe-ingopentowardstheantecedent,ifthatopennesshasbeenexplicitlydenied?Indicativeconditionalsdonotseemtobeboundtobeopenconditionals.(ThispointissharedbyWoods[1997,p.54].)

Hereisaperfectlystraightforwardaccountofthisexamplewithinmyframework,whichneitherinvolvescontextshiftnordifferenttheo-riesofconditionals,butonlyonerankingfunctionandtwoexpressiveoptions.First,itisclearthatwebelieveOandrejectS.Hence,ourtwo-sidedrankingfunctiontissuchthat:

(21) t(O)>0andt( )>0.

Let’skeepthingssimpleanddisregardrelevanceconsiderations.Then(19),or>1,ismostplausiblyinterpretedaccordingtotheRamseytest(I.1)andthusexpresses:

(22)t(S | )>0.

It doesnotmerely express, though, a belief in thematerial implica-tion → S.Thisissobecausetheantecedentisbelievedtobefalseaccordingto(21),andhencetheconditionfortheequivalencein(11)isnotsatisfied.



vastlyunderdeterminestheentireinductiveordynamicbehaviorbuiltintoarankingfunction.

Bycontrast,thefactthatapropositionreceivesaspecificpositiveornegativetwo-sidedrankisnotobjectivizableintherequiredsense.Of course, itmaybe anobjective fact about thedoxastic state of agivensubjectatagiventime;butnothingcorrespondstothatfactinreality.Thepropositionmaybetrueorfalseinsofarasitspositiveornegative rankmaybe correct.However, thenumerical valueof therankrepresentsthestrengthof(dis)beliefandcannotbecalledtrueorfalseassuch.(Asiswellknown,thisobservationequallyappliestosubjectiveprobabilities.)

Our interest focuses on conditional belief. Indeed, CBnoTC, thethesis thatconditionalbeliefhasnotruthconditions,wascentral tothispaper.Insection2,IhavereferredtothetrivializationtheoremsofLewis(1976)andGärdenfors(1986)forprima faciejustification(though,asmentioned, thedialecticsituation is tricky).AndIhave indicatedtherethatthethesisisconfirmedbythisobjectivizationtheory.Condi-tionalbeliefturnsouttobeobjectivizableonlytoaverylimitedextent,i. e., only for ranking functionswith a devastatingly poor and unac-ceptable inductive behavior.As to the relationof being a sufficientreason,mattersareevenworse;iftherearemorethantwopossibili-tiesinthepossibilityspace,itisnotobjectivizablewithrespecttoanyrankingfunctionnotstrictlyidenticalto0(cf.Spohn[2012,theorem15.5]).Thisnegativeresultextends,ofcourse,topositiverelevanceorbeingareasoningeneral.

Theseobservationsalreadyclearupourissuefortheexpressiveop-tions(I)–(IV).Insofarastheschematicconditional“j > y”expresses(dis)beliefinthepropositionsrepresentedbytheantecedentjandtheconsequenty according to (II)and (III), thisbelief isobjectivizableandtheconditionaltrueorfalse.Thisisnotveryinteresting,ofcourse;wewouldratherspeakofwhetherornotthepresuppositionsorimpli-caturesoftheconditionalaresatisfied.(ButIhadmadeclearthatweneed not distinguish between assertion, presuppositions and impli-caturesaslongasonlytheexpressionofepistemicstatesisatissue.)

containedin ,becausewerathersticktothebeliefthatKennedywasindeedkilled.

8. Truth

Letusfinallyturntothetruth-evaluabilityofconditionalsintheem-phatic sense. This issue has caused the deepest divide. There arestrong intuitions and good arguments on either side. In section 6,whenintroducingthe“circumstancesaresuchthat”readings(V)and(VI),Iindicatedthatthisreadinghasthepotentialofbridgingthedeepdivide.So,howisthisreconciliationsupposedtogo?

Thebackgroundofmyreconciliationisthatrankingtheorycomesalongwith a rigorous and sophisticated objectivization theory. Thecoreideaofthistheoryistostudytheextenttowhichaspectsorfea-turesofrankingfunctionsuniquelycorrespondtotruthconditionsorpropositions(Iwillgiveexamplesofwhatismeantbythis)andmaythusbecalledtrueorfalse(intheemphaticsense).Thisispreciselywhat is required here. I argued that conditionals may express (I)–--(VIII),whicharefeaturesofourrankingfunctions.(It isafeatureofarankingfunctiontosatisfy(I.1),(I.2),etc.)Ifthosefeaturesweretouniquelycorrespondtopropositions,thecorrespondingconditionalswoulddoaswellandwouldthusbetruth-evaluable.

Oneimportantpointisthatobjectivizationisnotanall-or-nothingaffair,buthasasurprisinglydifferentiatedanswer,whichextendstotruthconditionsofconditionalsinthesamedifferentiatedway.Thisseemspreciselywhatwewant.Anotherpointisthatthisobjectiviza-tiontheorybecomesquiteinvolved.HereIcanonlymentionthebasicresultsweneed;foralldetailsIhavetorefertoSpohn(2012,ch.15).

Afirstpositive,thoughtrivialresultisthis:thefeatureofanegativerankingfunctionthatconsistsinhavingcertain(dis)beliefsisobjectiv-izable:beliefsuniquelycorrespondtopropositionsbelieved;proposi-tionsare(objectively)trueorfalse,andsoarebeliefs.Tothatextent,arankingfunctioncanbecalledtrueorfalseaswell.However,havingbeliefsissimplyamatterofnegativerank0or≠0,andsothisfeature



conditionalsisapeculiarlymixedaffairaccordingto(V)and(VI).Thispeculiaritydoesnotseemtohavereceivedproperrecognitionintheliteratureonconditionals.

The problem can only be avoidedwhenwe presuppose a fixedranking function,orat leastfixedconditionalbeliefs,onwhichcon-ditionalassertionsarebased.Relativetothem,conditionalsaretruth-evaluable; and ifwehave thoseconditionalbeliefs in common,dis-puteaboutconditionalsispurelyfactualdispute.However,nogeneralstatementispossibleabouttheextenttowhichthiscommonalitycanbepresupposed.

This treatment resembles thenotionof objectiveprobability putforwardbyJeffrey(1965,sec.12.7).Hesaysthatobjectiveprobabilityisjustsubjectiveprobabilityconditionalonthetruecellofarelevantpartition.Thisisstillamixtureofobjectivityandsubjectivity,asJeffreywaswellaware.However,insofarasconditionalsubjectiveprobabili-ties agree,disputes aboutobjectiveprobabilities aredisputes aboutthetruecelloftherelevantpartition — justasinthecaseof(V)and(VI).

Thetruth issuegetsstillmore involvedandinterestingwhenwelookatthespecialinterpretations(VII)and(VIII).However,Ishouldfirstaddressaconcernaboutmyaccountofthetruth-evaluabilityof(V)and(VI),whichwillapplyto(VII)and(VIII)aswell.Theconcernisthis:

WhenIsaidthatunderoptions(V)and(VI)aconditionalexpress-esmybelief that“circumstancesaresuchthat…,”thismayhaveap-pearedacceptable.However,nowIsaythatthisisthetruthconditionofaconditionalunder(V)and(VI).Andthismaysoundodd;thatcir-cumstancesarerightdoesnotseemtobethecontentoftheconditional,orwhattheconditionalasserts.Inotherwords,whenweargueaboutaconditionalanditscircumstancesinthewayindicated,weseemtoex-change reasons for taking the conditional to be trueandnottomakeclaimsaboutwhattheconditionalasserts.Isthereasubtleconfusion?

Idon’tthinkso.First,notethatthereisasubjectivecorrelatetothistalkofreasonsforaconditional.Wemightextendournotionofareasoninsection5andsaythatCisareasonfortheconditionalbeliefinBgivenA

Bycontrast,insofarastheconditional“j > y”isunderstoodaccordingtotheRamseytest(I)andsimplyexpressesconditionalbelief,itcangenerallynotbecalledtrueorfalse.A fortiori,thisassertionextendstothecasewheretheconditional“j > y”isusedtoexpressanykindofrelevanceassessmentaccordingto(IV).

However,thisisnottosaythatthecaseisentirelylost.Thebasicthesishereis thatweusetheconditionaltoexpresssomething about ourconditionalsbeliefs,and thismaystillbeobjectivizable,even ifconditionalbeliefby itself isnot.This is indeed thecase for theop-tions(V)–(VIII),andindeedtoalargerextentthanimmediatelymeetstheeye.

Let us first look at the general cases (V) and (VI). According tothem,theconditional“j > y”simplyexpressesabelief,abeliefaboutthecircumstancesunderwhichtheconditionalbeliefinBgivenA canbemaintained.Thisbeliefcanbetrueorfalse,andsotheconditionalcanbeaswell.Thisexplainswhywecanhavefactualdisputesaboutconditionals thusunderstood; theyaredisputesabout thosecircum-stances,asdisplayedinthevariousexamplesabove.

However, the objectivity of those conditionals is partial andnotstraightforward.ItfirstpresupposesagreementaboutwhichissueorpartitionP is thecontextuallygiven focusofargument. If this isun-clear, it isunclearwhichbelief isexpressedaccordingto(V)or(VI).More importantly,even if thepartition is clear, thebeliefexpressedaccordingto(V)or(VI)secondlydependsonone’srankingfunction,morespecifically,onparticularconditionalbeliefsorrelevanceassess-mentscontainedinit,whichare,asInotedabove,notgenerallyob-jectivizable.Thoseconditionalbeliefsareindirectlyexpressedaswell.Tothisextent,conditionalsinterpretedaccordingto(V)or(VI)arenotobjectivizable.

This fact is also reflected inourdisputes.Wemay think thatwehaveafactualdisputeandthendiscoverwearetalkingatcross-pur-posesbecausewejudgetheconditionalathandrelativetodivergingconditional beliefs. That divergence ismuch harder to discuss andmaynotbeobjectivelydecidable.Therefore,thetruth-evaluabilityof



Thus,accordingtothelatterthreeways,thetruthconditionof“j > y”isnotsimplyafixedproposition,asinthefirstway.Rather,thecon-ditional“j > y”assertsthattheworldissuch,andtheepistemicstateshouldbesuch,thatthe conditional belief in B given A can be maintained in that world.Accordingtotheseways,then,thebeliefsexpressedbytheconditional “j > y”underoption(V.1)mightbemoreappropriatelyconceivedasreasonsfortheconditional,i. e.,formaintainingtherel-evantconditionalbelief,orasthebaseofaconditionalinthesenseofEdgington(1995,p.283)andBennett(2003,ch.22).

However, I cannot see any substantial difference between thosealternatives;theyarejustvariationsoftheinitial,triplyrelativedefini-tionof thetruthofaconditional.Theimportantpoint iswhat isob-jectiveandwhatissubjectiveinthoseshallowtruthconditions.Thefactualworldwisobjective,andtherankingfunctiont(andtheparti-tionP)issubjective.Andthisholdsforallfourrepresentationsoftruthconditions.Inanycase,onemustnotassumethattheworldwsome-howdeterminestherankingfunctiontappropriatetoitoreventhetruerankingfunction(asLewis[1986,p.22]doeswithrespecttothesimilarityspheres,whenheclaimsthemtosuperveneonthecharacteroftheactualworld).Thisideaissofarwithoutanyfoundation.

Itwillreceivepartialfoundation,though,whenwefinallylookatthe special interpretations (VII) and (VIII) of the temporally loadedconditional“j > y,”wherejrepresentsAtandyrepresentsBt ’.Theaboveobservationsaboutthegeneralcases(V)and(VI)applytothesespecial cases aswell.However, the situation improves further.OnepointisthattheuncertainreferencetothecontextuallygivenpartitionPinthegeneralcaseisreplacedbyatextuallygivenpartition ofhistoriesinthespecialcase.

Themoreimportantpointconcernsthespecialformofthecondi-tionalbeliefsorconditionalranksreferredtoin(VII)and(VIII).Theircondition consists in a full possible history up to t’ includingAt or

t;andthepropositionBt ’conditionallyrankedisaboutt’.So,wemaycall thempast-to-present conditionalbeliefsor ranks(withavariable

(relativetoarankingfunctiont)ifft(B | A ∩ C)>t(B | A ∩ ).Thereby,wecanrepresentwhatIsofardescribedasadisputeaboutthetruthofaconditionalunderoption(V.1)asanexchangeofreasonsfortherelevantconditionalbelief.However,thisisjustaredescriptionofthesamematter.

Still,arethetruereasonspartofwhattheconditionalasserts?Wecanputthingseitherway.Letusstillrefertooption(V.1),andstillas-sumethatjandy,respectively,representpropositionsAandB.Thenwemightsaythattheconditional“j > y”istrueinworldwrelativetothepartitionPandwithrespecttotherankingfunctiontiffwehavet(B | A ∩ C)>0fortheuniquecellC ∈ Pforwhichw ∈ C, i. e.,whichistrueinw.Somuchisclear.Notethattruthof“j > y”isthustriplyrelative,tow, P,andt.Thisisawayofassigningtruthintheshallowsensereferredtoinsection2,whichItaketobewellcompatiblewiththeexpressivisticstrategyadoptedhere.However,whatshouldwesaynowisthetruthconditionof“j > y”?Therearefouroptions:

First,wemightsay,asIdidsofar,thatthetruthconditionof“j > y”isthesetofallworldswinwhich“j > y”istrue,keepingtfixed(andneglectingtherelativitytoP).Stillkeepingtfixed,wemightsecondlysay that the truthconditionof “j > y”varieswith theworldw andsimplyconsistsinthecellCofPthatistrueinw,provided“j > y”isatalltrueinw.Wemightthirdlyincludetinthetruthconditionof“j > y”,whichthenconsistsofallpairs⟨w, t⟩ in which “j > y”istrue.Orwemight,fourthly,combinethesecondandthethirdpossibility.

Eitherwayisfine.Accordingtothefirstway,thetruthconditionofaconditionalisindeedthatthecircumstancesareright;thisishowIhavedescribedthemabove.Accordingtothesecondway,thefactualtruthconditionmaychange (because itdependson the truecellC)andtheconditionalbelief iswhat iskeptconstantacrossthechang-ingtruthconditions.Accordingtothethirdway,thetruthconditionisnotmerelyfactualandincludestheappropriateconditionalbeliefs,onwhichthefactualpartofthetruthconditiondepends.Andthefourthwaycombinesthetwodependencies.



abroadlyfactualdisagreement(thoughthebeliefsaboutthepastalsoinvolvelotsofinductiveinferences).If,ontheotherhand,wedisagreeinourinductivestrategies,thiscannotbeafactualdisagreementsince,as stated,noargumentabout thehistoricpart can changeanythingabout our inductive strategies. A dispute about the latter must besettled in a differentway, and itmaynot be resolvable at all, as inourdiscussionswith thoseguysstubbornlymaintainingGoodman’soddgrue-hypothesisconcerningafutureemerald.Reversely,thismayraisethehopethatweagree inourmorestable inductivestrategies,confiningdivergencetoourhistoricparts,whereitisquitecommon.

Ifthisisso,schemes(VII)and(VIII)makealotofintersubjectivesense.Accordingtothe“historyissuchthat”readingtheconditionalbeliefsindirectlyexpressedarepartoftheinductivestrategyembod-iedinarankingfunction.Wemaythenreasonablyhopetoshareourinductivestrategies,even ifwedonotshare(conditional)beliefs ingeneral. If so, conditionals fallingunder (VII) and (VIII)maybeun-derstoodasreferringtoafixedcommoninductivestrategy,relativetowhichtheyarefullytruth-evaluable.Hence,whatmaybeadubitablepresuppositioninthegeneralcaseisperhapsaplausibleassumptioninthisspecialcase.

Thefinalcrucialstepisthatthenegativeresultsconcerningtheob-jectivizabilityofconditionalbeliefsingeneraldonotcarryovertothespecialconditionalbeliefscontainedinaninductivestrategy.Theis-sueiscomplex,butthenetresultisthatundercertainconditions—inSpohn(2012,sec.15.4–5),Idescribetwodifferentsetsofsuchcondi-tions—inductivestrategiesareobjectivizable, i. e.,dohaveobjective,emphatictruth-valuesafterall.

Letmedescribethispointinslightlydifferentterms.Intheprevi-oussection,Ihaveexplainedonthebasisofmyexplication(18)ofdi-rectcausationhowscheme(VIII.1)isappropriateforcausalcondition-als.Thisapproach,however,seemedtobestuckwithanepistemicallyrelativized notion of causation, something intuition revolts against.Therefore, I was keen on obtaining positive objectivization resultsconcerningdirectcausation(orwhatcomestothesame,concerning

presentt’).Variouspointsareremarkableaboutthosepast-to-presentconditionalbeliefs:

First,wemightsaythattheyembodyourinductivestrategy,whattoexpectnext ifhistoryhadbeensuchandsuch.Slightlymoreex-plicitly, define the inductive strategy as from t* (relative to a givenrankingfunction)asthesetofallpast-to-presentconditionalranksforall t’≥ t*.This inductivestrategydoesnotrepresent theactualpredictions,whichsuffer,ofcourse,fromincompleteknowledgeofhistoryuptot*;andouractualpredictionsmaywelldivergebecauseourhistoricbeliefsdiverge.Still,togetherwithsuchhistoricbeliefs,or rather adoxastic stateor a ranking function concerninghistory,theinductivestrategydeterminestheactualpredictionsorexpecta-tions.Moreprecisely,thehistoric partofarankingfunctionaboutthehistoryuptot*andtheinductivestrategyasfromt* togetherdeter-minethefullrankingfunctionconcerningpast,present,andfuture.(This is adirect consequenceof an iteratedapplicationof (3), justas the probabilistic counterpartwould result from the generalizedmultiplicationtheorem.)

Moreover, the past-to-present conditional ranks have a peculiarstability.Thatis,noinformationaboutthehistoryuptot*canchangeanythingabouttheinductivestrategyasfromt*.So,thesetwopartsplaytwoverydifferentepistemologicalroles.Welearnabouthistoryallthetime;allourexperiencesareabouthistory.Bycontrast,ourin-ductivestrategiesforthefutureare,inaway,experientiallystable.Thisisnottosaythatwecannotchangeourinductivestrategiesatall;oneshouldthinkthatourinductivestrategiesareabletochangeorlearnaswell.However,howtoaccountforthisabilityisadifficultissuenottobeaddressedhere;inmyview,thisabilitycanbeaccountedforonlyinanindirectwayasasecond-orderaffair(inSpohn[2012,sec.12.5],whenaddressingtheranking-theoreticconfirmationanddisconfirma-tionoflaws,Isaymoreaboutthisissue).

Thus,thepointofthisdecompositionisthatwemayagreeordis-agreeabouteitherpart,andthatthesedisagreementsareofanentirelydifferentnature.Ifwedisagreeinourhistoricparts,thismaybecalled



inamorestraightforwardandembraciveway than ishitherto tobefoundintheliterature.Ifsomuchwereacknowledged,wewouldhavegoodreasontoelaborateonallthemissingdetails.

References

Adams, Ernest W. (1965), “The Logic of Conditionals.” Inquiry 8,166–197.

—. (1970), “Subjunctive and Indicative Conditionals.” Foundations of Language6,89–94.

—.(1975),The Logic of Conditionals,Reidel:Dordrecht.Alchourrón,CarlosE.,Gärdenfors,Peter,andMakinson,David.(1985),

“OntheLogicofTheoryChange:PartialMeetFunctions forCon-tractionandRevision.”Journal of Symbolic Logic50,510–530.

Anderson,AlanR.(1951),“ANoteonSubjunctiveandCounterfactualConditionals.”Analysis 12,35–38.—andBelnap,NuelD.(1975),En-tailment. The Logic of Relevance and Necessity.Princeton:UniversityPress.

Bamber,Donald.(2000),“EntailmentwithNearSuretyofScaledAs-sertions of High Conditional Probability.” Journal of Philosophical Logic 29,1–74.

Bennett,Jonathan.(2003),A Philosophical Guide to Conditionals.Oxford:OxfordUniversityPress.

Burge,Tyler. (1979) “Individualismand theMental.”Midwest Studies in Philosophy, Vol. IV: Metaphysics.P.A.French,T.E.UehlingJr.,andH.K.Wettstein(eds.)Minneapolis:UniversityofMinnesotaPress,pp.73–121.

Carnap,Rudolf.(1971),“ABasicSystemofInductiveLogic,PartI.”Stud-ies in Inductive Logic and Probability, Vol.I.R.CarnapandR.C.Jeffrey(eds.)UniversityofCaliforniaPress,Berkeley,pp.33–165.

Collins, John,Hall,Ned,andPaul,LaurieA.(eds.)(2004),Causation and Counterfactuals,Cambridge,Mass.:MITPress.

Douven,IgorandVerbrugge,Sara(2010),“TheAdamsFamily”.Cogni-tion117,302–318.

inductivestrategies in thespecificsensedefinedabove),and to theextentIhavesucceededIcanspeakofcausationasanobjectiverela-tionevenwithinmyepistemicapproach.This ismyattemptofcon-structivelycarryingouttheprojectionstrategywhichStalnaker(1984,ch.7)hadarguedtoberequiredandhadattemptedtoexplicate.(Infact,ItakeallofthistobeessentiallyanexplicationofHume’sideasoncausationinpresent-dayterms.)

Whatdoesallofthismeanforacausalconditional“j > y,” inter-pretedaccordingto(VII)and(VIII)?Itcannotonlybetrueorfalseinits factualclaims;what the factualclaim iscanalsobebasedonanobjectivelytrueorfalseinductivestrategy.Historymightactuallybesuchthattheglasswouldhavebrokenifdropped.So,“ifIhaddroppedtheglass,itwouldhavebroken”mightmakeatrueclaim.Anditdoessonotjustonthebasisofmysubjectiveinductivestrategy;whetheritisbasedonacorrectstrategy,isitselftruth-evaluable.

Ishouldemphasize,though,thatthedetailsoftheobjectivizationtheoryneedtobestudied.ThedescriptionIhavegivenisas fairasitcanbeontheinformallevel;butitisnosubstitutefortheprecisetheory.Also,IshouldpointtoHuber(2014),whopresentsanalterna-tiveandquitedifferentprojectionorobjectivizationstrategyforrank-ing functions and for conditionals explained in ranking terms.ThisstrategyproceedsbytranslatingDavidLewis’PrincipalPrincipleintorankingtheory.However,thisisnottheplaceforcomparison.

9. Conclusion

Wehavegonealongway — and it was still much too short. I have left open very many details, and too often I have referredtoSpohn(2012)forfurtherexplanation.Comparativediscussionswiththevastrelevantliteraturecould and should have beenmuchmore extensive and systematic.Giventhetopic,thisisunavoidable.Still,evenwithoutallthosemiss-ingparts,mytwobasicpointsshouldhaveunfoldedtheirpersuasivepower: thatweshouldbase the topiconranking theoryas thebestaccountofconditionalbelief,andthatbystudyingthemanywaysinwhichconditionalsexpressconditionalbeliefwecanunify thefield



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