Introduction
Thelatterhalfofthetwentiethcenturywitnessedan‘intensional revolution’:agreatcollectiveefforttoanalysenotionswhichare absolutelyfundamentaltoourunderstandingoftheworldandof ourselves–frommeaningandinformationtoknowledge,belief,causation,essence,supervenience,conditionality,aswellasnomological, metaphysical,andlogicalnecessity–intermsofasingleconcept. Thiswastheconceptofa possibleworld:awaythingscouldhave been.
Possibleworldsfoundapplicationsinlogic,metaphysics,semantics, gametheory,informationtheory,artificialintelligence,thephilosophy ofmindandcognition.In1986,in OnthePluralityofWorlds,David Lewiscalledpossibleworlds‘aphilosophers’paradise’.Whatever viewonehadonthekindsofthingspossibleworldsare,therewas widespreadagreementontheirbeinganindispensabletheoretical tool.
Thatparadisehasturnedouttobefullofproblems.These haveemergedinpiecemealfashion,asdifficultiesforthisorthat applicationofthepossibleworldsparadigm.Itseemstous,however, thatthedifficultiesrevolvearoundasingleissue.Mostofthose fundamentalnotionsare hyperintensional:theyrequiredistinctions thestandardpossibleworldsapparatuscannoteasilymake.
Whenwesetouttowriteaboutimpossibleworlds–waysthings could not havebeen–wedecidedtosetournarrativeagainstthe backgroundofanenvisagedtwenty-firstcentury‘hyperintensional revolution’.Anumberofaccountshavebeendeveloped,which
qualifyashyperintensionalinsomesense.Theyrangefromtwodimensionalsemantics(Chalmers2006),totheoriesofaboutness (Yablo2014),truthmakersemantics(Fine2017),metaphysical grounding(CorreiaandSchnieder2012),structuredpropositions (King2011),transparentintensionallogic(Duzietal.2010),and variousnon-classicallogicalapproaches(DunnandRestall2002). Howsuchtheories,orfamiliesthereof,areconnectedtoeachother andhowtheirrelativemeritscanbeassessed,areatpresentlargely openquestions.Butwhateverpositionimpossibleworldstakeinthis landscape,webelievethattheywillplayaroleintherevolution,and wefeltthetimewasripeforabookprovidingguidancethroughthe burgeoningliteratureonthesubject.
Thisbookincludesanopinionatedintroductiontotheoriesand usesofimpossibleworlds.(Ashorterandsimplifiedpresentation canbefoundinour‘ImpossibleWorlds’entryinthe Stanford EncyclopediaofPhilosophy.)Wehaveourownpreferencesonthe metaphysicsofimpossibleworldsandthelogicalandphilosophical applicationstheyafford.Wedon’thidethosepreferences;butwe havetriedtoprovidefairaccountsofthealternativeviewsandto assesstheminabalancedway.
Thebookalsoincludesourownoriginalproposalsonanumberof topicsinvolvingimpossibleworlds.Someofthesehaveappeared previouslyinprint,althoughoftennotintheformtheyappear here.WehavedrawnonmaterialfromBerto’spapers‘Impossible WorldsandPropositions’(ThePhilosophicalQuarterly,2010),‘On ConceivingtheInconsistent’(ProceedingsoftheAristotelianSociety, 2014),‘ImpossibleWorldsandtheLogicofImagination’(Erkenntnis, 2017),‘ConceivabilityandPossibility:SomeDilemmasforHumeans’ (withTomSchoonen, Synthese,2018),‘TruthinFiction,Impossible Worlds,andBeliefRevision’(withChrisBadura, AustralasianJournal ofPhilosophy,2018),‘WilliamsononCounterpossibles’(withRohan French,GrahamPriest,andDaveRipley, JournalofPhilosophical Logic,2018),andonBerto’sbook OntologyandMetaontology (with MatteoPlebani,Bloomsbury,2015).Wehavedrawnonmaterialfrom Jago’spapers‘AgainstYagisawa’sModalRealism’(Analysis,2013),
‘TheContentofDeduction’(JournalofPhilosophicalLogic,2013), ‘RecentWorkinRelevantLogic’(Analysis,2013),‘TheProblemof RationalKnowledge’(Erkenntnis,2013),andonJago’sbook The Impossible (OxfordUniversityPress,2014).Weareverygratefulto alltheeditorsandpublishersforpermissiontousetheseworks.
OutlineoftheBook
Thebookisdividedintothreeparts.PartIdealswithfoundational issues.InChapter1,wesurveyanumberofapplicationsofpossible worlds;findthemallwanting;tracetheproblembacktohyperintensionality;andsuggestthatimpossibleworldsmayhelp.Wepresent variousdefinitionsofthenotionofanimpossibleworldfromthe literature.Suchworldsmakesenseonlyifwecangenuinelythink abouttheimpossibilitiestheyrepresent.Wearguethatwecan.
Acentralphilosophicalissuewithworlds,possibleorimpossible,is howtheyrepresentwhattheyrepresent.Thisisobviouslyconnected totheproblemofwhatkindofthingstheyare.InChapter2,we discussanumberofdifferentproposals.Perhapsimpossibleworlds aremetaphysicallydifferentfrompossibleworlds,andrepresentin adifferentway.Orperhapstheyaremetaphysicallyonaparwith possibleworlds.Impossibleworldsmaybetakenas‘genuine’entities which,likeLewisianpossibleworlds,representsomethingasbeingan F byhavingareal F asapart.Or,theymaybetakenasnon-existent objects.Orasabstractentitieswhich,liketheobjectsofgeneralobject theory,representbyencoding.Ortheymaybetakenasprimitive entities,withnoquestionsaskedonhowtheyrepresent.Ormaybe therearenosuchworlds:weshouldtakeafictionaliststance,and justmakebelievethatthereare.
Wearguethatallsuchviewsfacedifficulties,andconclude thatsome ersatz approachfaresthebest.Aftercharacterizingthe notionofanersatzworldingeneralterms,wenoticethatthere aredifferentwaystospecifytheview.Wedelveintotheoptionsin Chapter3.Ersatzpossibleworldscanbeunderstoodasmaximal
statesofaffairs,maximalproperties,recombinationsofbitsof actuality,maps,orthingsbuiltoutofpropositionsorsentences. Wearguethat,whenextendedtoimpossibleworlds,mostofthese approachesfaceissues:theyeithercollapseintootherviews,orare notgeneralenoughtoaccommodatealltheimpossibiliteswemay want.Weconcludethatlinguisticersatzism,whichviewsworldsas constructionsfromsentencesofa‘worldmaking’language,isthemost promisingmetaphysicsofimpossibleworlds.WecloseChapter3by discussingaproblemit,togetherwiththeothervariantsofersatzism, faces:theproblemofaliens.
PartsIIandIIIofthebookareaboutthelogicalandphilosophical applicationsofimpossibleworlds.Theboundarybetweenlogicand philosophyistosomeextentarbitrary,asisourpartitionofthe topics.PartIIcoversepistemic,doxastic,andvariousnon-classical logics.PartIIIcoversapplicationsconnectedtoissuesinmainstream epistemology,informationtheory,thephilosophyoffiction,and topicsinsemanticsandthephilosophyoflanguage.ButPartIIis notcompletelyfreefromphilosophicaldiscussionandPartIIIis notcompletelydevoidofformalism,althoughwehavetriedtokeep technicalitiesundercontrolthroughoutthebook.
InPartII,Chapter4,weintroducenormalmodallogicsand theirframesemantics.Wethenshowhowimpossibleworldscan beusedtomodel non-normal modallogics,inwhichtheRuleof Necessitationisnotvalid.Wediscussfurtheruses,involving nonadjunctiveness and non-primeness.Twogeneralpatternsemergein theseapplications.Firstly,impossibleworldsaregenerallyunderstood as‘logicviolators’:worldswheresomelogicallawfails.Secondly, insemanticsofthiskindtruthconditionsareoftennotspelledout uniformly:theydifferbetweenpossibleandimpossibleworlds.This raisesaphilosophicalproblem,whosediscussionispostponeduntil PartIII:whatofcompositionality,abasicrequirementforatheory ofmeaning?
Chapter5dealswithapplicationsinepistemicanddoxasticlogic. Herethecentraltopicistheproblemoflogicalomniscience.The standardviewmodelsagentsasknowingorbelievingalllogical
truthsandalllogicalconsequencesofwhattheyknoworbelieve.We discusssomeapproachestoavoidingthisconsequencewhichdon’t useimpossibleworlds,andfindthemwanting.Anaïveimpossible worldsapproachcaneasilydeliveraviewwhichavoidsthisproblem. Butitfacesadeeperproblemof boundedrationality:howshould theaccessibleimpossibleworldsbeconstrained,soastomodela moderatelyrationalthoughnotlogicallyomniscientagent?Weargue thatclosingworldsunderaweaker-than-classicallogicwon’thelp.We alsocriticallydiscussadynamicapproachusingimpossibleworlds, onwhichepistemicstatesevolvegraduallytowardsclosure.
Chapter6dealswiththeroleimpossibleworldsplayinthe semanticsofrelevantlogics.Thesearenon-classicallogicsthataim toavoidtheparadoxesofthematerialandstrictconditional.The mainstreamsemanticshereincludesnon-normalpointsofevaluation, whicharenaturallyinterpretedasimpossibleworlds.Thediscussion hasrevolvedaroundmakingsenseofthetruthconditionsforthe relevantconditionalandnegation.Wediscussinformation-theoretic interpretationsofimpossibleworldsinthissetting,andraisesome issues.Wealsodiscussinterpretationsguidedbygeneralviewson conditionalityandaninterpretationintermsoftruthmaking.
Chapter7presentsanapplicationofimpossibleworldstomodellingactsofimagination.Wefocusonasemanticsforhyperintensionaloperatorscapturingakindofmentalsimulation.Wediscuss anumberofplausibleconstraintsonsuchoperators,including non-monotonicity,non-primeness,anda‘PrincipleofImaginative Equivalents’thatlimitsthehyperintensionalanarchyofimagining.
InPartIII,Chapter8revolvesaroundaverygeneralphilosophical issue:ishyperintensionalityagenuinephenomenon?Orisitafeature tobeexplainedaway,andwhichthereforedoesnotrequireus toamendthestandardpossibleworldsapparatus?Weconsider argumentsforthelatterview,andfindthemunsuccessful.Wethen focusonageneralnotionofhyperintensionalcontent,anddiscusstwo issuesconcerningit.Firstly,anyhyperintensionaltheoryofcontent mustaddresstheproblemofgranularity:howfine-grainedmustthe relevanthyperintensionaldistinctionsbe?Secondly,wereturntothe
issue,flaggedinChapter4,ofnon-uniformtruthconditions,which raisesacompositionalityobjectionfortheoriesofcontent.Weargue thatimpossibleworldsaccountscandeliverafullycompositional theoryofcontent.
Chapter9isaboutinformation,whichweconceptualizesemantically,intermsofrulingoutscenarios.WearguethatFrege’spuzzleof informativeidentities,andtheinformativenessoflogicalinferences, canbeaccountedforhyperintensionally,usingimpossibleworlds.In ourfavouriteanalysis,itmaybeindeterminatewhetheragivenlogical inferenceisinformative.Wealsosketchananalysisofinformative contentintermsofwhatissaidbyaspeakermakinganutterance.
Chapter10dealswithepistemicanddoxasticcontents.Here wefocusonhowtomodelarealisticcognitiveagent,strikinga balancebetweentheimplausibleextremesoflogicalomniscience andcompletelogicalignorance.Thisistheproblemofbounded rationality,flaggedinChapter5.Thebeliefstatesofsuchanagent seemtobeclosedunder‘easy’,triviallogicalconsequence,butnot underfulllogicalconsequence.Yettheformerseemstoimplythe latter.Oursolutionisthat,whilesometrivialclosureprinciplemust failinabeliefstate,itisindeterminatejustwhereanysuchfailure occurs.Wegiveformalmodelsofbeliefstatessostructured.These entailthatnobodygenuinelybelievesanoutrightcontradiction.We closethechapterdiscussingtheissueofpeoplewhoclaimtheydo.
Chapter11,writtenwithChrisBadura,appliesimpossibleworlds totheanalysisoftruthinfictionandthemetaphysicsoffictional objects.Weshowthatinconsistentfictionsarenaturallyhandledviaa spaceofworldsincludingimpossibleworlds,andthattruthinfiction canbeunderstoodasakindofsimulatedbeliefrevisionoversuch aspace,triggeredbythefiction’sexplicitcontent.Wethendiscuss fictionalist,realist,andMeinongianaccountsoffictionalcharacters, theirproblems,andtheirrelativemerits.Weshowhowimpossible worldscanhelptoimproveonsomeoftheseaccounts.
Chapter12,writtenwithRohanFrench,GrahamPriest,andDave Ripley,isaboutcounterfactuality.Thestartingpointhereisthe intuitiveviewthatcounterpossibles–counterfactualconditionals
withimpossibleantecedents–arenotallvacuouslytrue,independentlyofthetruthvalueoftheconsequent.Wediscussobjections totheeffectthatthisintuitionshouldbeexplainedaway,and findthemunconvincing.Wethenofferanon-vacuistsemanticsfor counterpossiblesthatresortstoimpossibleworlds.Thistriggersa discussionoftheso-called‘StrangenessofImpossibilityCondition’ (SIC).Thisrelatestotheideathatsomepairsofworldsarecloser tooneanotherthanothers,andthatweevaluatecounterfactuals byconsideringtheclosestworlds.The(SIC),then,holdsthat,for anygivenpossibleworld,anyimpossibleworldisfurtherawayfrom itthananypossibleworldis.Inthesemantics,thesubstitutivityof rigidlycoreferentialtermsfailsincounterfactualcontexts.Thisis arguablyaproblem.Anotherobjectionrevolvesaroundmakingsense ofargumentsby reductioadabsurdum inmathematicalpractice.We arguethatbothobjectionscanbemet.
Acknowledgements
Versionsofpapersrelevanttothebookhavebeenpresentedbyus inanumberofworkshops,seminars,andconferencesinAustralia, theCzechRepublic,France,Germany,Italy,Japan,theNetherlands, Slovakia,Sweden,Switzerland,theUK,andtheUS.Thosewhoasked goodquestionsandcameupwithusefulcommentsincludeJcBeall, ThomasBrouwer,ColinCaret,RobertoCiuni,AaronCotnoir,Franca D’Agostini,JohnDivers,MikeDunn,RichardDietz,CatarinaDutilh Novaes,RohanFrench,EmmanuelGenot,VittorioHoesle,Justine Jacot,JonathanJenkins-Ichikawa,IraKiourti,BarteldKooi,Ernie Lepore,TitoMagri,DiegoMarconi,FriederikeMoltmann,Bence Nanay,DanielNolan,LucyO’Brian,HitoshiOmori,FrancescoOrilia, MichelePaoliniPaoletti,MatteoPlebani,ShahidRahman,Stephen Read,ManuelRebuschi,GregRestall,DaveRipley,MaciejSendlak, SebastianSequoiah-Grayson,JeroenSmid,MatthewSoteriou,Bob Stalnaker,JacopoTagliabue,StephanTorre,GiulianoTorrengo, MartinVacek,MarkVanAtten,AlbertoVoltolini,HeinrichWansing,
ZachWeber,DavidWiggins,CrispinWright,anddoubtlessmany others.
SpecialthanksgotoChrisBadura,JCBjerring,JorgeFerreira,Ed Mares,GrahamPriest,TomSchoonen,andtherefereesofOxford UniversityPress,whoread(variouspartsof)themanuscriptand providedhelpfulcomments.Anyremainingerrorsaftertheirscanning areentirelytheirfault.
1 FromPossibletoImpossible Worlds
1.1WorldsasWays
Thingsmighthavebeenotherwise.DavidBowiemaystillhavebeen withus,thesunmayhavebeenshiningonNottingham,andtheAxis powersmayhavewontheSecondWorldWar.Suchalternativeways wecall possibleworlds.Eachpossibleworldisawaythingscould havebeen.(Thisinitialcharacterizationsaysnothingofwhatpossible worldsare,metaphysicallyspeaking.That’sthetopicofChapters 2and3.)Theactualworldisthemostgeneralandcomprehensive wayinwhichthingsinfactare.Intheactualworld,theNazislost theSecondWorldWar,theskyoneofusseesfromhisofficein Nottinghamiscloudy,andDavidBowiediedatthebeginningof 2016.
Waysthingscouldhavebeencanresemblethewaythingsactually are.AworldwheretheAxispowerswontheSecondWorldWaris stillaworldwheretherewasawarinwhichtheNazisfought,though withadifferentoutcomefromtheactualworld.Somepossibleworlds involveonlysmallchangesfromours:thinkofaworldexactlylike theactualone,exceptthatyouareoneinchtaller.Othersarevery different:thinkofonewherethelawsofbiologyandphysicsare turnedupsidedown,sothatyoucanbeborntwice,ortravelfaster thanthespeedoflight.Aswewillsee,theideathatitmakessenseto speakofrelationsofsimilaritybetweenpossibleworldsisimportant forsomeapplications.
Possibleworldshavea vast arrayofapplications.Accordingto some,thisisthemainreasonforacceptingthem:‘itmaybethatthe bestphilosophicaldefencethatonecangiveforpossibleworldsisto usetheminthedevelopmentofsubstantivetheory’(Stalnaker1991, 141).SincethelatetwentiethcenturyrejectionoftheQuineanand Davidsonianideathatonlyextensionalconceptsshouldbeallowed inseriousphilosophicalinquiry,thenotionofpossibleworldhas becomeubiquitousincontemporaryphilosophy.Itplaysakeyrole inmostbranchesofthediscipline,rangingfromlogictometaphysics andontology,thephilosophyofmind,thephilosophyofinformation, moralandpoliticalphilosophy,andaesthetics.Butithasbeenused alsooutsideofphilosophy,infieldsthatrangefromthesemanticsof naturallanguagetogametheory,artificialintelligence,andcognitive science.Westartwithanoverviewoftheseapplications.(Partsofthe followingsectiondrawonBertoandPlebani2015,chapter11.)
1.2PossibleWorldsatWork
PossibilityandNecessity
Perhapsthemosttypicalapplicationofpossibleworldsisinmodal logic.Thisis,firstofall,thelogicofexpressionslike‘necessarily’, ‘possibly’,‘contingently’.Suchexpressionsareusedintwodifferent ways.Afirstuseconsistsinqualifyingthetruthofasentence,orof thepropositionexpressedbythesentence:
(1.1) Itisnecessarythat 7+5=12.
(1.2) ItispossiblethatScotlandleavestheUK.
(1.3) Possibly,Annawinsthemusiccontest.
(1.4) Necessarily,Valeriaishuman.
Modalitiesofthiskindarecalled dedicto.Expressionslike‘necessarily’or‘itispossiblethat’,ortheconceptstheyexpress,areattached
to dicta,thatis,topiecesoflanguage,orlanguage-likeentities,such assentencesorpropositions.Theyexpressthewaythatsentence,or proposition,bearsitstruthvalue.Thus,accordingto(1.1),thatseven plusfiveistwelveisnecessarilytrue,andaccordingto(1.3),that Annawinsthemusiccontestispossiblytrue.
Modalexpressionscanalsobeusedtoqualifythefeaturesof objects:
(1.1a) Sevenisnecessarilyanoddnumber.
(1.2a) ScotlandissuchthatitcouldleavetheUK.
(1.3a) Annaisapossiblewinnerofthemusiccontest.
(1.4a) Valeriaisnecessarilyhuman.
Modalitiesofthiskindarecalled dere,forthemodalsareusedhere toexpressthewayinwhichathing,a res,hassomefeature.Thus, accordingto(1.1a)and(1.4a),sevenhasthepropertyofbeingodd, andValeriathatofbeinghuman,inanecessaryway.
ContemporarylogiciansandphilosophersfollowLeibniz’sinsight thatthenecessaryiswhatholds nomatterwhat,inanywaythings couldhavebeen:thatis,inallpossibleworlds.Whatispossible,onthe otherhand,iswhatholdsatsomepossibleworld.Whatiscontingent iswhatholdsatsome,butnotall,possibleworlds.Necessityand possibilityarethusinterpretedasquantificationsoverpossibleworlds. Using‘◻’for‘necessarily’,‘◇’for‘possibly’,‘iff’for‘ifandonlyif’, andletting W bethetotalsetofpossibleworlds,weget:
‘◻A’istrueatworld w iff A istrueatallworlds w1 ∈ W
‘◇A’istrueatworld w iff A istrueatsomeworld w1 ∈ W
(Thenotation‘w1 ∈ W’heremeansthat w1 isamemberoftheset W.It’sawayofexpressingthat w1 isapossibleworld.)
Thetwonotions ◻ and ◇ are duals ofoneother,justastheuniversal andparticularquantifiers, ∀x and ∃x,areofoneanother.Eachmodal canbedefinedviatheotherandnegation.Thatitisnecessarilythe
frompossibletoimpossibleworlds
casethat A meansthatitisnotpossiblethat ¬A (‘not-A’).Andthat itispossiblethat A meansthatitisnotnecessarythat ¬A. Necessityandpossibilityarehighlyambiguousnotions.(Fora taxonomy,seechapter1ofDivers2002.)Althoughthereisno universalconsensusonthis,manyphilosophersadoptthreekindsof absolutenecessity,holdinginallpossibleworldsunrestrictedly:
Logicalnecessity fixedbythelawsoflogicbroadlyconceived (e.g.,thatif A,theneither A or B);
Mathematicalnecessity fixedbymathematicaltruths(e.g.,that 7+5=12);and
Metaphysicalnecessity fixedbytheidentityandnatureofthings (e.g.,thatwaterisH2O;thatValeriaisahumanbeing).
Wewillnotgetintotheissueofwhetheroneoftheseisreducibleto another(e.g.,themathematicaltothelogical,asclaimedby logicists inthephilosophyofmathematics,includingDedekind(1901),Frege (1879),Peano(1889),andRussell(1903)).
Wealsotalkofthingsbeingnecessary,orimpossible,onlyina relativesense,orfromacertainviewpoint.Wearestuckinatraffic jaminParisat2pm;ourflightisleavingfromDeGaulleairport at2:10pm.Wemoan:‘There’snowaythatwecanmakeittothe airportintime’.Whatwemeanisthat,giventhetiming,themeans oftransportavailable,andthelawsofphysicsofourworld,itis impossibleforustoreachtheairportintime.Itisnotunrestrictedly, absolutelyimpossible:ifwehad StarTrek’stransporter,wecould makeit.Buta StarTrek worldinwhichonecanbeinstantaneously disassembledintoatomsandreassembledexactlywiththesame atomicstructureinadifferentplaceisaworldquitedifferentfrom ours.Onemaydoubtthatsuchaworldisevenphysicallypossible, thatis,compatiblewithourlawsofphysics.
Othermodalnotions,thus,arenaturallyunderstoodasrestricted formsofnecessityorpossibility.Somethingcancountas R-necessary, forsomerelativizedmodalnotion R,evenifitfailstoholdatsome
possibleworldorother.Accordingly,thecorrespondingmodals areunderstoodasrestrictedquantifiersoverpossibleworlds.Thus nomologicalnecessity,compliancewiththelawsofnatureofthe actualworldoroftheworldunderconsideration,isoften(no universalconsensushereeither)takentobearelativeorrestricted necessity.Itisbiologicallyimpossiblebutnotabsolutelyimpossible forahumanbeingtojumponemileupintheair;itisphysically impossible(ifEinsteinwasright)butnotabsolutelyimpossiblefora bodytotravelfasterthanthespeedoflight.
Propositions
Possibleworldsareextremelyimportantfortheoriesof representation,bothinlanguageandthought,andhavebeenusedtoanalysekey notionsfromthephilosophyoflanguage.Manyoftheseapproaches buildonWittgenstein’sinsightthatunderstandingthemeaningofa sentenceisgraspingitstruthconditions:‘tounderstandaproposition meanstoknowwhatisthecaseifitistrue’(Wittgenstein1921/1922, §4.024).Montague(1970)andStalnaker(1976a)haveclaimedthat propositions,themeaningsorcontentsexpressedbysentencesand theprimarybearersoftruthvalues,shouldbeunderstoodassetsof possibleworlds.ThepropositionexpressedinEnglishby‘raccoons liketosomersault’is,onthatview,thesetofpossibleworldswhere raccoonsliketosomersault,preciselythesamesetofpossibleworlds makingforthepropositionexpressedinItalianby‘aiprocionipiace farelecapriole’.
KnowledgeandBelief
Anothernotionanalysedviapossibleworldsis knowledge.Following Hintikka(1962),knowledgehasbeencharacterizedintermsofwhat istruethroughoutallthewaysthingscouldbe,foralltheagentin questionknows.Onthisapproach,thepossibleworldsaccessibleto anagentrepresenther epistemicpossibilities.Knowledgecanthen betreatedasanotherrestrictedquantifieroverpossibleworlds.If K standsforagivenagent’sstateofknowledge,and R isabinary
frompossibletoimpossibleworlds
accessibilityrelationonthespaceofworlds W,theHintikka-style characterizationgoesthus:
(H) KA istrueat w iff A istrueatall w1 suchthat Rww1
Thisthoughtisatthecoreofcontemporaryepistemiclogic(see, e.g.,Blackburnetal.2002,Faginetal.1995,VanBenthem2003). Butseveralresearchprogramsinmainstreamepistemologyalsorely onasimilarviewpoint.Dretske’s relevantalternatives approachtakes knowledgeas‘anevidentialstateinwhichallrelevantalternatives(to whatisknown)areeliminated’(Dretske1981,367).Lewis(1996) discussesasimilarapproach.Alternativeshereworksimilarlyto possibleworlds,andtheuneliminatedrelevantalternativeswork similarlytoaccessibleworlds.
Necessity(whetherlogical,mathematical,metaphysical,ornomological)andknowledgesharethefeatureofbeing factive:whatis necessary,andwhatisknown,istrue.Factivitycanbeexpressedby claimingthattheactualworldmustalwaysbeoneofthe(accessible) possibleones,withrespecttotherelevantkindofpossibility.Unlike otherfactivemodalities,though,knowledgeisan intentional state: astateoftheminddirectedtowardsacertaincontent.Thereare alsonon-factiveintentionalstates,includingbelief,desire,fear,hope, andimagination.Thesehavealsobeenunderstoodusingrestricted quantifiersoverpossibleworlds,wheretheaccessibleworldsarethe oneswherethingsareastheagentbelieves(imagines,etc.)themtobe. (Faginetal.1995isacomprehensiveguidethroughepistemicand doxasticlogics.Niiniluoto(1985)andWansing(2017)eachdiscuss theapplicationtoimagination;Berto(2018)givesasemanticsfor imaginationusinganenrichedpossibleworldsapproach.)
Knowing(orbelieving,imagining,etc.)that A isoftentakentobea mentalstatewhosecontentisthepropositionexpressedby A.Aswell asbeingtheprimarybearersoftruthvalues,thus,propositionshave beenunderstoodasthecontentof(dedicto)intentionalstates:they arewhatisknown,believed,feared,orimaginedwhenoneknows, believes,fears,orimaginessomething(dedicto).Justasdifferent
sentenceslike‘raccoonsliketosomersault’and‘aiprocionipiace farelecapriole’cansharethesamecontent,socandifferentpeople’s mentalstatessharethesamecontent:Johnbelieves,andMaryfears, thatMarinewillwintheelections.Thereissomedoubtthatone andthesamekindofentitycancoverboththeroleofprimarytruth bearersandtheroleoftargetsof dedicto intentionalstates(Jago 2018b,Lewis1986b).Nevertheless,possibleworldsstakeagood claimatgivingaunifiedaccountofabroadrangeoflinguisticand mentalcontents.
Information
Informationisconnectedtoknowledge,orpotentialknowledge.Ifa sentenceorpropositionisinformative,thenonecancometoknow thatinformation(say,byhearingthesentenceutteredtruthfullyby atrustedspeaker).Somethingmaybeinformativeevenifnoone yetknowsit,however.Wemightthinkofinformationasembodying potentialknowledgeforsomesuitablecognitiveagentintheright circumstances.Ifweanalyseknowledgeintermsofpossibleworlds, weshouldexpectasimilarapproachtoinformationtobeavailable.
Accordingtothe Bar-Hillel-Carnaptheoryofinformation (BarHillelandCarnap1953,Bar-Hillel1964),theinformativejobofa sentence A consistsinpartitioningthetotalityofpossibleworldsinto thosewhere A istrueandthosewhereitisfalse.Wemayidentify theinformationwiththepartitioningfunction,whichineffectsays ‘yes’tosomepossibleworldsand‘no’toalltheothers.Orwemight identifytheinformationwiththesetof‘yes’worlds.(Mathematically speaking,theformeristhe characteristicfunction ofthelatterset. Thetwoapproachesare,inastraightforwardmathematicalsense, equivalent.)
Youmightnoticethesimilaritytothepossibleworldsaccount ofpropositions:botharetreatedassetsofpossibleworlds.That’s nocoincidence.Onthisapproach,theinformationcontainedina sentence(inacontext)ispreciselythepropositionexpressedbyan utteranceofit(inthatcontext).
frompossibletoimpossibleworlds
Thisgivesusastaticnotionofinformation,assomethingthat’s possessedbyasentenceorproposition.Thepossibleworldsapproach alsoallowsustoaccountforadynamicnotionofinformation,of becominginformed ofsuch-and-such.Whenacognitiveagentgains theinformation(and,let’ssuppose,therebylearns)thatraccoonslike tosomersault,wecanmodelthisintermsofrulingouttheworlds whereitisnotthecasethatraccoonsliketosomersault.(Perhaps theraccoonsofthoseworldshavedifferenttastes;perhapsthereare noraccoonsthereatall.)By‘ruleout’,wedon’tmeanthattheagent therebytreatsthoseworldsasimpossibilities.Rather,sherulesthem outascontendersforactuality:thewaysthingsare,forallsheknows.
IndicativeConditionals
Conditionality mayalsobedealtwithusingpossibleworlds.Many philosophersandlogiciansareunsatisfiedwiththematerialconditional,‘⊃’,takenastheoperatorgivenbytheusual(two-valued)truth table:‘A ⊃ B’isfalsewhen A istrueand B false,trueotherwise.This deliverstwoinferenceswhichhavesometimesbecalled‘paradoxes ofthematerialconditional’(AndersonandBelnap1975,MacColl 1908,Routleyetal.1982):
(1.5) If ¬A,then A ⊃ B
(1.6) If B,then A ⊃ B
IfwetrytounderstandtheEnglishindicativeconditional‘if… then’intermsof ⊃,manyseeminglyfalseconditionalswillcomeout true,justbecausetheirantecedentisfalse,ortheirconsequenttrue:
(1.7) IfObamaisCanadian,thentheMoonismadeofgreencheese.
(1.8) IfStrasbourgisinGermany,thenObamaisAmerican.
Onereasontorejecttheseisthatthereisnorelevantconnection betweentheirantecedentandconsequent:what’sObama’snationality gottodowiththeconstitutionofthemoon,orthelocationof
Europeancities?Anotherreasontoreject(1.7)and(1.8)isthat anyconnectionbetweentheirantecedentandconsequentseemsfar toocontingent.EvenifObamawereCanadian,themoonwouldnot beagiantcheeseball.
Thissuggestsanalternativeconceptionoftheconditionalonwhich, for‘if A,then B’tobetrue,it cannot bethecasethat A istruewhile B isfalse.Thisanalysisgivesusthe strictconditional,‘ ’.‘A B’ istruejustincasethereisnopossibleworldinwhich A istruebut B isnot.Thestrictconditionalisthenecessitationofthematerial conditional: A B isunderstoodas ◻(A ⊃ B).It’seasytoseethat (1.5)and(1.6)areinvalidwhenwereplace‘⊃’with‘ ’.(Whether thismovereallyavoidstheworriesissomethingwe’llcomebackto below,in§1.3andChapter6).
CounterfactualConditionals
Possibleworldshavealsobeenusedtogiveasemanticsfor counterfactual conditionals.Theseareconditionalsoftheform‘ifitwere (or,hadbeen)thecasethat A,thenitwouldbe(or,havebeen) thecasethat B’,symbolizedas‘A ◻→ B’.Counterfactualsaresocalledbecause,inatypicaluse,theyhaveafalseantecedent, contra factum.Inexplainingwhykangarooshavetails,forexample,we mightsay‘well,ifkangarooshadnotails,theywouldtoppleover’ (Lewis1973b).
(Manyphilosophersuse‘counterfactual’foranyconditionalofthe form,‘ifitwere…,thenitwouldbe…’,eveniftheantecedentis true(Bennett2003,Lewis1973b,Williamson2007).Othersprefer totalkof‘subjunctiveconditionals’.We’llwillsticktothestandard ‘counterfactuals’terminologyforallsuchconditionals.)
Counterfactualsareextremelyimportantinourcognitivelives. Weconceivecounterfactualalternativestorealityinordertoexplore whatwouldorwouldnothappen,werethosealternativesrealized. WouldJohnnothavebeeninjured,hadheavoidedcrossingtheroad? Theyarealsoimportantinunderstandinghistory(Nolan2016b): whatifHitlerhadhadtheA-bombin1944?Theymayhelpusto understandtheconceptofcausationbetter(Lewis1973a,Paul2004;
20 frompossibletoimpossibleworlds
seePaul2009,PaulandHall2013forin-depthdiscussion).Soit’s importanttogiveasemanticanalysisofcounterfactuals.
How?Thatcounterfactualsmustbemodalconditionalscan bearguedbycomparingthemtothecorrespondingindicative conditionals:
(1.9) IfKateBushdidn’twrite‘TheKickInside’,someoneelsedid.
(1.10) IfKateBushhadn’twritten‘TheKickInside’,someoneelse wouldhave.
Thesehavethesameantecedentandconsequent(indifferentmoods), butdifferenttruthvalues.(1.9)seemstrue.Weknowthatsomeone wrote‘TheKickInside’,soifitwasn’tKateBush,itmusthavebeen someoneelse.Bycontrast,(1.10)seemsfalse:‘TheKickInside’might neverhavebeenwritten,ifithadn’tbeenforKateBush.Soevenifone insiststhat(1.9)betakenasamaterialconditional,(1.10)seemstobe ofadifferentkind.Thedifferenceinmoodbetween(1.9)and(1.10) hasbeenunderstoodasgettingustoevaluate(1.10)bylookingat alternativewaysthingscouldhavebeen,thatis,atalternativepossible worlds.
Whichworlds?Themainstreamtreatmentofcounterfactuals,due toStalnaker(1968)andLewis(1973b),saysthatweshouldevaluate ‘ifkangarooshadnotails,theywouldtoppleover’bylookingtothe closest possibleworldswherekangarooshavenotails.Wethensee whetherkangaroostoppleoverthere. Closeness betweenworldsis understoodasinvolving(contextuallydetermined)similarityinthe relevantrespects.Soevaluatingacounterfactualwilltypicallyinvolve the minimalchange (withrespecttotheworldofevaluation)required toverifytheantecedent.Wedisregardworldswherekangarooshave notailsbuthelpthemselveswithcrutches,orhaveevolvedwings. Overall:‘A ◻→ B’istrue(atworld w)ifftheclosest(-to-w)possible A-worldsare B-worlds.
(Whatifseveralpossibleworldstieforcloseness?Dowerequire all closest A-worldstobe B-worlds?Orsome?Ormost?Whatifthe A-worldsgetforevercloserandclosertoours,withnonebeingthe
closest?Thesearetrickyquestions:Kratzer(1981),Lewis(1973b, 1981),andNute(1975)discussthemindetail.Wewon’tgetinto themhere.)
Possibleworldshavealsobeenusedintheanalysesof essence and deremodality (Lewis1986b),andof metaphysicaldependence and supervenience (Bennett2004,Davidson1970).Manyphysicalist philosophersofmind,includingHorgan(1982,1993),Kim(1982, 1993),andLewis(1983),expresstheircommitmenttophysicalism aboutmentalstatesintermsofsupervenience,cashedoutinterms ofpossibleworlds.But,forreasonswewon’tgointohere,wedon’t thinkaworlds-basedapproachisthebestwaytocapturenotions ofessenceordependence.(We’relargelypersuadedbyFine’s(1994) arguments.)Sowewon’tdiscusstheseapplicationsanyfurther.
Possibleworldsareasuccessstoryofphilosophicaltheorizing. Still,mostoftheaccountsusingthem,whichwehavejustsketched, faceissues.Theumbrellaunderwhichmanyofthesecanbegathered istheconceptof hyperintensionality,towhichwenowturn.
1.3TheProblem(s)ofHyperintensionality
Hyperintensionalitycanbecharacterizedasafeatureofconcepts.A conceptishyperintensionalwhenitdrawsadistinctionbetween necessarilyequivalentcontents,wheretherelevantnecessityis unrestricted:logical,mathematical,ormetaphysical,ifwestickto thethreefolddistinctionmentionedabove.Iftherelevantconcept isexpressedbyanoperator ℋ,then ℋ ishyperintensionalwhen ℋA and ℋB candifferintruthvalue,inthefaceof A and B’sbeing necessarily(logically,mathematically,ormetaphysically)equivalent.
(Cresswell(1975)originallydefined‘hyperintensional’topickouta positioninasentenceinwhichlogicalequivalentscannotbereplaced salvaveritate.But,asNolan(2014,151)notes,itisnowcommonto usethetermmorebroadly,with‘necessaryequivalence’inplaceof ‘logicalequivalence’.)
