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OXFORDPHILOSOPHICALMONOGRAPHS
EditorialCommittee
WilliamChild,R.S.Crisp,A.W.Moore,StephenMulhall,ChristopherG.Timpson
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Everything,More orLess
ADefenceofGeneralityRelativism
J.P.Studd
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Inlovingmemoryofmyparents
PeterandElizabeth
Preface
Everything,moreorless?
Isay: more.Thisbookdefendsanexpansionistviewof‘everything’.Theexpression‘everything’isstandardlygroupedwithrelatedquantity-indicatingexpressions: quantifiers (e.g.‘nothing’,‘something’,‘atleastonething’,‘fewdonkeys’,‘exactlyfour philosophers’).Quantifierspermitustomakeutterancesandstatetheorieswhich reachoutbeyondtheparticularand generalize.Buthowgeneralcanwebe?Isthere awidest,maximallyinclusive,mostgeneralsenseof‘everything’?Notaccordingto expansionism.Onthisview,nomatterhowbroadourtopicofconversationorwhat itemsourtheorygeneralizesabout,what‘everything’andotherquantifiersencompass isalwaysopentoexpansion.Thereisalways,sotospeak,amoreinclusive‘everything’.
Morethaneverything?Crudelystated,theviewmayseemabsurd.Giventhechoice, manyinsteadplumpforasecondview: everything.It’shardtodenytheprimafacie plausibilityoftheprincipalviewIdefendexpansionismagainst.Theabsolutistabout quantifierstakesthesober-seemingviewthatthereisawidestsenseof‘everything’. Theone,theabsolutistmightadd,where‘everything’reallymeans everything :everythingwhatsoever,everythingintheentireuniverse,everythingwithoutrestriction, everythingwithoutexception, absolutelyeverything.What‘everything’thenencompasses,theabsolutistclaims,isnotopentoexpansionbecausethereisnothing— absolutelynothing —lefttoadd.
Athirdviewisalsoavailable: less (althoughfewofitsadvocateswouldputit thisway).Restrictionism,likeexpansionism,opposesabsolutism,butfordifferent reasons.Thereiswidespreadagreementthatwhat‘everything’encompasses—the domainquantifiedover —mayvaryaccordingtowhatisunderdiscussion,orsalient inoursurroundings,andsoon.Anoccurrenceof‘everything’postedinashop window,forexample—‘40%offeverything!’—isnotintendedtogeneralizeaboutthe sameitemsasthe‘everything’onitscompetitor’sposter:‘60%offeverything!’.But therestrictionistgoesfurther.Onthisview,what‘everything’encompassesisalways subjecttorestriction.
Thetwoviewsopposingabsolutismaboutquantifiersmaybegroupedtogether asversionsof relativismaboutquantifiers.Theimmediateimportanceofthedisputedquestion—doquantifiersachieveabsolutegenerality?—comesoutinthefact thatabsolutegeneralityappearstobedeeplyembeddedinourtheorizing.Evenif it’scomparativelyuncommonineverydayconversation,howarewetoengagein metaphysics,logic,orsettheorywithoutquantifyingoverabsolutelyeverything? Williamson(2003)arguesthatrelativismaboutquantifierswouldseverelydiminish ourcapacityforsystematicenquirywellbeyondthesedisciplines:itwouldleave usunabletoadequatelycapturethesemanticsofquantifiersandpreventourfully expressingevensomekindsof limited generalizationwhichareamainstayoflarge swathesofscientifictheorizing.
Ontheotherhand,absolutismhasbeencontestedondiversegrounds,some broadlysemantic,othersmoremetaphysical.Inmyview,however,themostcompellingreasontodoubtabsolutismcomesfromphilosophicalissuessurrounding thefoundationsofmathematics.Thethoughtthatabsolutegeneralityissomehow culpableintheset-theoreticparadoxesisanoldone,almostasoldastheparadoxes themselves.Someofthechaptersthatfollowexaminerelatedlinesofthoughtinthe workofRussell,Zermelo,andDummett.Butmyprimaryconcernisnotexegetical. Itistodefendanexpansionistversionofrelativism.Marshallingconsiderations relatingtotheparadoxesintoarigorousanddialecticallyeffectiveargumentagainst absolutismisanimportantpartofthisdefence.Andevenifnosuchargumentis ultimatelytobefoundinthewritingsoftheseauthors,therelativisthasagreatdeal tolearnfromtheirwork.
BlamingabsolutismforRussell’sparadox(orBurali-Forti’s)maysuggestthatthere isawould-bedecisiveargumentagainstthisviewintheoffing.SomeofDummett’s remarkssuggestthistoo:thefailureofabsolutism,hetellsus,isthe‘primelesson’of theset-theoreticparadoxes(1981,p.516).ButIhavenosuchargument(andnordoes Dummett).Theprincipal,broadlyDummettian,argumentsIofferagainstabsolutism relyonnon-logicalassumptions,specificallyonliberalplenitudeassumptionsabout, sotospeak,whichpluralitiesofzeroormoreitemscanbecollectedtogetherasthe elementsofasingleset.Doubtless,wecansaveabsolutegeneralitybysufficiently limitingcollectability.Butwhyprefertheformeroverthelatter?
Inexaminingthisquestion,Iadoptamodestmethodologicalnaturalismwhich limitsthescopeofthisbook.Iassumethatsettheory,aspractisedbysettheorists, isbroadlyalongtherighttrack.Iapplythesameassumption,moreover,towidely acceptedsemantictheories,castinsettheory.ThisisnottosaythatIfetishizeclassical logic(oreventhatIfullyacceptit—seeAppendixA.3).ButitdoesmeanthatIset asideresponsestotheset-theoreticparadoxes,suchasDummett’sintuitionism,which seektoradicallyreformmathematicalpractice,andthose,suchasthedeploymentof paraconsistentlogic,whichareyettomakesubstantialprogresstowardsrecovering classicalsettheory.Evenifsettheoryormodel-theoreticsemanticsarenotbeyond philosophicalinterpretationorcriticism,wemust,Iassume,ultimatelyleavethings substantiallyaswefoundthem.Animportantpartofmydefenceofrelativismis consequentlytoshowthatitisconsonantwithmathematicalandsemanticpractice.
Ifneitherabsolutismnorrelativismisopentoaknock-downobjection,weareleft todecidethematteronpoints.Inmyview,relativism,initsexpansionistversion, avoidsorsubstantiallymitigatesthemainobjectionsraisedagainstrelativisminthe literature(often,itseems,withrestrictionisminmind).Atthesametime,absolutismfaresworsethanitsadvocatesgenerallyacknowledge.Overall,expansionism outperformsbothitsabsolutistopponentanditsrelativistrival.Thatsaid,Ihave nosystematicaccounttoofferofphilosophicalcost–benefitaggregation.Perhaps someonewhodifferedsufficientlyonherweightingoftheissuescould,inprinciple, agreewitheveryspecificconclusioninthisbookandstilldisagreewithmeoverall. Inpractice,however,Iimaginethatanti-expansionistswillfindplentytocontestin particularaswellasingeneral.
Indeed,theabsolutistmaybeunwillingtoconcedethatthere’sagenuinetradeofftobehadbetweenabsolutegeneralityandtheunlimitedcollectabilitypromised
byhisopponents.Afterall,hemayclaim,thelessonofRussell’sparadoxisthatthe liberalattitudetowardscollectabilityisincoherent.Thecontradictionensuesdirectly fromtherelevantplenitudeassumptions,withouttheneedtocallonabsolutism.The incoherencechargeissometimespushedfurtherstill.NotonlydoestheDummettian caseforrelativismturnonincoherentassumptions,incoherenceseepsintorelativism itself.Crudestatementsoftheviewreinforcethesuspicionthatrelativismcannotbe coherentlymaintained.
Inorderfortherelativisttosuccessfullydefendherview,then,sheneedstoshow thatthereisacoherentviewtobedefended.Tothisend,therelativistcanfruitfully deploytoolsfromphilosophicallogic.Withthehelpofrelativist-friendlyschemas or,moreambitiously,suitablyinterpretedmodaloperators,therelativistisableto coherentlyframebothherpositionandtheargumentforit.
Inadditiontophilosophicalquestions,theuseoftheseresources,especiallythe modaloperators,raisessomeinterestingtechnicalquestions.Thetechnicaldevelopmentsinthisbooklargelytakeplaceoutsidethemaintext,predominantlyinthethree appendices.Themainchaptersarenotsymbol-free.But,asfaraspossible,I’vetried toprovideplainEnglishglossesofformalstatementsandtheorems.Myhopeisthat readerswhoarenotsymbol-loversshouldbeabletogetthethrustoftheargument, withouttheneedtodeciphertheformalism.Thesmallamountofsettheoryneeded tounderstandtheabsolutegeneralitydebateisintroducedinaninformalwayin Chapter2.
TheEnglishglosses,however,raiseissuesoftheirown.Inlightofthemanifest differencesbetweenEnglishandtheextensionsandvariationsoffirst-orderlanguages deployedinthisbook,thenaturallanguageglossesshouldnotbebegrudgedtheir granumsalis.Readersoftheabsolutegeneralityliterature,however,mayfindthe needtokeepthesaltcellarcloseathand.Thisbookisnoexception.Thechapters thatfollowmakeliberaluseofnumeroustermswhichcannotbeaccordedaliteral, face-valueinterpretationbybothsidesinthedebate:‘domain’,‘universe’,‘extension’, ‘semanticvalue’,‘absolutelycomprehensive’,‘absolutelyeverything’,‘collectable’,‘plurality’,‘superplurality’,‘propositionalfunction’,‘entity’,‘howeverthelexiconisinterpreted’,amongothers.
Thefactthattheproblematictermsincludemostofthekeytermsintheabsolute generalitydebatehighlightstheutilityoftheformalism.Whatwemayatbesthopeto metaphoricallyconveywithalooseuseofthesetermsmayadmitofaprecise, metaphor-free,statementinasuitablyinterpretedsymboliclanguage.ItisconsequentlytheseregimentationsthatItaketogivetheofficialstatementofthetheses putforwardinwhatfollows.TheaccompanyingEnglishglossmaythenbetreated simplyasasuggestive,reader-friendlyshorthandfortheofficialregimentation.
Ofcourse,whenitcomestotheintelligibilityoftherelevantnotions,thisparaphrasestrategyonlypushestheissuebackintothesymboliclanguage.Butatleast thiswaywesharplydelineateexactlywhichtermsofartwemustmakesenseof ifwearetounderstandthekeythesesdisputedbyabsolutistsandrelativists.In somecases(e.g.‘plurality’),theregimentinglanguageadmitsofareasonablyfaithful translationbackintoEnglish.Thispermitsustocomefullcircleandinterpret theregimentationviaitsnaturallanguagetranslation,effectivelytreatingtheloose ‘plurality’-talkasellipticalforapluralparaphrasegiveninEnglish.Inothercases
(e.g.‘superplurality’),wherenosuchtranslationisavailable,theintelligibilityissue mustbesettledanotherway.
ThisbookoriginatedinaDPhilthesisentitled‘AbsoluteandRelativeGenerality’, submittedlatein2011andexaminedearlyin2012,whichinturnderivedfromaBPhil thesiswiththesametitle,submittedin2008.Boththeseswerecompletedunderthe primarysupervisionofGabrielUzquiano,thenattheUniversityofOxford.Hardlyany completeparagraphssurviveintactfromthisearlierwork,butmanyoftheleading ideasdo.Thethrustoftheargumentinfavourofexpansionismoverrestrictionism derivesfrommyBPhilthesis,andthematerialonmodalsettheory,andthemodaland schematicformulationsoftheargumentfromindefiniteextensibilitymaybefound, inmoreorlesstheircurrentform,inmyDPhilthesis.Thegenesisandflourishingof theseideasowesaverygreatdealtoGabriel’sinspirationalsupervision.
TheothertwopeopletowhomIowethegreatestintellectualdebtareØystein LinneboandTimWilliamson,whoexaminedmyDPhil.IencounteredØystein’swork onmodalsettheoryshortlyafterIembarkedonmyBPhilthesis,inatalkhegaveata workshopinOxfordinMarch2008entitled‘PluralitiesandSets’.Whilethebimodal theorypresentedinthisbookdiffersinsomeimportantwaysfromtheunimodal theoryØysteindeveloped,theinfluenceofhisapproachshinesthrough.Ontheother sideoftheabsolutism–relativismdebate,Tim’santi-relativist tourdeforce ‘Everything’ isoneofthemostfrequentlycitedworksinthisbook.AndevenifIdon’texpecthim torenounceabsolutismanytimesoon,seekingtoformulateaversionofrelativism thatcouldrespondtosomeofhismoredamagingcriticismsplayedanimportantrole inshapingmyview.
AlongwithGeoffreyHellman,ØysteinwasalsooneoftheOUPReaders.Alex PaseauworkedthroughtheentireDPhilthesisinasingletermasmysecondary supervisor.AndTheaGoodsellandLorenzoRossipainstakinglytrawledthrough themanuscriptofthecurrentwork.Theircriticismsandsuggestionshavegreatly improvedthisbook.Thanksarealsoduetothemanyotherswhohavegenerously givenmetheirhelp,especiallytoSolveigAasen,TomAinsworth,DenisBonnay, HannahCarnegy,RoyCook,VeraFlocke,SalvatoreFlorio,RachelFraser,Peter Fritz,KentaroFujimoto,MarcusGiaquinto,VolkerHalbach,LeonHorsten,Torfinn Huvenes,DanIsaacson,GrahamLeigh,GuyLongworth,BeauMount,CarloNicolai, JonathanPayne,OliverPooley,AgustínRayo,SamRoberts,StefanSienkiewicz,Rob Watt,PhilipWelch,andJuhaniYli-Vakkuri.
2.Russell,Zermelo,andDummett21
2.1Self-reproductiveprocessesandclasses
6.ModalOperators142
7.RussellReductioRedux178
8.HowUniversesExpand214
8.1Theexplanatorychallenge
8.4TheExpansionQuestion
8.5Idealization
8.6Closingsummary
Appendices
Acknowledgements
Thesectionsofthisbooklistedbelowdrawon,orreproduce,materialthatIhave previouslypublishedelsewhere:
•Sections6.2and6.4andAppendicesA.3,B.1–B.2,andC.ReprintedbypermissionfromSpringerNature, JournalofPhilosophicalLogic,Volume42,Issue5, J.P.Studd,‘Theiterativeconceptionofset:(bi)-modalaxiomatisation’,697–725, Copyright©2012,SpringerScienceBusinessMediaB.V.
•Sections3.2,3.5,4.1,and4.3.ReprintedbypermissionfromSpringer,J.P. Studd,‘Absolutegeneralityandsemanticpessimism’in:TorzaA.(ed.) Quantifiers, Quantifiers,andQuantifiers:ThemesinLogic,Metaphysics,andLanguage,SyntheseLibrary(StudiesinEpistemology,Logic,Methodology,andPhilosophy ofScience),Volume373,Springer,Cham,339–66,Copyright©2015,Springer InternationalPublishingSwitzerland.
•Sections2.5,5.1,7.1,and7.3–7.4.ReproducedfromJ.P.Studd,‘Generality, extensibility,andparadox’, ProceedingsoftheAristotelianSociety,Volume117, Issue1;reprintedbypermissionofOxfordUniversityPressandcourtesyofthe EditoroftheAristotelianSociety:©2017.
Ithankthepublishersfortheirpermissiontoreproducethismaterialhere.
AbsolutismandRelativism
Canwetheorizeaboutabsolutelyeverything?Doweeversucceedinbeingmaximally general,insomeinterestinglymaximalsenseof‘maximal’?Itmayseemobviousthat wecananddo.Afterall,Englishisequippedwithquantifierssuchas‘everything’ whichpermitustomakegeneralclaims. Absolutismaboutquantifiers maintains further,withconsiderableplausibility,thatwesometimesusethesequantifiersto makeclaimsthatareasgeneralascanbe:sometimesweuse‘everything’totalk about—quantifyover—absolutelyeverything.
Nonetheless,despitetheobviousappealofabsolutismaboutquantifiers,diverse groundshavebeenforthcomingfortheopposingview, relativismaboutquantifiers. Thisintroductorychapteraimstogiveanoverviewoftheabsolutegeneralitydebate andtosetthesceneforthedefenceofrelativismlaterchapterspursue.Section1.1 elaboratesonabsolutism.Wethentakeupsomeofthemainargumentsthathavebeen giveninfavourofrelativism:theargumentfromsortalrestriction(Section1.2),the argumentfrommetaphysicalrealism(Section1.3),andtheargumentfromindefinite extensibility(Section1.4).Twoimportantobjectionsagainstrelativism,andanumber ofrelativistresponses,followinSection1.5andSection1.6.
1.1Absolutism
Toafirstapproximation, absolutismaboutquantifiers istheviewthatsometimes— whensubjecttonoexplicitortacitrestrictions—quantifierssuchas‘everything’or ∀x rangeoveranabsolutelycomprehensivedomain.1 Thekeynotionstandsinneed ofexplanation.Whatisittoquantifyoveran absolutelycomprehensivedomain?The absolutistmayexpandonhisviewasfollows:2
Toquantifyoveranabsolutelycomprehensivedomainissimplytoquantifyoverabsolutely everythingthereis.‘Absolutelyeverything’meansjustthat:absolutelyeverythingwhatsoever intheentireuniverse.[Thumpingthetable:]NOEXCEPTIONS!
Theabsolutelycomprehensivedomainissimplythedomaincomprisingabsolutelyeverything.Itcontainseveryitemwecantalkabout,inthiscontextoranyother,inadditiontoevery itemspeakersofotherlanguages,naturalorartificial,generalizeoverusingtheirquantifiers.If thereareabstractobjects(suchasnumbersorsets),thesebelongtotheabsolutelycomprehensivedomain;similarly,iftherearetheoreticalobjects(suchaselectronsorquarks)orfictional
1 OurprincipaltemplatesforabsolutismaboutquantifiersareCartwright’s(1994)defenceofspeaking ofeverythingandWilliamson’s(2003)elucidationofgenerality-absolutism.
2 CompareCartwright(1994,p.1)andWilliamson(2003,p.415).
objects(suchasunicornsorhippogryphs)ormerelypossibleobjects(suchasWittgenstein’s possiblechildren),thenthesetoobelongtotheabsolutelycomprehensivedomain.Thesame goes,withoutexception,foreverythingelse.
Threefurtherpreliminaryclarificationsareinorder.First,theabsolutist’sthesisthat wesometimesquantifyoverabsolutelyeverything that thereistellsusnothingabout what thereis.Forinstance,theabsolutistneednotagreewiththeplatonistwhothinks thatthere are abstractobjects,norwiththosewhoposittheoretical,fictional,or merelypossibleobjects.(Notethe‘if’sintheabsolutist’selucidation.)Theabsolutistis freetoadoptasaustereorbloatedanontologyashepleases,solongasheclaimsthat wecanquantifywithoutrestrictionoverabsolutelyeveryitemitcomprises.3
Second,theabsolutist’sthesissaysonlythatwesometimesquantifyoverabsolutely everythingthat thereis.Forinstance,accordingtoaperennialfiction,thereareunicorns.Butthisdoesnotcommittheabsolutisttomaintainingthattheabsolutelycomprehensivedomaincontainsaunicornunlesshemaintains,further,thattherereally are unicorns.Nordoesthiscommithimtotheabsolutelycomprehensivedomain’s containingafictionalunicornunlesshemaintains,further,thatthere are fictional unicorns.Similarly,justbecausethere could beagoldenmountainorthere will one daybehumansonMars,let’sassume,doesn’tmeanthatagoldenmountainorahuman locatedonMarsisintheabsolutelycomprehensivedomain;norneedtheabsolutist maintainthatthisdomaincontainsa possible goldenmountainora futuretime-slice ofahumanlocatedonMarsunlesshemaintains,further,thatthere are suchitems.4
Third,tosaythatwe sometimes quantifyoverabsolutelyeverythingthereisisnot tosaythatwe always do.Forinstance,theEnglishquantifier‘nodonkey’isalways restrictedtorangeoveradomaincomprisingonlydonkeys.Theabsolutistmayalso maintainthatevenwhennorestrictionisexplicitinthesyntaxofthequantifier(as, forexample,in‘everything’,‘everyobject’,or‘everyitem’),itsdomainmaystillbe subjecttorestrictionssuppliedbythecontextofutterance.Imagine,forexample,that aftermuchtoil,havingpainstakinglymadethefinaladjustmentstoherapparatus,a scientistmakesthefollowingutterance,herhandpoisedoverthestartbutton:
(1)Everythingisready.
Ononewidely-heldview,5 thetruthofherutteranceiscompatiblewithagreatmany thingsnotbeingready(theexperimentpencilledin next month,forexample).All thesame,thesenon-readyitemsfailtobecounter-instancestothegeneralclaimshe makesbecausethecontextservestorestricttheoccurrenceof‘everything’inher utteranceof(1)torangeonlyoveritemsrelevanttothetaskathand.
Theoperationofanysortofquantifierdomainrestrictionisperfectlyconsonant withabsolutismprovideditcansometimesbelifted.Theabsolutistneedonlyclaim thatsomelanguagescontainquantifierswhichinsomecontextsrangeoverthe absolutelycomprehensivedomain.It’shelpfultosupposeheadds—ashetypically does—thatEnglishquantifierssuchas‘everything’aresuchquantifiersandthecontext
3 CompareWilliamson(2003,p.423).
4 CompareWilliamson(2003,pp.421–3).
5 Themechanismbehindquantifierdomainrestrictioniscontroversial.SeeSection4.2.
inwhichheexplainshisviewissuchacontext.(Indeed,hemustaddthisifhis elucidationistoachieveitsrequiredgenerality.)6
Furtherclarificationswillbenecessaryinduecourse.Butenoughhasalreadybeen saidtooutlinetheprimafacieappealofabsolutism.Veryofteninscience,philosophy, andeverydaylifeitsufficestoquantifyover less thaneverything.Theenterpriseat handmayonlycallforustogeneralizeabout,say,particlesinthestandardmodel, oragentswithfreewill,orthecontentsofone’sfridge.Butsometimesrestricted generalitydoesn’tseemtobeenough:somestatementsinlogic,settheory,and metaphysicsseemtocryoutforanabsolutelygeneralformulation.
Take,forinstance,mereologicalnihilism.Havingexplainedthatmereologically simplethingsarethosethathavenoproperparts,thenihilistattemptstostateher sparseviewofparthoodwiththefollowingutterance:
(2)Everythingismereologicallysimple.
Wellawareofthepotentialforsucharadicalclaimtoinvitemisunderstanding,the nihilistmaytakepainstoemphasizethatshedoesnotintendheruseof‘everything’ toberestricted.Ifsheismorecharitablyinterpretedtoquantifyoveralimited domain,shedoesn’twantourcharity.Tointerpretherwithanythinglessthanabsolute generalityseemstovitiatethestatementofherview.
AlogicianorsettheoristmightmakesimilareffortstoaccompanyinformalEnglish renderingsofthefollowingtheoremsofpredicatelogic(withidentity)andofset theory:7
(3)Everythingisself-identical.
(4)Everythingisthesoleelementofitssingletonset.
Ineithercase,interpreting‘everything’torangeoveraless-than-absolutelycomprehensivedomainappearstodeprivethetheoremofitsintendedgenerality. Withtheinitialquantifiersorestricted,anutteranceof(3)or(4)failstoruleout thepossibilityofnon-self-identicalthingsorsingletonlessitems outside thelimited domain.Tocapturethesetheoremsintheirintendedgeneralityseemstocall,onthe contrary,forquantificationoveranabsolutelycomprehensivedomain.8
Theprimafaciecaseforabsolutismaboutquantifiersisclear.Allthesame,absolutismhasbeenopposedondiversegrounds.Anti-absolutistargumentsdrawvariouslyonsemantic,metaphysical,andmathematicalconsiderations:(i)advocatesof ‘sortalrestriction’disputetheavailabilityofauniversalsenseof‘thing’;(ii)relativists waryof‘metaphysicalrealism’contendthatabsolutismleadstoobjectionableviews inmetaontology;(iii)friendsof‘indefiniteextensibility’maintainthattheavailability ofanabsolutelycomprehensivedomainisinconflictwiththeopen-endednatureof conceptssuchas set and interpretation
6 CompareCartwright(1994,p.1)andWilliamson(2003,n.1).
7 ComparetheexamplesinCartwright(1994,p.1)andWilliamson(2003,p.416).
8 Assumingdomainsareextensional,thereisatmostoneabsolutelycomprehensivedomain.But,except whenexpoundingtheviewsofabsolutists,wegenerallyeschewtalkof‘theabsolutelycomprehensive domain’toavoidanysuggestionofapresuppositionthatthereisanysuchdomain.
Inmyview,whenproperlydeveloped,considerationsfromindefiniteextensibility providebyfarandawaythemostpowerfulcaseagainstabsolutism.Nonetheless,a fairlybrisksurveyofsomeoftheothermainargumentsagainstabsolutismishelpful inordertobringthisviewandtheprincipalargumentagainstitintosharperrelief. Thenextthreesectionstakeupeachoftheseanti-absolutistargumentsinturn.9
1.2Theargumentfromsortalrestriction
Theargumentagainstabsolutismfromsortalrestrictiontradesonthedistinctive determiner–nominalstructureofquantifiersinnaturallanguage.InEnglish,and manyotherlanguages,quantifiersresultfromcombiningadeterminer(e.g.‘every’, ‘some’,‘no’,‘most’)withanominal(e.g.‘thing’,‘set’,‘donkey’).10 Thenominalservesto delimitthequantifier’srange.Forinstance,‘everydonkey’generalizesaboutdonkeys; ‘everyset’rangesonlyoversets;‘everything’quantifiesoverthings.Absolutismconsequentlycallsforuniversalnominals:inordertocontendthat‘everything’sometimes attainsabsolutegenerality,theabsolutistneedstoclaimthatthenominal‘thing’ appliesindiscriminatelytoanyitemwhatsoever,regardlessofitssort.Theargument fromsortalrestrictionconteststhisclaimonthetwingroundsthataquantifier’s nominalmustbea sortal termandthatnosortaltermisuniversal.11
Advocatesofthesortal–non-sortaldistinctiondifferonexactlywhatittakestobe sortal,buttermslike‘set’,‘cardinalnumber’,‘book’,and‘person’aretypicallytakento beclearcasesofsortalterms;termssuchas‘thing’or‘redthing’areusuallytakentobe clearcasesofnon-sortalterms.12 Ononeprominentview,asortaltermisequipped witha non-trivialcriterionofidentity whichgivesidentityconditionsforitemsofthe relevantsort.Inthecaseoftheterm‘set’,theAxiomofExtensionalityisoftengiven asaparadigmexampleofanon-trivialcriterionofidentity:13
AxiomofExtensionality. Asetisidenticaltoanotherifandonlyiftheyhavethe sameelements.
Theneedfornominalstobeequippedwithnon-trivialcriteriaofidentityis oftenmotivatedinconnectionwithcardinalityquestions.P.T.Geach,forinstance, maintainsthatthere’ssomethingproblematicaboutattributingacardinalnumberto theredthingsinagivenroom:
thetroubleaboutcountingtheredthingsinaroomisnotthatyoucannotmakeanend ofcountingthem,butthatyoucannotmakeabeginning;youneverknowwhetheryouhave countedonealready,because“thesameredthing”suppliesnocriterionofidentity.
(1968,pp.38–9)
AsMichaelDummettputsit,onthisview,inthecaseofanon-sortalsuchas‘red thing’:
9 RayoandUzquiano(2006b)andFlorio(2014a)surveysimilarterrain,alsoconsideringattackson absolutegeneralitybasedonSkolemitescepticism.
10 SeeLewis(1970,p.40)andBarwiseandCooper(1981,pp.161–2).
11 SeeRayoandUzquiano(2006b,sec.1.2.5).
12 See,forinstance,Wallace(1965).
13 SeeGeach(1968),Dummett(1981,ch.16),Wiggins(2001),andLowe(2009).
... itsimplymakesnosensetospeakofthenumberofredthings ... therearesomequestions ‘Howmany?’whichcanonlyberejected,notanswered ... (1981,p.547)
Forsortalrestrictiontothreatenabsolutismaboutquantifiers,itneedstobeargued furtherthatputativeuniversalnominals,suchas‘thing’,‘object’,‘item’,andsoon, eitherfailtobeuniversalorfailtoyieldmeaningfulquantifierswhencombined withdeterminerssuchas‘every’.Tothisend,thesortalistmaymaintainthatdespite functioningsyntacticallyasacountnoun,onaparwith‘set’or‘book’,thenominal ‘thing’andothersupposedlyuniversalnominalsfailtobeequippedwithanon-trivial criterionofidentity.
Facedwithsuchanobjection,manyabsolutists,Isuspect,willbealltoohappyto simplyrejecttheunderlyingmetaphysics.Ifwelackaneffectivemeanstodetermine whetherthisreditemisthesameasonewealreadycounted,wehavenowaytocome toknowhowmanyreditemsoccupytheroom.Butsurelyourignorance,evenif unavoidable,isnobartothereinfactbeing,say,exactly88 itemsintheroomlarge enoughtoreflectlightintheredpartofthevisiblespectrum.Norisitabartothe quantifier‘Exactly88 reditems’beinganon-semantically-defectiveEnglishquantifier. Contemporarysortalistsoftenprefertocastnon-trivialcriteriaofidentityinaless epistemic,moremetaphysicalrole.Onsuchviews,anon-trivialcriterionofidentity for F sneednotgiveanidealizeddecisionprocedurefordeterminingwhetherornot F s areidentical;insteaditservesassomethingapproachingaconceptualanalysisof being thesameF ,aninformativeaccountof whatitis thatmakes F sidenticalordistinct.14
Thedifficultyofthismetaphysicalenquirydependsonhowdemandinganotionof informativenessisinplay.But,onceagain,thelinkbetweenthesuccessofthisenquiry andquantificationovereverymemberoftherelevantsortisfarfromimmediate.Not everymeaningfultermcanbedefinedintermsofmorebasicones.Evenifweaccept tightlinksbetweenquantificationandidentity,whythinkthemeaningfulnessofthe relevantquantifiersturnsontherebeinga non-trivial criterionofidentity?
Thisisnottheplaceforafullevaluationofasortalistmetaphysics.Butit’sworth observingthat,evenforphilosophersgenerallywell-disposedtothisprogramme,two sizeablegapsneedtobefillediftheargumentfromsortalrestrictionistomakeawellsupportedcaseagainstabsolutism.
First,supposingweagreewiththesortalistthatnon-trivialcriteriaofidentityfor F sareneededinordertorendercontentful numericallydefinite quantifierssuchas ‘exactlyone F ’,whythinkthesameisrequiredinthecaseof universal quantifiers, suchas‘every F ’?Theconnectionbetweenidentityandquantificationismuchless apparentinthelattercase.Indeed,evenifwelackthemeanstocountthereditems, asinGeach’sexample,wemaystillbeabletostraightforwardlyobserve,forinstance, thateveryreditemintheroomfailstoexceedacubicmetreinvolume.
Second,assumingthefirstgapcanbefilled,thesortaliststillneedstoestablish thattheabsolutist’sputativelyuniversalnominal—‘thing’,let’ssay—lacksanon-trivial criterionofidentity.It’snotenoughheresimplytoobservethatnocandidatecriterion isimmediatelyapparent.Afterall,thesameistrueforsupposedlyparadigmexamples ofsortaltermssuchas‘person’and‘river’.Moreover,supposingthevarioussorts
14 See,forinstance,Lowe(2009,pp.18–19).SeealsoHorsten(2010).
everything,moreorless
exhaustthecontentsoftheuniverse,andthatweareoptimisticabouttheprospects offraminganon-trivialcriterionofidentityforeachlimitedsort,theabsolutistcan laydownacriterionofidentityfor‘thing’whichisparasiticontheothers:
CriterionofIdentityforThing. Onethingisidenticaltoanotherifandonlyif,for somesort F ,theyareboth F andmeetthenon-trivialcriterionfor F -identity.
Thiscriterionofidentityseemstobeasgoodaprimafaciecandidatefornon-triviality asany;it’scertainlynologicaltruth.Toclosethegap,thesortalistneedstoframea well-motivatedsenseofnon-trivial,anddemonstratethattheproposedcriterionfails tomeetit.
1.3Theargumentfrommetaphysicalrealism
Asecondanti-absolutistargumentconnectstheabsolutism–relativismdebateto issuesinmetaontology.Theabsolutegeneralitydebateisusuallysetupintermsofuniversalquantifiers:‘everything’or ∀x.Inontology,bycontrast,existentialquantifiers— ‘something’or ∃x—cometothefore.Butthedifferenceinfocusissuperficial. Existentialquantifiersachieveabsolutegenerality,iftheydo,inexactlythesameway universalonesdo:namely,byrangingoveranabsolutelycomprehensivedomain. Somerelativistshavearguedagainstabsolutismonthegroundsthatsuchgenerality leadstoanobjectionablepositioninmetaontology.
Tointroducetheobjection,let’srehearseawell-wornexamplefromPutnam(1987a, 1987b).Imaginetwolinguisticcommunitieswhosemembersappeartoespousedifferentviewsaboutmereology(whileretainingthesyntaxofEnglish).Membersofthe firstcommunityhavelongconsideredthemselvesstaunchmereologicalnihilists.The secondcommunity,ontheotherhand,appearstobemadeupofdevoutmereological universalists.ItsmembersupholdthePrincipleofUnrestrictedComposition:15 ‘for anyoneormorethings—nomatterhowscatteredorunrelated—somethingistheir mereologicalfusion’.
Imaginenowthatthetwocommunitiesmeetforthefirsttime,andamember ofthenihilistcommunityattemptstoconveyhermereologicalworldviewtothe universalistswiththeusualsortofnihilistutterance(havingfirstdoneherbestto removeanycontextualrestrictionsonherquantifiers):
(5)It’snotthecasethatsomethingisnon-simple. Theuniversalists’spokespersonrejoins:
(6)Something is non-simple.
Somephilosophersclaimthat,contrarytoappearances,thereisnosubstantivedisagreementbetweenthetwocommunities.Wecanlegitimatelyconceptualizereality withorwithoutnon-trivialmereologicalstructure.Themembersofbothcommunitiesspeaktrulyrelativetotheirlinguisticframework/conceptualscheme/language invirtueofoperatingwithdifferentinterpretationsoftheexistentialquantifier.
15 See,forinstance,Lewis(1986,pp.212–13).
Underthenihilist’sinterpretation,evenwhenshesucceedsinremovingallcontextual restrictions,theexistentialquantifierrangesonlyoversimples,rendering(5)true.The universalist’sutteranceof(6)islikewisetruewithherquantifierinterpretedtoexpress existentialquantificationoverawiderdomain.16
Thisapparentpossibilityofequivocatingonthe(unrestricted)existentialquantifier raisesametaontologicalquestion:arequestionsofthesortthatoccupyfirst-order ontologists—e.g.‘issomethingnon-simple?’—substantivequestionsforwhichthe worldsuppliesadefiniteframework/scheme/languageindependentanswer?
Somerelativists,apparentlyendorsinganegativeanswertothisquestion,argue againstabsolutismonthegroundsthatitiscommittedtoapositionthatsustainsa positiveanswer.CharlesParsonswrites:
Whatseemstomeapotentialproblem[forabsolutism]isthatifourquantifierscanreally captureeverythinginsomeabsolutesense,thensomeformofwhatHilaryPutnamcalls ‘metaphysicalrealism’seemstofollow.AsIunderstanditthatisthatthereissomefinalanswer tothequestionwhatobjectsthereareandhowtheyareindividuated.(2006,p.205)
GeoffreyHellmandeployssimilarconsiderationsinoneofhisargumentsagainst absolutism:
Theabsolutistmustinsistthat atmostoneoftheframeworksiscorrect,theone(ifany)that quantifiesoveronlythoseobjectsintherangeoftheabsolutequantifiers,theobjectsthat‘really exist’(‘REALLYEXIST ’?).(2006,p.87,emphasishis)
Inaslogan:absolutismimpliesmetaphysicalrealism.
Isacommitmenttoaspeciesofrealismofthiskindaproblemforabsolutism?One obviouspointhereisthatthedialecticaleffectivenessofthisconsiderationissensitive totheambientmetaphysics.Manyabsolutists,Isuspect,willfindametaontological outlookwheretheabsolutedomainservesastheultimatearbiterofontological questionsquitecongenial.
Second,andmoreimportantly,areParsonsandHellmanrighttomaintainthat absolutismimpliesmetaphysicalrealism?Toanswerthisquestion,weneedabetter griponsomeofthekeymetaontologicaltermsofart.What’ssupposedtobe‘final’ aboutthe‘finalanswers’toontologicalquestionssoughtbythemetaphysicalrealist (accordingtoParsons)?Whatistherelevantsenseof‘reallyexist’(orindeed‘REALLY EXIST ’)inHellman’sformulation?ThemetaontologyliteratureaboundswithtechnicaltermsandmetaphorsusedtodescriberealistattitudesofthekindParsonsand Hellmanseemtobedrivingat.Thefinalanswerastowhat‘reallyexists’isgivenusing ‘thequantifierthatGodwoulduse’,theonewhoseinterpretationis‘metaphysically privileged’;substantiveontologyisconductedusingquantificationthatbest‘carves natureatthejoints’orispartofthe‘fundamentalstructureofreality’.17
Thetermsofartpermitustorephraseourquestioninmoreevocativeterms: whythinkthatabsolutistsarecommittedtoametaphysicallyprivileged/joint-carving existential-quantifier-interpretation(thatGodwoulduse)?Theonlyapparentanswer
16 See,forinstance,Carnap(1950),Putnam(1987a,1987b)andHirsch(2002,2005,2009).
17 ThefirsttwoarefromHirsch(2002,p.61),thesecondtwofromSider(2011,pp.1,3).
isbecausetheabsolutistiscommittedtoattachingthisspecialstatustotheexistentialquantifier-interpretationthathetakestoachieveabsolutegenerality:
BiggestisBest. Ifthereisanabsolutelygeneralexistential-quantifier-interpretation, itistheuniquemetaphysicallyprivileged/maximallyjoint-carvingexistentialquantifier-interpretation.
ThisassumptionisclearlypresentinthepassagefromHellmanquotedabove.And assumingthatthebiggestinterpretationreallyis(metaphysically)best,theParsons–Hellmanimplicationfromabsolutismtometaphysicalrealismwouldseemtobean immediatecorollary.
ButshouldweaccepttheBiggestisBestassumption?Theassumptionclearlyhas someintuitivepull.Grantedtheavailabilityofanabsolutelycomprehensivedomain, whymakedowithaless-comprehensiveone?Toframemetaphysicaltheoriesby quantifyingoverthesmallerdomainmayseemto‘ignore’itemsthatareavailable tobequantifiedover.
Theusefulnessofpre-theoreticintuitionsinsuchdeepmetaphysicalterritory, however,islimited.Tomakefurtherprogressweneedtounpackthecrucialterms ofart.Hereweconfrontsomethingofanironyinthemetaontologydebate:intheir disputeoverwhethertheparticipantsinfirst-orderontologicaldisputesequivocateon theexistentialquantifier,metaontologistsoftenseemtoattachverydifferentmeanings totheirpreferredtermfor‘metaphysicallyprivileged’/‘joint-carving’.Significantly, however,prominentfiguresonbothsidesofthedebateelucidatethecrucialmetaontologicaltermsinwaysthatcallintoquestiontheBiggestisBestassumption.
EliHirsch(2002)isoneprominentopponentofmetaphysicalrealism.Heinstead advocates quantifiervariance,whichdeniesthatthereis‘onemetaphysicallyprivileged senseofthequantifier’(p.61).Onhisview,themarkofmetaphysicaldistinctionis expressivepower:anon-metaphysically-privilegedquantifier-interpretation‘would leaveuswithoutadequateresourcestostatethetruthproperly’(p.61).Heelaborates ontherelevantresourcesinglossingthe‘basicidea’ofhisview:
thebasicideaofquantifiervariancecanbenicelyformulatedbysayingthatthesame (unstructured)factscanbeexpressedusingdifferentconceptsof“theexistenceofathing”,that statementsinvolvingdifferentkindsofquantifierscanbeequallytruebyvirtueofthesame (unstructured)factsintheworld.(p.59)
Hirsch’sfocuson unstructured factsandpropositionssetstheexpressivebarcomparativelylowbypermittingsentenceswithradicallydifferentsyntacticstructures toexpressthesameproposition.Ontheunstructuredaccount,sentencesutteredin differentlanguages(andcontexts)expressthesamepropositioniftheyaretruein thesamepossibleworlds(asinterpretedaccordingtotheirrespectivelanguagesand contexts).18 Suppose,forinstance,thattheuniversalistmakesamodeststarttothe projectofcataloguingfactsbyuttering:
(7)Somethingisatable.
18 SeeHirsch(2009,p.234);compareHirsch(2002,p.57).
Thenihilistcanplausiblycapturethesame(unstructured)factbyutteringanihilisttranslationof(7),whichonlydeploysquantificationoversimplesbutexpressesthe same(unstructured)proposition:
(8)Somesimplesarearrangedtablewise.
Thistoyexampleisenoughtoshowthatthere’snoobviousconnectionbetweenhow widelyaquantifierrangesandwhetheritsinterpretationcountsas‘metaphysically privileged’onHirsch’saccount.Evenifwesupposethattheuniversalistsquantifyover adomainthatisabsolutelycomprehensive,whydoubtthatthenon-absolutely-general quantificationavailableinthenihilist’slanguagecanmatchtheuniversalist’sinterms ofcoarse-grainedexpressivepower,providedweallowforenoughothernihilistfriendlyexpressiveresources?19 Understanding‘metaphysicalprivilege’intermsof Hirsch’scoarse-grainedexpressivecriterion,then,it’sfarfromclearthattheBiggestis Bestassumptionholds.
Hirsch’saccountofthecrucialmetaontologicalterm,however,isbynomeansthe onlyoption.TheodoreSider,ontheopposingsideofthemetaontologydebate,offers aradicallydifferentaccountofwhatittakesforaquantifiertobe‘joint-carving’or ‘fundamental’(tousesomeofhispreferredterms).SiderfollowsDavidLewis(1983) intakingittobeabrutefactabouttheworldthatithasmetaphysicalstructure. Lewis’saccountpositsthatsomepropertiesaremorenatural,orjoint-carving,than others.Interalia,naturalnessunderwritesobjectivesimilarity.IfwefollowLewis inconceivingofpropertiesliberally,anytwoitems, a and b,howeversimilaror dissimilar,shareinfinitelymanyproperties(e.g. identical-to-a-or-b, identical-to-a-orb-or-pink,andsoon)anddifferoninfinitelymanymore(e.g. identical-to-a, nonidentical-to-b,andsoon).Butsharingnaturalpropertiesmakesforgenuinesimilarity. Twogluonsare, ceterisparibus,moreobjectivelysimilarthanagluonandaglue stickbecausetheproperty gluon sharedbythefirstpairismorenatural—betterjointcarving—thantheproperty gluon-or-glue-stick sharedbythesecondpair.20
Sider(2009,2011)generalizesLewis’saccountofnaturalnesstoalsoapplyto quantifier-interpretations.Onhisaccount,thereisawiderangeofexistentialquantifier-interpretationsavailabletous.Theseincludebothlesscomprehensive nihilist-friendlyinterpretationsthatrender(5)trueandmorecomprehensive universalist-friendlyinterpretationsthatrender(6)true.Nonetheless,itisa fundamentalfactabouttheworld’smetaphysicalstructurethatsomequantifierinterpretationsarebetterjoint-carvingthanothers.21
WhatbecomesoftheBiggestisBestassumptioninthismetaontologicalframework?Despiteitsapparentremotenessfromordinaryenquiry,Sider(2011,sec.2.3) contendsthatwecanfindoutabouttheworld’sfundamentalmetaphysicalstructurebyfollowingafamiliarsetofQuineancriteriafortheorychoice.Weshould acceptthetheorythatdoesbestoverallintermsoftheoreticalvirtuessuchas simplicity,explanatorypower,integrationwithothergoodtheories,andsoon.Sider,
19 ThisstyleoftranslationisduetovanInwagen(1990);seeUzquiano(2004)andSider(2009)for discussionoftherequiredexpressiveresources.
20 SeeLewis(1983,pp.346–7).
21 SeeSider(2009,esp.pp.392,407–8,2011,sec.9.2).
everything,moreorless
however,proposesonecrucialadditiontothismethodologyconcerningtheideology (i.e.primitiveterms)ofourbesttheories:
Agoodtheoryisn’tmerelylikelytobe true.Itsideologyisalsolikelytocarveatthejoints.For theconceptualdecisionsmadeinadoptingthattheory—andnotjustthetheory’sontology— werevindicated;thoseconceptualdecisionsalsotookpartinatheoreticalsuccess,andalso inheritaborrowedluster.SowecanaddtotheQuineanadvice:regardtheideologyofyour besttheoryascarvingatthejoints.Wehavedefeasiblereasontobelievethattheconceptual decisionsofsuccessfultheoriescorrespondtosomethingreal:reality’sstructure.(p.12)
Assumingweacceptthismethodology,it’sfarfromclearweshouldaccepttheBiggest isBestassumption.Sider(2011,ch.13)tentativelysketchesaworldviewwhich eschewsthemorecomprehensiveinterpretationsoftheexistentialquantifierthathe takestobeavailable.Instead,thisviewtakesamorelimitednihilist-friendlyinterpretationtobetheuniqueperfectlyjoint-carvingexistential-quantifier-interpretation. Onthisview,evenifthereisanabsolutelygeneralexistential-quantifier-interpretation availabletous,thelesscomprehensiveinterpretationisbetterjoint-carving.In theorizingabouttheworld’sfundamentalstructure,itmaybebettertoignoremereologicallycomplexobjectsifquantifyingoverthemrequiresustocarvethejointsless well.
DoubtlessthereismuchtoquestioninbothHirsch’sandSider’smetaontology.But we’vewadedintothemetametaphysicsdeepenoughtoseethattheBiggestisBest assumptionisfarfromimmediate.Andwithoutit,there’snoobviousreasontosustain theParsons–Hellmanimplicationfromabsolutismtometaphysicalrealism.
1.4Theargumentfromindefiniteextensibility
Intheprevioustwosections,wefailedtofindacompellingreasontorejectabsolutism aboutquantifiers.Ofcourse,thisbriefdiscussionoftheargumentsfromsortal restrictionandmetaphysicalrealismfarfromrulesouttheirbeingdevelopedinto effectiveanti-absolutistarguments.ButsinceIcanseenopromisingwaytodoso, theseargumentsaresetasideinthechaptersthatfollow.
Theprimaryanti-absolutistargumentthatisdevelopedinthisbookdrawson considerationsofaquitedifferentkind.AtleastsinceRussell(1908)therehasbeena suspicionthatexcessivegeneralityissomehowboundupwithRussell’sparadoxand theotherset-theoreticantimonies.Considerationsofthiskindareinfluentiallytaken upbyMichaelDummett(1981,chs.14–16,1991,ch.24)asthebasisofapopularand powerfulargumentagainstabsolutism.CentraltoDummett’sargumentisthethesis thatsomeconcepts F are indefinitelyextensible:toafirstapproximation,thisistosay thatgivenanydomaincomprising F s,howeverextensive,afurther F canalwaysbe specified,givingrisetoawiderdomain.22
Thissectionconsiderstwoexamplesofputativelyindefinitelyextensibleconcepts: collection and interpretation.It’sworthemphasizingattheoutsetthatourfirstpass presentationoftheargumenthereisnomorethanthat.Wereturntoconsidersome
22 FornowwesetasidesomeofthenuancesinDummett’spresentation.Wereturntohisviewinmore detailinSection2.5.
ofRussell’sandDummett’sargumentsinChapter2,andtooffermypreferredversion oftheargumentfromindefiniteextensibilityinChapter7.
Collection
Startwiththeconcept collection,insomethingapproximatingitspre-theoreticuse. Acollection,intherelevantsense,isanarbitraryextensionalcollectionofzeroor moremembers.Acollectionissaidto comprise certainitems,wheneachofthese items, andnothingelse, 23 isamemberofthecollection.
Tosaythatacollectionis arbitrary istosaythatthereneedbenonon-arbitrary relationbetweenthemembersitcomprises.Themembersofacollectionneednotbe thepropertyofasinglecollector;theyneednotberelevantlysimilarormetaphysically joint-carving;andtheyneednotbespecifiedbyaformulaofaformallanguage orapredicateofanaturalone.ThecollectioncomprisingWeston-super-Mare,the electronthirdclosesttothecentreofmassofthesolarsystem,andmyfavourite ordinalisnolessacollectionthanthecollectionofglassflowersintheHarvard MuseumofNaturalHistoryorthecollectionofnaturalnumbers.
Tosaythatacollectionis extensional istosaythatitisindividuatedaccordingtothe AxiomofExtensionality:acollectionisidenticaltoanotherifandonlyiftheyeach comprisethesamemembers.
Elucidatedinthisway,theconcept collection isaprimecandidatetobeanindefinitelyextensibleconcept.24 Forsupposeweinitiallyquantifyoveradomain D.No matterhowextensive D maybe,weseemabletospecifyacollectionthatisdemonstrablynotamemberof D.The‘new’collectioninquestionisthedomain’s Russell collection,thecollectionthatcomprisesthecollectionsin D whichlackthemselvesas members(i.e.thecollectionofnon-self-memberedcollectionsin D).Letuslabelthis collection rD .AdaptingthereasoningofRussell’sparadox,wecanthenshowthat rD isnotinthedomain D.
Ratherthanleadingtoanoutrightcontradiction,theRussellianargumentbecomes a reductio ontheassumptionthat rD isamemberof D.Indeedtheargumenthas muchincommonwiththeargumentZermelousestoshowthateveryset M hasa subset {x ∈ M : ¬x ∈ x} whichitlacksasanelement.25 Todistinguishtheargument fromRussell’sparadoxproper,let’scallitthe RussellReductio.Itsdemonstrationis straightforward.
therussellreductio
Notefirstthat rD onlycontains non-self-memberedcollections.Consequently,if rD itselfisself-membered,then rD is not amemberof rD —i.e. rD isnon-self-membered. Thissufficestoshow,outright,that rD isanon-self-memberedcollection.
Nowsupposefor reductio that rD is in D.Itfollowsfromourintermediateconclusionthat rD isa non-self-memberedcollection in D.But rD contains all non-selfmemberedcollectionsin D.So, rD isamemberof rD ,i.e. rD isself-membered.
23 Wehenceforthuse‘comprise’inthisexhaustivesense,usuallyleavingthe‘andnothingelse’clause tacit.
24 CompareDummett(1981,pp.530–1,1991,p.317).
25 SeeZermelo(1908,thm.10).Fortheset-theoreticnotation,seeTable2.1inChapter2.
