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TheManyandtheOne

TheManyandtheOne

APhilosophicalStudyofPluralLogic

SALVATOREFLORIOANDØYSTEINLINNEBO

GreatClarendonStreet,Oxford,OX26DP, UnitedKingdom

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ToAnetaandLaurel

I.PRIMITIVEPLURALS

II.COMPARISONS

III.PLURALSANDSEMANTICS

8.4Doesontologicalinnocenceensuredeterminacy?

9.Superplurals

IV.THELOGICANDMETAPHYSICSOFPLURALS

Preface

Plurallogichasbecomeawell-establishedsubject,especiallyinphilosophical logic.Wewanttoexploreitsbroadersignificanceforphilosophy,logic,and linguistics.Whatcanplurallogicdoforus?Aretheboldclaimsmadeonits behalfcorrect?

Differentreadersmaywanttofollowdifferentthreadsrunningthrough thebook.Readersinterestedintheapplicationofplurallogicinphilosophy willfindChapters1,2,and8especiallyrelevant.Wearguethatplurallogic hasusefulapplications,thoughnotallthoseitiscommonlythoughttohave. Next,questionsaboutthecorrectlogicofpluralsarediscussedinChapters1, 2,4,and9–12,wherewedefendanunconventionalview.Werejecttraditionalplurallogicinfavorofa“critical”alternative.Themoststrikingfeature ofthisalternativeisthatthereisnouniversalplurality.Chapters1–3,5, 7,and9discussthesignificanceofplurallogictolinguistics.Advocatesof plurallogicoftenclaimthatlinguisticsemanticsshouldavoid“singularist” prejudicesandbeformulatedtakingpluralsatfacevalue.Wecontestthis claim.

Afewwordsabouttheoriginoftheprojectmaybeappropriate.Both authorshaveforanumberofyearsbeeninterestedinquestionsaboutthe logic,meaning,andmetaphysicsofplurals.Manyoftheideasinthebook werefirstconceivedduringlongrunsalongtheRiverThamesintheperiod 2010–12.Afirstglimpseofthebookprojectaroseinconnectionwiththe course“PluralsinSemanticsandPhilosophicalLogic”taughtatESSLLI2012 inOpole,Poland.

Therearealotofpeopletothank.Thisbookhasbenefitedenormously fromextensivecommentsgivenbyPeterFritz,SimonHewitt,DavidNicolas, AlexOliver,AgustínRayo,SamRoberts,TimothySmiley,EricSnyder,Hans RobinSolberg,andGabrielUzquiano.Forusefulfeedbackanddiscussion, wearealsoindebtedtoColinCaret,AistėČelkytė,EyjólfurEmilsson,Vera Flocke,OlavGjelsvik,NicholasJones,JönneKriener,DanMarshall,Ian Rumfitt,StewartShapiro,SeanWalsh,TimWilliamson,thestudentsinour courseatESSLLI,andtheaudiencesofnumeroustalkswherematerialfrom thebookhasbeenpresented.PeterMomtchiloffhasprovidedinvaluablehelp asaneditor.

WegratefullyacknowledgefundingfromtheEuropeanResearchCouncil (Startinggrantnumber241098),theLeverhulmeTrust,theResearch CouncilofNorway’sToppforskprojectConceptLab,theUniversitiesof Birmingham,London,andOslo,andKansasStateUniversity.

Finally,wewouldliketoexpressourimmensegratitudetoourfamiliesfor theirsupportandpatience.

Acknowledgments

Somepartsofthebookarebasedonpreviouslypublishedwork.Sections 2.1–2.5overlapwithFlorioandLinnebo2018.MostofChapter8derives fromFlorioandLinnebo2016.Chapter10isarevisedandexpandedversion ofLinnebo2016.WethankJohnWileyandSons,Routledge,andTaylor& Francisforpermissiontore-usetherelevantmaterial.

1 Introduction

Englishandothernaturallanguagescontainpluralexpressions,whichallow ustotalkaboutmanyobjectssimultaneously,forexample:

(1.1)Thestudentscooperate.

(1.2)Thenaturalnumbersareinfinite.

Howshouldsuchsentencesbeanalyzed?Inrecentyears,therehasbeen asurgeofinterestinplurallogic,alogicalsystemthattakespluralsat facevalue.Whenanalyzinglanguage,thereisthusnoneedtoeliminate thepluralresourcesofEnglishinfavorofsingularresources.Rather,the pluralresourcescanberetainedasprimitive,notunderstoodintermsof anythingelse.

Plurallogichasemergedasanewtoolofgreatpotentialsignificancein logic,philosophy,linguistics,andbeyond.Whatisthisnewtool,andwhat isitgoodfor?Wewishtoprovideamorenuanceddiscussionthanhasbeen givensofar.

Threequestionsrunthroughourdiscussion.First:

Thelegitimacyofprimitiveplurals

ShouldthepluralresourcesofEnglishandothernaturallanguagesbe takenatfacevalueorbeeliminatedinfavorofthesingular?

Differentconsiderationspullindifferentdirections.Ontheonehand,there isthetremendoussuccessofsettheory,whichshowshowtorepresent manyobjectsbymeansofasinglecomplexobject,namelytheirset.This isapowerfultheory,whichhasproventobeofgreattheoreticalvalue. Whybotherwiththemanywhenwehaveasupremelysuccessfultheoryof complex“ones”?Ontheotherhand,thereisastrongcasefortakingplurals atfacevalue.Englishandmanyothernaturallanguagesallowustotalk aboutthemany,apparentlywithoutanydetourviacomplex“ones”suchas

TheManyandtheOne:APhilosophicalStudyofPluralLogic.SalvatoreFlorioandØysteinLinnebo, OxfordUniversityPress.©SalvatoreFlorioandØysteinLinnebo2021. DOI:10.1093/oso/9780198791522.003.0001

sets.Whynotutilizetheseexpressiveresourcesinoursystematictheorizing? Moreover,attemptstoeliminatethepluralinfavorofthesingularappearto leadtoparadoxes.WeareallfamiliarwithRussell’sparadoxofthesetofall setsthatarenotelementsofthemselves.Whilethissetleadstoparadox,its manyelements—consideredasmany,notone—donot.Itthusappearsto followthattalkaboutthemanyelementscannotbeeliminatedinfavorof talkabouttheirset.Theseconsiderationsencouragetheviewencapsulated inBertrandRussell’strenchantremarkthat“themanyareonlymany,and arenotalsoone”(Russell1903,Section74).

Whileweendupfavoringa“pluralist”view,whichtakespluralresources atfacevalue,thisbooktriestogivetheopposing“singularist”viewafair hearing.Ourreasonsforendorsingpluralismaresomewhatunconventional. Werejectmanyoftheusualargumentsagainstsingularismand,inparticular, arguethatlinguistsareoftenentitledtotheirpredominantlysingularist approach.Weplacegreaterweightonalessfamiliarargumentforpluralism, namelythatprimitivepluralsareofgreatvaluefortheexplanationofsets andsettheory.

Supposeweacceptprimitiveplurals.Thisgivesrisetooursecondoverarchingquestion.

Howprimitivepluralsrelatetothesingular

Whatistherelationbetweenthepluralandthesingular?Weare particularlyinterestedinthecircumstancesunderwhichmanyobjects correspondtoasingle,complex“one”andwhetheranysuchcorrespondencecanshedlightonthecomplex“ones”.

Considerallthestudentsatthenearestuniversity.Presumably,theyarevery many.Itisnatural,however,tothinkthattheyalsocorrespondtovarious singleobjects,suchasasinglegroup,orset,ofstudents.Thequestionthus ariseswhatkindsofsingularizingtransformationsthereareandwhether suchtransformationsmightbeusedtoshedlightontheresulting“ones”. FollowingGeorgCantorandothers,wefinditilluminatingtoexplainasetas anobjectthatissomehow“constituted”byitsmanyelements.Thissuggests anon-eliminativereductionofcertain“ones”tothecorresponding“many”; thatis,weretainthe“ones”asobjectsingoodstandingbutseekanaccount ofthemintermsofthecorresponding“many”.Itisimportanttonoticethat thisnon-eliminativereductionwouldproceedintheoppositedirectionof thesingularists’proposedeliminationofthepluralinfavorofthesingular.

Thus,ourproposalisnottoeliminatethemanybut,onthecontrary,toput themtouseinexplainingcertaincomplex“ones”.

Asiswellknown,however,singularizingtransformationsarefraughtwith danger.IfyouknowCantor’stheorem,youwon’tbesurprisedtolearnthat traditionalplurallogicenablesustoprovethattherearemorepluralitiesof objectsthansingleobjects.(Ifyoudon’tknowthetheorem,don’tworry— itwillbeexplainedinduecourse.)ThisgeneralizationofCantor’stheorem appearstoshowthatitisimpossibleforevery“many”tocorrespondtoa unique“one”.Fortherearemore“manys”thanthereare“ones”!Thisresult appearstolimitseverelywhatsingularizingtransformationscanexist— andthusalsotothreatentheexplanatoryvaluethatsuchtransformations mighthave.

Whenexaminingtherelationbetweenthepluralandthesingular,we faceconflictinglogicalandmetaphysicalpressures.Ontheonehand,the traditionalandmostintuitiveplurallogicseverelyrestrictswhatsingularizingtransformationstherecanbe.Ontheotherhand,suchtransformations areintuitivelyplausibleintheirownrightand(moreimportantly)promise tobeofgreattheoreticalvalue.Howarewetonegotiatetheseconflicting pressures?FollowinganapproachrecentlydefendedbyTimothyWilliamson (2013,2014),werejecta“logicfirst”orientationaccordingtowhichwe first chooseaplurallogicand then requireeveryothertheorytoconformtothis logic.Instead,wearguethatthechoiceofaplurallogicisentangledwith commitmentsinmetaphysics,semantics,andthephilosophyofmathematics.Wemustthereforechoosebetweenvarious“packagedeals”thatinclude notonlyaplurallogicbutalsocommitmentsfarbeyond.

Threesuchpackagedealswillbeexamined.Oneisbasedon generality relativism,whichrejectsthepossibilityofquantificationoverabsolutely everything.Thissurprisingrejectionofabsolutegeneralityhasthebenefit ofreconcilingtraditionalplurallogicwiththeavailabilityofsingularizing transformations.Whenweapplysuchtransformations,therangeofour quantifiersexpandsinawaythatenablesustoavoidparadox.Theothertwo packagedealsholdontoabsolutegeneralitybutdifferonhowtoaddress theconflictingpressuresidentifiedabove.Themorefamiliarversionof absolutegeneralityretainstraditionalplurallogicandthereforelimitswhat singularizingtransformationstherecanbe.Wealsoexplorealessfamiliar versionofabsolutegeneralitywhichismoreliberalconcerningsingularizing transformationsandinsteadrestoresconsistencybydevelopingamore “critical”plurallogic.Inthefinalpartofthebook,wearguethatthefirst

twopackagedealssufferfromanalogousexpressibilityproblemsandshould thereforeberejectedinfavorofthethirdpackagedeal.

Finally,thereisourthirdoverarchingquestion.

Thesignificanceofprimitiveplurals

Whatarethephilosophicaland(morebroadly)scientificconsequences oftakingpluralsatfacevalue?

Theveryfactthatprimitivepluralresourcesareavailableinthoughtand languageisitselfhighlysignificant.Manyrecentwritersonthissubject, especiallyphilosophers,haveclaimedthattherearemajorfurtherconsequencesaswell.Forexample,weencounterclaimstotheeffectthatprimitiveplurals:(i)helpuseschewproblematicontologicalcommitments,thus greatlyaidingmetaphysicsandthephilosophyofmathematics;(ii)ensure thedeterminacyofhigher-orderquantification;and(iii)requireustoreformulatethesemanticsofnaturallanguageusingprimitivepluralsnotonlyin theobjectlanguagebutalsointhemetalanguage.Wearguethattheseclaims areseverelyexaggerated.Whileprimitivepluralsareindeedlegitimateand oftenveryuseful(especiallyfortheexplanationofsets),manyotherdebates areunaffectedbyourchoiceofwhetherornottoacceptprimitiveplurals.In particular,wearguethat(i)theuseofpluralquantifiersincursaformofcommitmentanalogoustoontologicalcommitmentastraditionallyunderstood; (ii)primitivepluralsprovidenoadditionalassuranceofthedeterminacyof higher-orderquantification;and(iii)linguistsare,forthemostpart,fully withintheirrightstocontinueintheirold“singularizing”ways.

Thetitleofourbookmightenticesomereaderswhopondertheancient questionofwhetherrealityisfundamentallyaunityoramultiplicity.Parmenidesfamouslyviewsrealityasaunity,assertingofit:

Norisitdivisible,sinceitisallalike,andthereisnomoreofitinone placethaninanother,tohinderitfromholdingtogether,norlessofit,but everythingisfullofwhatis.Whereforeitiswhollycontinuous;forwhatis, isincontactwithwhatis.(Fragment8,translatedinBurnet1920,262)

Russellvehementlydisagrees:

Academicphilosophers,eversincethetimeofParmenides,havebelieved thattheworldisaunity.[…]Themostfundamentalofmyintellectual beliefsisthatthisisrubbish.Ithinktheuniverseisallspotsandjumps,

withoutunity,withoutcontinuity,withoutcoherenceororderlinessorany oftheotherpropertiesthatgovernesseslove.(Russell1949,98)

WeshallnottakeastandonParmenides’squestionaboutthefundamental natureofreality.Butwefullyendorsetheancientviewthattherelation betweenthemanyandtheoneisofprofoundphilosophicalimportance.As Russellobserves,therearemanyobjects(whetherfundamentalornot).Our discussion—andbooktitle—thereforestartwiththemany.Butasweshall see,therearesomesurprisinglyhardpuzzlesandproblemsconcerningthe relationbetweenthemanyandtheone.Ouranalysisofthesepuzzlesand problemsleadsustoproposeanunconventionalsolution,namelytoreplace thetraditionalplurallogicwithamore“critical”alternative.

I PRIMITIVEPLURALS

2 TakingPluralsatFaceValue

2.1Someprominentviewsofpluralsentences

Manynaturallanguagescontainagrammaticaldistinctionbetweensingular andpluralexpressions.∗ Considertheseexamples:

(2.1)Johnishunting.

(2.2)Thegnusaregathering.

Whenavailable,pluralexpressionscanplayacriticalroleinthoughtand language.Ontheonehand,bygraspingtheirmeaninganddeploying them,weareabletothinkandspeakaboutmanyaswellasaboutone. Forinstance,weareabletosortobjectsintocollectionsandcommunicate importantinformationaboutsuchcollections.Ontheotherhand,plural expressionshavelogicalpropertiesthatgeneratevalidpatternsofreasoning throughwhichweorganizeandextendourknowledgeaboutcollectionsof objects,forexample:

(2.3)(a)Thegnusaregathering.

(b)Thegnusaretheanimalsbeinghunted.

(c)Theanimalsbeinghuntedaregathering.

Thesepatternsofreasoninggobeyondthosestudiedandsystematizedin traditionalfirst-orderlogic,formingthesubjectmatterofanewbranchof logicknownas plurallogic.

FollowingtheleadofGeorgeBoolos’sseminalwork,researchonplurallogichasflourishedinrecentdecades.1Ithasalsobeguntoinfluence

∗ Sections2.1–2.5drawfromFlorioandLinnebo2018.

1 See,e.g.,Boolos1984b,Boolos1985a,Yi1999,OliverandSmiley2001,Rayo2002,Linnebo 2003,Yi2005andYi2006,McKay2006,andOliverandSmiley2016.

TheManyandtheOne:APhilosophicalStudyofPluralLogic.SalvatoreFlorioandØysteinLinnebo, OxfordUniversityPress.©SalvatoreFlorioandØysteinLinnebo2021. DOI:10.1093/oso/9780198791522.003.0002

linguisticsemantics,wherepluralshavereceivedconsiderableattention sincethe1980s.2

Althoughthisfocusonpluralsisarelativelyrecentphenomenon,semanticquestionsconcerningpluralswerealreadyentertainedbythefounders ofmodernlogic.3GottlobFrege,forinstance,addressedthequestionofthe properlogicalanalysisofsentenceswithapluralsubject,suchas:

(2.4)SocratesandPlatoarephilosophers.

Hewrites:

Herewehavetwothoughts:SocratesisaphilosopherandPlatoisa philosopher,whichareonlystrungtogetherlinguisticallyforthesakeof convenience.Logically,SocratesandPlatoisnottobeconceivedasthe subjectofwhichbeingaphilosopherispredicated.

(LettertoRussellof28July1902,inFrege1980,140)

Ineffect,Fregeproposestoeliminatepluralsandanalyze(2.4)as:

(2.5)SocratesisaphilosopherandPlatoisaphilosopher.

However,herealizesthatthisstrategyisn’talwaysavailable.Sentencessuch as(2.6)and(2.7)arenotamenabletotheconjunctiveanalysisproposed for(2.4).

(2.6)BunsenandKirchhofflaidthefoundationsofspectralanalysis.

(2.7)TheRomansconqueredGaul.

Fregeremarks:

Herewemustregard BunsenandKirchhoff asawhole.‘TheRomans conqueredGaul’mustbeconceivedinthesameway.TheRomanshere aretheRomanpeople,heldtogetherbycustom,institutions,andlaws.

(Frege,ibidem)

2 SeeNicolas2008andMoltmann2016forapplicationsofplurallogictolinguisticsemantics. Forsomeresearchinlinguisticsemanticsparticularlyrelevanttoourproject,seeLink1983,Link 1998,Schein1993,Schwarzschild1996,Moltmann1997,andLandman2000.

3 Forhistoricaldetails,seeOliverandSmiley2016,Chapter2.

Elsewhereheexplainsthat,in(2.7),‘theRomans’mustberegardedasa propernamewhoselogicalfunctionistostandforanobject(Frege1980,95).

WhileFregeunderstands“wholes”inbroadlymereologicalterms—an approachtowhichwewillreturnshortly—variousalternatives,suchassets andgroups,havebeensuggestedinthesubsequentliterature.Letusbriefly considertheappealtosets.

ThemostfamousadvocateofthisapproachisWillardVanOrmanQuine. Oneofthesentenceshegrappleswithisknownasthe Geach-Kaplan sentence:⁴

(2.8)Somecriticsadmireonlyoneanother.

AccordingtoQuine,by“invokingclassesandmembership,wecandojustice to[theGeach-Kaplansentence]”(Quine1982,293).Heproposeswhat amountstothefollowinganalysis,or,asheputsit,“regimentation”:⁵

(2.9)Thereisanon-emptysetsuchthateveryelementofthesetisacritic whoadmiressomeoneandeveryonesheadmiresisanelementofthe setotherthanherself.

Quine’suseofsettheorytoeliminatepluralsexposeshimtoanobjection (seeBoolos1984b,440).Considerthefollowingsentence,whichappearsto beaset-theoretictruism:

(2.11)Therearesomesetssuchthatanysetisoneofthemifandonlyif thatsetisnotanelementofitself.

⁴ AsshownbyBoolos,whocreditsDavidKaplan,thereisnocorrectparaphraseofthis sentencescomprisingonlysingularvocabularyandthepredicatesoccurringinit(Boolos1984b, 432–3).

⁵ WereturninSections2.7and3.1toadiscussionoftheimportantQuineannotionof regimentation,whichdiffersfromthefamiliarphilosophicalnotionofanalysis.