TheManyandtheOne:APhilosophicalStudyof PluralLogicSalvatoreFlorio
https://ebookmass.com/product/the-many-and-the-one-aphilosophical-study-of-plural-logic-salvatore-florio/
Instant digital products (PDF, ePub, MOBI) ready for you
Download now and discover formats that fit your needs...
Kant and the Science of Logic: A Historical and Philosophical Reconstruction Huaping Lu-Adler
https://ebookmass.com/product/kant-and-the-science-of-logic-ahistorical-and-philosophical-reconstruction-huaping-lu-adler/
ebookmass.com
Bias: A Philosophical Study Thomas Kelly
https://ebookmass.com/product/bias-a-philosophical-study-thomas-kelly/
ebookmass.com
The Historical and Philosophical Significance of Ayer’s Language, Truth and Logic 1st Edition Adam Tamas Tuboly
https://ebookmass.com/product/the-historical-and-philosophicalsignificance-of-ayers-language-truth-and-logic-1st-edition-adam-tamastuboly/ ebookmass.com
New Feminist Perspectives on Embodiment 1st Edition Clara Fischer
https://ebookmass.com/product/new-feminist-perspectives-onembodiment-1st-edition-clara-fischer/ ebookmass.com
Advanced Financial Accounting Theodore E. Christensen
https://ebookmass.com/product/advanced-financial-accounting-theodoree-christensen/
ebookmass.com
Sports and The Global South: Work, Play and Resistance in Sri Lanka 1st Edition S. Janaka Biyanwila
https://ebookmass.com/product/sports-and-the-global-south-work-playand-resistance-in-sri-lanka-1st-edition-s-janaka-biyanwila/
ebookmass.com
Advances in Parasitology, Volume 117 David Rollinson
https://ebookmass.com/product/advances-in-parasitologyvolume-117-david-rollinson-2/
ebookmass.com
Information Privacy Law (Aspen Casebook Series) 6th Edition, (Ebook PDF)
https://ebookmass.com/product/information-privacy-law-aspen-casebookseries-6th-edition-ebook-pdf/
ebookmass.com
Who Are We Now? Jason Cowley
https://ebookmass.com/product/who-are-we-now-jason-cowley/
ebookmass.com
https://ebookmass.com/product/radiation-therapy-study-guide-aradiation-therapists-review/
ebookmass.com
TheManyandtheOne
TheManyandtheOne
APhilosophicalStudyofPluralLogic
SALVATOREFLORIOANDØYSTEINLINNEBO
GreatClarendonStreet,Oxford,OX26DP, UnitedKingdom
OxfordUniversityPressisadepartmentoftheUniversityofOxford. ItfurtherstheUniversity’sobjectiveofexcellenceinresearch,scholarship, andeducationbypublishingworldwide.Oxfordisaregisteredtrademarkof OxfordUniversityPressintheUKandincertainothercountries
©SalvatoreFlorioandØysteinLinnebo2021
Themoralrightsoftheauthorshavebeenasserted
FirstEditionpublishedin2021
Impression:1
Somerightsreserved.Nopartofthispublicationmaybereproduced,storedin aretrievalsystem,ortransmitted,inanyformorbyanymeans,forcommercialpurposes, withoutthepriorpermissioninwritingofOxfordUniversityPress,orasexpressly permittedbylaw,bylicenceorundertermsagreedwiththeappropriate reprographicsrightsorganization.
Thisisanopenaccesspublication,availableonlineanddistributedunderthetermsofa CreativeCommonsAttribution–NonCommercial–NoDerivatives4.0 Internationallicence(CCBY-NC-ND4.0),acopyofwhichisavailableat http://creativecommons.org/licenses/by-nc-nd/4.0/.
Enquiriesconcerningreproductionoutsidethescopeofthislicence shouldbesenttotheRightsDepartment,OxfordUniversityPress,attheaddressabove
PublishedintheUnitedStatesofAmericabyOxfordUniversityPress 198MadisonAvenue,NewYork,NY10016,UnitedStatesofAmerica
BritishLibraryCataloguinginPublicationData
Dataavailable
LibraryofCongressControlNumber:2020951752
ISBN978–0–19–879152–2
DOI:10.1093/oso/9780198791522.001.0001
Printedandboundby
CPIGroup(UK)Ltd,Croydon,CR04YY
LinkstothirdpartywebsitesareprovidedbyOxfordingoodfaithand forinformationonly.Oxforddisclaimsanyresponsibilityforthematerials containedinanythirdpartywebsitereferencedinthiswork.
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.
