In The Foundations of Arithmetic and The Basic Laws of Arithmetic, Frege held the view that

January14,2002

InTheFoundationsofArithmeticandTheBasicLawsofArithmetic,Fregeheldtheviewthatnumber-termsrefertoobjects.1Laterinhislife,however,heseemstohavebeenopentootherpossibilities:

Sinceastatementofnumberbasedoncountingcontainsanassertionaboutaconcept,inalogicallyperfectlanguageasentenceusedtomakesuchastatementmustcontaintwoparts,firstasignfortheconceptaboutwhichthestatementismade,andsecondlyasignforasecond-orderconcept.Thesesecond-orderconceptsformaseriesandthereisaruleinaccordancewithwhich,ifoneoftheseconceptsisgiven,wecanspecifythenext.Butstillwedonothaveinthemthenumbersofarithmetic;wedonothaveobjects,butconcepts.Howcanwegetfromtheseconceptstothenumbersofarithmeticinawaythatcannotbefaulted?Oraretheresimplynonumbersinarithmetic?Couldthenumbershelptoformsignsforthesesecond-orderconcepts,andyetnotbesignsintheirownright?2

ToillustrateFrege’spoint,letusconsiderthenumber-statement‘therearethreecats’.Itmightbeparaphrasedinafirst-orderlanguageas:3(1)(∃3x)[Cat(x)].

Ifitslogicalformistobetakenatfacevalue,(1)canbedividedintotwomainlogicalcomponents:first,thepredicate‘Cat(...)’,whichforFregereferstothe(first-order)conceptcat;and,second,thequantifier-expression‘(∃3x)[...(x)]’,whichforFregereferstoasecond-orderconcept(specifically,thesecond-orderconceptwhichistrueofthefirst-orderconceptsunderwhichprecisely3objectsfall).4Significantly,Fregewouldregardneitherofthesecomponentsasreferringtoanobject.

Letusnowconsideraclosecousinof‘therearethreecats’,namely,‘thenumberofthecatsisthree’.Thissentencemightbeparaphrasedas:(2)thenumberofthecats=3.

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Ifitslogicalformistobetakenatfacevalue,(2)cannotbedividedintoapredicateandaquantifier-expression,like(1).Instead,Fregewouldtake‘thenumberofthecats’and‘3’tobenames,referringtonumbers(whichheregardedasobjects).

Fregesawadeepconnectionbetweensentenceslike(1)—inwhichsomethingispredicatedofaconcept—andsentenceslike(2)—inwhichsomethingispredicatedofthenumberassociatedwiththatconcept.Anefforttoaccountforthisconnectionwasamainthemeinhisphilosophyofarithmetic.But,afterthediscoverythatBasicLawVleadstoinconsistency,hefoundmuchreasonfordissatisfactionwithhisoriginalproposal.Asevidencedbythequotedpassage,henolongerfeltconfidentaboutthepossibilityofgettingfromconceptstotheirnumbers‘inawaythatcannotbefaulted’.

Towardstheendofthepassage,Fregeconsidersanalternative:theviewthattherereallyarenonumbersinarithmetic,andthat—appearancestothecontrary—numeralsarenotnamesofobjects.Theydonoteveninstantiatealegitimatelogicalcategory,theyaremerelyorthographiccomponentsofexpressionsstandingforsecond-orderconcepts.Thegrammaticalformofasentencelike(2)isthereforenotindicativeofitslogicalform.Presumably,‘thenumberofthecats=3’istobedividedintotwomainlogicalcomponents.First,theexpression‘...cats’,whichreferstothe(first-order)conceptcat;and,second,theexpression‘thenumberofthe...=3’,whichreferstoasecond-orderconcept(specifically,thesecond-orderconceptwhichistrueofthefirst-orderconceptsunderwhichprecisely3objectsfall).Thenumeral‘3’ismerelyanorthographiccomponentof‘thenumberofthe...=3’,inmuchthesamewaythat‘cat’isanorthographiccomponentof‘caterpillar’.Theoutermostlogicalformof(2)isthereforeidenticaltothatof(1).If,inaddition,itturnsoutthatthelogicalformof‘thenumberofthe...=3’correspondstothatof‘(∃3x)[...(x)]’,thenthelogicalformof(1)isidenticaltothatof(2).

ItisunfortunatethatFregeneverspelledouthisunofficialproposal(asweshallcallit)inanydetail.Inparticular,hesaidnothingabouthowfirst-orderarithmeticmightbeunderstood.Luckily,HaroldHodeshasdevelopedanddefendedaversionoftheUnofficialProposal.5OnHodes’sreconstruction,asentence‘F(n)’ofthelanguageoffirst-orderarithmeticistoberegardedasabbreviatingahigher-ordersentence‘(FX)((∃nx)[Xx])’,where‘(∃nx)[...x]’referstoasecond-orderconcept,and‘(FX)(...X...)’referstoathird-orderconcept.Forinstance,thefirst-ordersentence‘Prime(19)’abbreviatesacertainhigher-ordersentence‘(PrimeX)((∃19x)[Xx])’.

OnHodes’sversionoftheUnofficialProposal,quantifiedsentencesinvolvequantificationoversecond-orderconcepts.Morespecifically,theyinvolvequantificationoverfinitecardinalityobject-quantifiers:thereferentsofquantifier-expressionsoftheform‘(∃nx)[...x]’.6Thus,thefirst-order‘∃zPrime(z)’wouldabbreviatetheresultofreplacingthepositionoccupiedby‘(∃19x)[...x]’in‘(PrimeX)((∃19x)[Xx])’

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byavariablerangingoverfinitecardinalityobject-quantifiers,andbindingthenewvariablewithaninitialexistentialquantifier.Hodes’saccountoffirst-orderarithmeticthereforerequiresthird-orderquantification.Andtheobviousextensiontonth-orderarithmetic(forn≥2)wouldcallfor(n+2)th-orderquantification.Suchlogicalresourcesareincreasinglyproblematic.7

Hereweshallseethatmoremodestresourceswilldo.WewilldevelopaversionoftheUnofficialProposalwithinasecond-orderlanguage,andshowthatitcanbeusedtoaccountfornthorderarithmetic(foranyfiniten).This,initself,isasurprisingresult.Butitisespeciallyimportantinlightofthefactthat,althoughtheuseofhigher-orderlanguagesisoftenconsideredproblematic,recentworkhasdonemuchtoassuageconcernsaboutcertainsecond-orderresources.8WewillalsoseethattheUnofficialProposalhasimportantapplicationsinthephilosophyofmathematics.

1ATransformation

Wewillseethatthereisageneralmethodfor‘nominalizing’arithmeticalformulasassecond-orderformulascontainingnomathematicalvocabulary.Asanexample,consider‘Thenumberofthecatsisthenumberofthedogs’.Thissentencemightbenominalizedas‘Thecatsarejustasmanyasthedogs’,or:

xˆ[Cat(x)]≈xˆ[Dog(x)],9

where‘≈’expressesone-onecorrespondence.10

Considernowthesentence‘thenumberofthecatsis3’.Itcanbenominalizedas:3f(ˆx[Cat(x)]);

wherenumeral-predicatesaredefinedintheobviousway:•0f(X)≡df∀v¬X(v);

•1f(X)≡df∃W∃v(0f(W)∧¬W(v)∧∀w(X(w)↔(W(w)∨w=v)));•2f(X)≡df∃W∃v(1f(W)∧¬W(v)∧∀w(X(w)↔(W(w)∨w=v)));•etc.

Thissortofnominalizationcaneasilybegeneralized.Inordertodoso,weworkwithinatwo-sortedsecond-orderlanguageLcontainingthefollowingvariables:first-orderarithmeticalvariables,‘m1’,‘m2’,...,monadicsecond-orderarithmeticalvariables‘M1’,‘M2’,...,first-ordergeneralvariables,‘x1’,‘x2’,

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nn

󰀂,....11Weassume...,and,fornapositiveinteger,n-placesecond-ordergeneralvariables󰀁X1󰀂,󰀁X2

thatLhasbeenenrichedwithasinglehigher-levelpredicate‘N’takingamonadicsecond-ordergeneralvariableinitsfirstargument-placeandafirst-orderarithmeticalvariableinitssecondargument-place.12Thewell-formedformulasofLaredefinedintheusualway,withtheprovisothatanatomicformula

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cancontainarithmeticalvariablesonlyifitisoftheform󰀁mi=mj󰀂,󰀁Mimj󰀂or󰀁N(Xi,mj)󰀂.13

Ontheintendedinterpretation,arithmeticalvariablesaretakentorangeoverthenaturalnumbers,

1andgeneralvariablesaretakentohaveanunrestrictedrange.14Inaddition,‘N(Xi,mj)’istruejustin1casethenumberoftheXisismj.Consider‘Thenumberofthecatsisthree’asanexample.Itcanbe

formalizedinLas:

x1[Cat(x1)],m1)∧3(m1));(3)∃m1(N(ˆ

where,again,thenumberpredicatesaredefinedintheobviousway:•0(m)≡df∃W(0f(W)∧N(W,m));•1(m)≡df∃W(1f(W)∧N(W,m));•2(m)≡df∃W(2f(W)∧N(W,m));•etc.15

Arithmeticalpredicatessuchas‘Successor’,‘Sum’and‘Product’caneasilybedefinedintermsof‘N’andpurelylogicalvocabulary.16So,withoutappealingtoarithmeticalprimitivesbeyond‘N’,thewholeofpureandappliedsecond-orderarithmeticcanbeexpressedwithinL.

Itwillbeconvenienttointroducethefollowingdefinitions,whicharecouchedinpurelylogicalvocabulary:

Definition1F(X)≡df

¬∃W(∃w(¬Ww∧∀v(Xv↔(Wv∨v=w)))∧W≈X)

(thereareatmostfinitelymanyXs)

Definition2∃fXφ(X)≡df

∃X(F(X)∧φ(X))

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Ournominalizationmethodcannowbegeneralizedtoencompassthewholeoffirst-orderarithmeticbywayofthefollowingtransformation:17

•Tr(󰀁∃mi(φ)󰀂)=󰀁∃fZi󰀂󰀔Tr(󰀁φ󰀂);•Tr(󰀁mi=mj󰀂)=󰀁Zi≈Zj󰀂;•Tr(󰀁N(Xi,mj)󰀂)=󰀁Xi≈Zj󰀂.

Intuitively,thetransformationworksbyreplacingtalkofthenumberoftheFsbytalkoftheFsthem-selves.Asanexample,letusreturnto‘thenumberofthecatsisthree’.ItcanbeformalizedinLas:

x1[Cat(x1)],m1)∧3(m1));∃m1(N(ˆwhichTrconvertsto:

∃fZ1(ˆx1[Cat(x1)]≈Z1∧3f(Z1));or,equivalently:

3f(ˆx1[Cat(x1)]).

Forfurtherillustration,notethat‘thenumberofthecatsisthenumberofthedogs’canbeformalizedinLas:

∃m1[N(ˆx1[Cat(x1)],m1)∧(N(ˆx1[Dog(x1)],m1)].whichTrconvertsto:

ˆ1[Dog(x1)]≈Z1],∃fZ1[ˆx1[Cat(x1)]≈Z1∧xor,equivalently:

ˆ1[Cat(x1)]≈xxˆ1[Dog(x1)].

Itisworthemphasizingthatmixedidentitystatementssuchas‘mi=xj’arenotwell-formedformulasofL,soourtransformationhasnotbeendefinedforthem.Intuitively,thismeansthatthetransformationisundefinedforsentencesalongthelinesof‘Thenumber2isJuliusCaesar’,whichdonotexpressinternalpropertiesofamathematicalstructure.WecallsuchsentencesCaesarsentences.

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Thisisasitshouldbe.TheviewthatnumbersareobjectsledFregetotheuncomfortablequestionofwhetherthenumberbelongingtotheconceptcatis,forinstance,JuliusCaesar.Butinthecontextofournominalizations,suchquestionsneverarise,becausenumber-termsdonotrefertoobjects.‘Thenumberbelongingtotheconceptcatisthenumberbelongingtotheconceptdog’isnominalizedas‘theobjectsfallingundertheconceptcatareinone-onecorrespondencewiththeobjectsfallingundertheconceptdog’,and‘thenumberbelongingtotheconceptcatis3’isnominalizedas‘therearethreeobjectsfallingundertheconceptcat’.

ThequestionwhetherJuliusCaesaristhenumberbelongingtotheconceptcatisn’tonlyuncomfort-ablebecauseitappearstobenonsensical.ItalsounderscoresaproblemPaulBenacerrafmadefamous,thatifmathematicaltermsrefertoobjects,thennothinginourmathematicalpracticedetermineswhichobjectstheyreferto.18AremarkablefeatureoftheUnofficialProposalisthatitavoidsBenacerraf’sProblemaltogether.Itwould,however,beamistaketoconcludefromthisthattheUnofficialPro-posalisthelastwordonBenacerraf’sProblem,sincetheinscrutabilityofreferencepervadesfarbeyondarithmetic.

2Second-orderArithmetic

OntheassumptionthatthereareinfinitelymanyobjectsintherangeofthegeneralvariablesofL,acertainkindofcodingcanbeusedextendTrsothatitencompassessecond-orderarithmetic(thankshereto...).Intuitively,thecodingworksbyrepresentingeacharithmeticalconceptMibyadyadicrelationRi.Specifically,werepresentthefactthatanumbermjfallsunderMibyhavingitbethecasethatsomeconceptWunderwhichpreciselymjobjectsfallbesuchthatsomeindividualvbearsRitoallandonlytheindividualsfallingunderW.19

Weimplementthecodingbyenrichingourtransformationwiththefollowingtwoclauses:20•Tr(󰀁∃Mi(φ)󰀂)=󰀁∃Ri󰀂󰀔Tr(󰀁φ󰀂);

ˆ[Ri(v,u)])󰀂.u[Ri(v,u)])∧Zj≈u•Tr(󰀁Mimj󰀂)=󰀁∃v(F(ˆ

3Higher-orderArithmetic

Itispossibletoexpressany(non-Caesar)formulainthelangaugeofn-thorderarithmeticasaformulaofLforwhichTrisdefined,providedthattherangeofthegeneralvariablescontainsatleast󰀁n−2manyobjects.

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Considerthecaseofthird-orderarithmetic.Intuitively,weproceedbypairingeachsecond-orderconceptαiwithatriadicrelationSiinsuchawaythatasetofnumbersMjfallsunderαijustincasethereissomeobjectxwiththefollowingproperty:

(∗)Foranynumbern,Mjnholdsjustincasethereissomeobjectysuchthatthereareexactlynvs

satisfyingSi(x,y,v).21

Sothatthe‘empty’second-orderconcept(i.e.thesecond-orderconceptunderwhichnofirst-orderconceptfalls)mayberepresented,weletSirepresentthefactthatMjfallsunderαionlyifthereisanobjectxsuchthatitisboththecasethat(∗)issatisfied,andthatthereisnoysuchthatSi(x,y,x).The‘empty’second-orderconceptcanthenberepresentedbyanyrelationSisuchthatforeveryxthereissomeysuchthatSi(x,y,x).

Formally,if‘αi’isamonadicthird-ordervariablerestrictedtothenaturalnumbers,22wedefineatransformationCasfollows:23

•C(󰀁∃αiφ󰀂)=󰀁∃Si󰀂󰀔C(󰀁φ󰀂)

•C(󰀁αi(Mj)󰀂)=

󰀁∃x[∀y(¬Si(x,y,x))∧∀m(Mjm↔∃y(N(ˆv[Si(x,y,v)],m)))]󰀂

Ontheassumptionthattherangeofthegeneralvariablescontainsleastcontinuummanyobjects,itiseasytoverifythat,foranyformulaofthird-orderarithmetic,φ,onwhichCisdefined,φ↔C(φ).Byusingn-adicrelationsinsteadoftriadicones,thisprocedurecanbeextendedton-thorderarithmetic.And,ontheassumptionthattherangeofthegeneralvariablescontainsatleast󰀁n−2objects,itwillbethecasethat,foranyformulaofn-thorderarithmetic,φ,onwhichCisdefined,φ↔C(φ).

4NumberingNumbers

Onewouldliketobeabletonumbercats.Butonewouldalsoliketobeabletonumbernumbers.Onewouldliketosay,forexample,thatthenumberofprimessmallerthantenisfour.And,unfortunately,anexpressionsuchas‘N(mˆi[Prime-less-than-10(mi)],mj)’isnotwell-formedformulaofLbecause‘N’canonlyadmitofageneralvariableinitsfirstargument-place.24Toremedythesituation,wemaydefineapredicate‘NN(Mi,mj)’,byappealingtothesamesortofcodingasbefore.

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Informally,‘NN(Mi,mj)’istoabbreviateaformulaofLtotheeffectthatthereisabinaryrelationRwiththefollowingproperties:

•Foranynumbern,MinholdsjustincasesomememberofthedomainofRispairedwithexactlyn+1objects;25

•everymemberofthedomainofRispairedwithfinitelymanyobjects;

•foranyxandyinthedomainofR,iftheobjectspairedwithxareasmanyastheobjectspairedwithy,thenx=y;

•thedomainofRcontainsexactlymjobjects.26

Thenewpredicateallowsustosaythatthenumberofprimessmallerthantenisfour.Italsoallowsustosaythatthenumberofprimessmallerthanthreeisthenumberofobjectsfallingundertheconceptcat:

x1[Cat(x1)],m2)).27∃m2(NN(mˆ1[Prime-less-than-6(m1)],m2)∧N(ˆ

And,asdesired,ouranyexpressionoftheform󰀁NN(Mi,mj)󰀂isdefinitionallyequivalenttoawell-formedformulaofL.

5FormulasofLandtheirTransformations

Ournominalizationmethodisnowcomplete.28Caesarsentencesaside,anyformulainthelanguageofn-thorderappliedarithmeticcanbeexpressedasaformulaofLforwhichTrisdefined.AndtheresultofapplyingTrisalwaysaformulawithnomathematicalvocabulary.

Wemaynowgiveageneralcharacterizationoftherelationshipbetweenaformulaanditstrans-formation.Inordertodoso,considerthefollowingfiveprinciples,allofwhichholdontheintendedinterpretationofL:

1.∀X(∃m(N(X,m))→∃!m(N(X,m)))

(IfmisanumberoftheXs,thenmisthenumberoftheXs.)2.∀m∃XN(X,m)

(Givenanynumberm,therearesomeobjectssuchthatmbelongstothoseobjects.)3.∀X(∃m(N(X,m))↔F(X))

(AnumberbelongstotheXsjustincasethetheyareatmostfiniteinnumber.)

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4.∀X∀Y[∀m(N(X,m)→(Y,m))↔X≈Y)].

(AnumberbelongingtotheXsisalsoanumberbelongingtotheYsjustincasetheXsareinone-onecorrespondencewiththeYs.)5.∃X¬F(X)

(Thereareinfinitelymanythingsintherangeofthegeneralvariables)

LetAbetheconjunctionofthesefiveprinciples,andlet󰀁φTr󰀂beanotationalvariantforTr(󰀁φ󰀂).Itispossibletoshowthat,foranysentenceφofL,29

A󰀐φ↔φTr

where‘󰀐’expressesderivabilityinastandardsecond-orderdeductivesystem.Inordertoprovethisresult,afewpreliminariesarenecessary.Definition3N(Ri,Mj)≡df

u[Ri(v,u)],m))).∀m(Mjm↔∃v(N(ˆ

Definition4Ifmi1,...,mik,Mj1,...,Mjlarearithmeticalvariables,welet

mi1,...,mik,Mj1,...,Mjl

abbreviatethefollowing:

(N(Zi1,mi1)∧...∧N(Zik,mik)∧N(Rj1,Mj1)∧...∧N(Rjl,Mjl)).

Definition5If󰀁φ󰀂isaformulaofL,withfreearithmeticalvariablesmi1,...,mik,Mj1,...,Mjl,welet󰀁φ↔∗φTr󰀂abbreviatetheuniversalclosureofthefollowing:

mi1,...,mik,Mj1,...,Mjl→(φ↔φTr).

If󰀁φ󰀂containsnofreearithmeticalvariables,welet󰀁φ↔∗φTr󰀂be󰀁φ↔φTr󰀂.Finally,weproceedtoourmainresult:

TheoremIf󰀁φ󰀂isawell-formedformulaofL,thenA󰀐φ↔∗φTr.

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Seeappendixforproof.[AninterestingfeatureoftheproofisthatthefifthconjunctofAisrequiredonlytoensuretheadequacyofthecodingforsecond-ordervariablessetforthinsection2.Inparticular,thefifthconjunctisnotrequiredtoproveaversionofthetheoremrestrictedtofirst-orderarithmetic.Ontheotherhand,withoutitsfifthconjunct—or,alternatively,withoutaprincipleguaranteeingtheexistenceofinfinitelyobjectsintherangeofthearithmeticalvariables—thestandardarithmeticalaxiomsdonotfollowfromA.]

Corollary1(CompletenessofAwithrespecttoappliedarithmetic.)If󰀁φ󰀂isasentenceofLandTisthesetoftruesentencesofLwhichdonotcontain‘N’,theneitherA∪T󰀐φorA∪T󰀐¬φ.Proof:Let󰀁φ󰀂beasentenceofL.Itiseasytoverifythat󰀁φ󰀂Trdoesnotcontain‘N’.Therefore,eitherT󰀐φTrorT󰀐¬φTr,sinceeitherφTr∈Tor¬φTr∈T.But,since󰀁φ󰀂containsnofreevariables,itfollowsfromourTheoremthatA󰀐φ↔φTr.So,eitherA∪T󰀐φorA∪T󰀐¬φ.󰀃

Corollary2SupposeAholdswhen‘N(X,m)’isinterpretedas‘thenumberoftheXsism’.Letφ(mi)beawell-formedformulaofL,andletψ(Zi)beTr(φ(mi)).IfthereareatmostfinitelymanyFs,thenφ(mi)istrueofthenumberoftheFsjustincaseψ(Zi)istrueoftheFs.30Proof:Immediatefromtheorem.

6InterpretingSecond-OrderLanguages

Wehavetakencaretoensurethattheoutputsofourtransformationarealwayssecond-orderformulas.Soaninterpretationforsecond-orderquantifiersisallweneedtomakesenseofournominalizations.Fregetooksecond-orderquantifierstorangeoverconcepts,butFregeanconceptsmightbeconsideredproblematiconthegroundsthattheyconstitute‘items’whicharenotobjects.

Notanyalternativewilldo.OnQuine’sinterpretation,second-orderlogicis‘set-theoryinsheep’sclothing’.Sowewouldhavesucceededineliminatingnumber-termsfromarithmeticonlybymakinguseofset-terms.And,fromtheperspectiveoftheUnofficialProposal,set-termsarepresumablynolessproblematicthannumber-terms.Norisanyprogressmadebyinterpretingsecond-orderlogicasBooloshassuggested.31Someofourdefinitionsmakeessentialuseofpolyadicsecond-orderquantifiers,whichBoolostreatsasranging(plurally)overorderedn-tuples.And,again,fromtheperspectiveoftheUnofficialProposal,ordered-pair-termsarepresumablynolessproblematicthannumbers-terms.SomedeviousnessisneededtoavoidFregeanconceptswithoutbetrayingthespiritoftheUnofficialProposal.Onewayofdoingsoisbydefiningsecond-orderquantifiersimplicitly,intermsofanopen-10

endedschema,asinMcGee’s‘Everything’.Anotherisbyinterpretingsecond-orderlogicasinRayoandYablo’s‘NominalismthroughDe-Nominalization’.Alternatively,onemightarguethatgenuinesecond-orderquantificationistobeacceptedasaprimitive.

7Applications

Frege’sUnofficialProposal—theviewthatnumber-statementsaretobeeliminatedinfavoroftheirtrans-formations—cantakeseveraldifferentforms,dependingonthesortofeliminationonehasinmind.OnanapproachlikeHodes’s,number-statementsaretakentoabbreviatetheirtransformations.Asaresult,number-termsdonotrefertoobjects,andthereisroomforrejectingtheexistenceofnumbersaltogether.TheUnofficialProposalmightthereforeprovideabasisforanominalistphilosophyofarithmetic.Itshouldbenoted,however,thatunlesstheuniverseisinfinite,φTrwillnotalwayshavethetruth-valuethatφreceivesonitsstandardinterpretation.Inordertoavoidinfinityassumptions,anominalistmightclaimthatanumber-statementφabbreviates‘necessarily,(ξ→φTr)’,where‘ξ’isasentencestatingthatthereareinfinitelymanyobjects,suchas‘∃X¬F(X)’.Ontheplausibleconditionthatitispossiblefortheuniversetoinfinite,‘necessarily,(ξ→φTr)’istrueifandonlyifφistrueonitsstandardinterpretation.32

AdifferentapproachtowardstheUnofficialProposalmightservethepurposesoftheNeo-FregeanProgram,championedbyBobHaleandCrispinWright.Neo-FregeansbelievethatHume’sPrincipleallowsustoreconceptualizethestateofaffairswhichisdescribedbysayingthattheFsareasmanyastheGs,andthat,onthereconceptualization,thatsamestateofaffairsisrightlydescribedbysayingthatthenumberoftheFsisthenumberoftheGs.33AversionoftheUnofficialProposalmightallowNeo-Fregeanstomakethemoregeneralclaimthateverynumber-statementφdescribes—ontheappropriatereconceptualization—thestateofaffairswhichisotherwisedescribedbyφTr.

EveniftheUnofficialProposalistobeabandonedaltogether,itwouldbeamistaketoneglecttheconnectionbetweennumber-statementsandtheirtransformationsdescribedinsection5.Fornon-nominalistaccountsofmathematicsmustyieldtheresultthatthereisnospecialmysteryabouthowonemightcometoknowwhatthetruth-valuesofmathematicalsentencesare.But,ontheassumptionthatAcanbeknowntobetrue,ourtheoremensuresthatthisgoalcanbeachievedforthecaseofpureandappliedarithmetic.LetφbeanarithmeticalsentenceofL.WhenAisknown,itfollowsfromourtheoremthatoneisinapositiontoderiveφ↔φTr.So,insofarasoneisinapositiontoknowthetruthofφTr,whichcontainsnoarithmeticalvocabulary,oneisalsoinapositiontoknowthetruthofφ.34(Of

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course,onemaynotbeinapositiontoknowthetruthofφTr.Inthatcaseoneisnot,forallthathasbeensaid,inapositiontoknowφ.Butthatcannotbeusedasanobjectionagainstanon-nominalistaccountofmathematicalknowledge.Suchanaccountisrequiredtoshowthatmathematicalknowledgeisnomoremysteriousthannon-mathematicalknowledge,notthatallknowledgeisunproblematic.)

8Logicism

OurtheoremprovidesuswithapartialvindicationofLogicism.Forwheneverφisasentenceofpurearithmetic(appropriatelyexpressedinL),φTrisasentenceofpuresecond-orderlogic.Moreover,Trallowsustoexpressformulasofpurearithmeticasformulasofpuresecond-orderlogicinawaywhichpreservescompositionality.35ThiswouldconstituteacompletevindicationofLogicismifitweretrueasamatterofpurelogicthat,foreveryappropriateφ,Tr(φ)hasthetruth-valuethatφwouldreceiveonitsstandardinterpretation.Unfortunately,thegeneralequivalenceintruth-valueholdsonlyiftheuniverseisbigenough,andthesizeoftheuniverseisnotamatterofpurelogic.Trdoesn’treducearithmetictologic—butitcomesclose.

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Appendix

Thetheoremisprovedbyinductiononthecomplexityof󰀁φ󰀂.Trivialcasesareomitted.

•Assume󰀁φ󰀂=󰀁N(Xi,mj)󰀂.Then󰀁φ↔∗φTr󰀂istheuniversalclosureof

N(Zj,mj)→(N(Xi,mj)↔Xi≈Zj),

whichisanimmediateconsequenceofA(firstandfourthconjuncts).•Assume󰀁φ󰀂=󰀁mi=mj󰀂.Then󰀁φ↔∗φTr󰀂istheuniversalclosureof

(N(Zi,mi)∧N(Zj,mj))→(mi=mj↔Zi≈Zj),

whichisanimmediateconsequenceofA(firstandfourthconjuncts).•Assume󰀁φ󰀂=󰀁Mjmi󰀂.Then󰀁φ↔∗φTr󰀂istheuniversalclosureof

mi,Mj→(Mjmi↔∃v(F(ˆu[Ri(v,u)])∧Zj≈uˆ[Ri(v,u)])).

Wemakethefollowingtwoassumptions:

N(Zi,mi),N(Rj,Mj).

Recallthat(2)isshorthandfor

∀m(Mjm↔∃v(N(ˆu[Ri(v,u)],m))),

fromwhichitfollowsimmediatelythat

(Mjmi↔∃v(N(ˆu[Ri(v,u)],mi))).

From(1)and(4),togetherwithA(firstandfourthconjuncts),itfollowsthat

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(1)(2)

(3)

(4)

Mjmi↔∃v(Zj≈uˆ[Ri(v,u)]).

Andfrom(1)and(5),togetherwithA(first,thirdandfourthconjuncts),itfollowsthat

(5)

Mjmi↔∃v(F(ˆu[Ri(v,u)])∧Zj≈uˆ[Ri(v,u)]).

Dischargingassumptions(1)and(2)weget:

(6)

ˆ[Ri(v,u)])).mi,Mj→(Mjmi↔∃v(F(ˆu[Ri(v,u)])∧Zj≈u

Andthedesiredresultfollowsfrom(7)byuniversalgeneralization.

(7)

•Assume󰀁φ󰀂=󰀁∃miψ(mi)󰀂.Letψhavefreearithmeticalvariablesmi1,...,mik,Mj1,...,Mjl

distinctfrommi.36Then󰀁φ↔∗φTr󰀂istheuniversalclosureof:

mi1,...,mik,Mj1,...,Mjl→(∃miψ(mi)↔∃fZiψTr(Zi)).

Byinductivehypothesis,thefollowingisprovablefromHP:

mi,mi1,...,mik,Mj1,...,Mjl→(ψ(mi)↔ψTr(Zi)).

Wemakethefollowingtwoassumptions:

(1)

mi1,...,mik,Mj1,...,Mjl,

(2)

∃miψ(mi).

ByA(secondandthirdconjuncts),itfollowsfrom(3)that

(3)

∃mi∃W(F(W)∧N(W,mi)∧ψ(mi)).

So,byexistentialinstantiation,

(4)

14

F(C)∧N(C,c)∧ψ(c).

Butby(1)wehave:

c,mi1,...,mik,Mj1,...,Mjl→(ψ(c)↔ψTr(C)).

Andfrom(2),(5)and(6)wemayconclude

ψTr(C).

Thus,makingagainuseof(5),

∃fZiψTr(Zi),

and,dischargingassumption(3),

∃miψ(mi)→∃fZiψTr(Zi).

Conversely,assume

∃fZiψTr(Zi).

Byexistentialinstantiation:

F(C)∧ψTr(C).

Itisaconsequenceof(10)andA(thirdconjunct)that

∃mN(C,m).

From(12)weobtainthefollowing,byexistentialinstantiation:

N(C,c).15

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

Butby(1)wehave:

c,mi1,...,mik,Mj1,...,Mjl→(ψ(c)↔ψTr(C)).

Andfrom(2),thesecondconjunctof(11),(13)and(14)wemayconclude

(14)

ψ(c).

Thus,

(15)

∃miψ(mi),

and,dischargingassumption(10),

(16)

∃fZiψTr(Zi)→∃miψ(mi).

Finally,wecombine(9)and(17),anddischargeassumption(2):

(17)

mi1,...,mik,Mj1,...,Mjl→(∃nmiψ(mi)↔∃fZiψTr(Zi)).

Thedesiredresultisthenobtainedbyuniversalgeneralization.

(18)

•Assume󰀁φ󰀂=󰀁∃Mjψ(Mj)󰀂.Letψhavefreearithmeticalvariablesmi1,...,mik,Mj1,...,Mjl

distinctfromMj.37Then󰀁φ↔∗φTr󰀂istheuniversalclosureof:

mi1,...,mik,Mj1,...,Mjl→(∃Mjψ(Mj)↔∃RjψTr(Rj)).

Byinductivehypothesis,thefollowingisprovablefromHP:

Mj,mi1,...,mik,Mj1,...,Mjl→(ψ(Mj)↔ψTr(Rj)).

Wemakethefollowingtwoassumptions:

(1)

mi1,...,mik,Mj1,...,Mjl,

(2)

16

∃Mjψ(Mj).

ByA(second,thirdandfifthconjuncts),itfollowsfrom(3)that

∃Mj(∃R(N(R,Mj))∧ψ(Mj)).

So,byexistentialinstantiation,

N(P,C)∧ψ(C).

Butby(1)wehave:

C,miTr1,...,mik,Mj1,...,Mjl→(ψ(C)↔ψ(P)).

Andfrom(2),(5)and(6)wemayconclude

ψTr(P).

Thus,

∃RjψTr(Rj),

and,dischargingassumption(3),

∃Mjψ(Mj)→∃RjψTr(Rj).

Conversely,assume

∃RjψTr(Rj).

Byexistentialinstantiation,

ψTr(P).

Thefollowingisalogicaltruth:

17

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

∃M∀m(Mm↔∃v(N(ˆu[P(v,u)],m))).

But(12)isdefinitionallyequivalentto

∃MN(P,M).

So,byexistentialinstantiation,

N(P,C).

Butby(1)wehave:

C,mi1,...,mik,Mj1,...,Mjl→(ψ(C)↔ψTr(P))

Andfrom(2),(11),(14)and(15)wemayconclude

ψ(C).

Thus,

∃Mjψ(Mj),

and,dischargingassumption(10),

∃RjψTr(Rj)→∃Mjψ(Mj).

Finally,wecombine(9)and(18),anddischargeassumption(2):

mi1,...,mik,Mj1,...,Mjl→(∃Mjψ(Mj)↔∃RjψTr(Rj)).

Thedesiredresultisthenobtainedbyuniversalgeneralization.󰀃

18

(12)

(13)

(14)

(15)

(16)

(17)

(18)

(19)

Notes

1

Thisisreflectedinhisdefinitionofnumber.See,forinstanceFrege(1884)§67.

NotesforLudwigDarmstaedter,pp.366-7.Ihavesubstituted‘second-order’for‘second-level’.Asusual,‘(∃1x)[φ(x)]’isdefinedas‘∃x(φ(x)∧∀y(φ(y)→x=y))’,and(forn>1)‘(∃nx)[φ(x)]’is

2

3

definedas‘∃x(φ(x)∧(∃n−1y)[φ(y)∧y=x])’.

4

ForFrege,afirst-orderconceptisaconceptthattakesobjectsasarguments,andan(n+1)th-order

conceptisaconceptthattakesnth-orderconceptsasarguments.SeeFrege(18931903),§21.Unlessotherwisenoted,weshalluse‘concept’tomean‘first-orderconcept’.

5

SeeHodes(1984).SeealsoWright(1983)pp.36-40andBostock(1979),volumeIIchapter1.SeeHodes(1990)§3.

Hodes(1990),observation5,offersanominalizationofsecond-orderarithmeticwhichdoesnot

6

7

exceedtheresourcesofsecond-orderlogic.ButitproceedsbyencodingRamseysentences,andisthereforenotaversionofFrege’sUnofficialProposal.

8

SeeBoolos(1984),Boolos(1985a),Boolos(1985b),McGee(2000)andRayoandYablo(2001).Syntactically,anexpressionoftheform‘ˆx[φ(x)]’takestheplaceofamonadicsecond-ordervariable.

9

Buttheresultofsubstituting‘ˆx[φ(x)]’for‘Y’inaformula‘Ψ(Y)’istobeunderstoodasshorthandfor:

∀W(∀x(Wx↔φ(x))→Ψ(W)).

10

Thatis,‘X≈Y’abbreviates

∃R[∀w(Xw→∃!v(Yv∧Rwv))∧∀w(Yw→∃!v(Xv∧Rvw))]

11

Asaprecautionagainstvariableclashes,wedividemonadicsecond-ordergeneralvariablesintwo:the

11󰀁X2i󰀂—whichweabbreviate󰀁Zi󰀂—willbepairedwithfirst-orderarithmeticalvariables;the󰀁X2i+1󰀂—

whichweabbreviate󰀁Xi󰀂—willbeusedformoregeneralpurposes.Alsotoavoidvariableclashes,we

2dividedyadicsecond-ordergeneralvariablesintwo:the󰀁X2i󰀂—whichweabbreviate󰀁Ri󰀂—willbe

19

22

pairedwithsecond-orderarithmeticalvariables;the󰀁X2i+1󰀂—whichweabbreviate󰀁Ri󰀂—willbeused3formoregeneralpurposes.Finally,wedividetriadicsecond-ordergeneralvariablesintwo:the󰀁X2i󰀂—3whichweabbreviate󰀁Si󰀂—willbepairedwiththird-orderarithmeticalvariables;the󰀁X2i+1󰀂—whichn3

󰀂asaterminological󰀂—willbeusedformoregeneralpurposes.Forn>3,weuse󰀁Riweabbreviate󰀁Ri

n

variantof󰀁Xi󰀂.Wewillsometimesappealtotheintroductionofunusedvariables.Weemploy‘m’

asanunusedfirst-orderarithmeticalvariable,‘w’,‘v’and‘u’asunusedfirst-ordergeneralvariables,‘M’asanunusedsecond-orderarithmeticalvariable,‘W’,‘V’and‘U’asunusedmonadicsecond-ordergeneralvariables,and,foreachn>1(tobedeterminedbycontext),weemploy‘R’asanunusedn-placesecond-ordergeneralvariable.(Itisworthnotingthatappealtounusedvariablescouldbeavoidedbyrenumberingsubscripts.)Itwilloftenbeconvenientregard‘x’,‘y’,and‘z’asarbitraryfirst-ordergeneralvariablesand‘X’,‘Y’and‘Z’asarbitrary(monadic)second-ordergeneralvariables.

12

Foradiscussionofhigher-orderpredicatesseemy....

1

Formally,thewell-formedformulasofLcanbecharacterizedasfollows:(a)󰀁N(Xi,mj)󰀂and

13

󰀁mi=mj󰀂areformulas;(b)foranyn-placeatomicpredicate󰀁P󰀂otherthan‘N’,󰀁P(xi1,...,xin)󰀂isa

nformula;(c)󰀁Mimj󰀂and󰀁Xi(xji,...,xjn)󰀂areformulas;(d)if󰀁φ󰀂and󰀁ψ󰀂areformulas,then󰀁¬φ󰀂,n

φ󰀂areformulas;and(e)nothingelseisaformula.󰀁(φ∧ψ)󰀂,󰀁∃miφ󰀂,󰀁∃Miφ󰀂,󰀁∃xiφ󰀂and󰀁∃Xi14

Moreprecisely,first-orderarithmeticalvariablesaretakentorangeoverthenaturalnumbers,and

first-ordergeneralvariablesaretakentohaveanunrestrictedrange.Therangeofthesecond-ordervariablesistobecharacterizedaccordingly.Forinstance,onaFregeaninterpretationofsecond-orderquantification,second-orderarithmeticalvariablesaretakentorangeoverfirst-orderconceptsunderwhichnaturalnumbersfall,andsecond-ordergeneralvariablesaretakentorangeoverfirst-orderconceptsunderwhicharbitraryobjectsfall.

15

Weusenumber-predicatesratherthannumeralsforthesakeofsimplicity,butitisworthnoting

thatournominalizationcouldbecarriedoutevenifLwasextendedtocontainnumerals.Toseethis,notethat—usingstandardtechniques—anyformulaφoftheextendedlanguagecanbetransformedintoanequivalentformulaφ∗oftheoriginallanguageinwhichnumeralshavebeeneliminatedinfavorofcorrespondingnumber-predicates(definedasabove).Onecanthenidentifythenominalizationofφwiththatofφ∗.

16

Thedefinitionsrunasfollows:

20

•Successor(mi,mj)≡df

∀V∀U[(N(V,mi)∧N(U,mj))→∃u(Uu∧wˆ[Uw∧w=u]≈V)];•Sum(mi,mj,mk)≡df

∀V∀U∀W[(N(V,mi)∧N(U,mj)∧N(W,mk)∧∀w(Vw→¬Uw))→wˆ[Vw∨Uw]≈W];•Product(mi,mj,mk)≡df

∀V∀U∀W[(N(V,mi)∧N(U,mj)∧N(W,mk))→∃R[∀v∀u((Vv∧Uu)→∃!w(Ww∧Rvuw))∧∀w(Ww→∃!v∃!u(Vv∧Uu∧Rvuw))]].

17

Theremainingclausesaretrivial:

•Tr(󰀁¬φ󰀂)=‘¬’󰀔Tr(󰀁φ󰀂);

•Tr(󰀁φ∧ψ󰀂)=‘(’󰀔Tr(󰀁φ󰀂)󰀔‘∧’󰀔Tr(󰀁ψ󰀂)󰀔‘)’;•Tr(󰀁∃xi(φ)󰀂)=󰀁∃xi󰀂󰀔(Tr(󰀁φ󰀂));•Tr(󰀁∃Xi(φ)󰀂)=󰀁∃Xi󰀂󰀔(Tr(󰀁φ)󰀂);•Tr(󰀁Xixj󰀂)=󰀁Xixj󰀂;

nn

(φ)󰀂)=󰀁∃Ri•Tr(󰀁∃Ri󰀂󰀔(Tr(󰀁φ)󰀂);nn

(xj1,...,xjn)󰀂;(xj1,...,xjn)󰀂)=󰀁Ri•Tr(󰀁Ri

•Tr(󰀁xi=xj󰀂)=󰀁xi=xj󰀂;

n

•Tr(󰀁Pnj(xi1,...,xin)󰀂)=󰀁Pj(xi1,...,xin)󰀂.

18

SeeBenacerraf(1965).

WerepresentthefactthatthenumberzerofallsunderMibyhavingitbethecasethatsomeobject

19

bearsRitonothing.Thus,inordertorepresentthefactthatzerodoesnotfallunderMiwemusthaveitbethecasethateveryobjectbearsRieithertonobjectsforsomen>0fallingunderMi,ortoinfinitelymanyobjects.

20

Polyadicsecond-orderquantificationcanbedefinedasmonadicsecond-orderquantificationover

sequences,whichcanbesimulatedwithinfirst-orderarithmetic.

21

21

WerepresentthefactthatthenumberzerofallsunderMjbyhavingitbethecasethatsomeobject

yissuchthattherearenovssatisfyingSi(x,y,v).Thus,inordertorepresentthefactthatzerodoesnotfallunderMjwemusthaveitbethecasethateveryobjectyiseithersuchthatthattherearenvssatisfyingSi(x,y,v)forsomen>0fallingunderMj,orsuchthatthereareinfinitelymanyvssatisfyingSi(x,y,v).

22

Forinstance,onaFregeaninterpretationofthird-orderquantification,‘αi’rangesoversecond-order

conceptsunderwhichfallfirst-orderconceptsunderwhichfallnaturalnumbers.

23

Theremainingclausesaretrivial.

Inanalogywiththeabove,welettheresultofsubstituting‘mˆi[φ(mi)]’for‘Mj’inaformula

24

‘Ψ(Mj)’beshorthandfor

∀M(∀mi(Mmi↔φ(mi))→Ψ(M)).

25

WerequirethatamemberofthedomainofRbepairedwithn+1objectsratherthannobjects

inordertoaccommodatethefactthatthenumberzeromightfallunderMi,sinceeverymemberofthedomainofRmustbepairedwithatleastoneobject.

26

Moreprecisely,‘NN(Mi,mj)’istoabbreviate:

∃R[∀mk(Mimk↔∃w∃W∃u(Rwu∧∀v(Wv↔(Rwv∧v=u))∧N(W,mk)))∧∀w∀v(Rwv→∃fW∀u(Wu↔Rwu))∧

∀w∀v∀W∀V((∃u(Rwu)∧∀u(Wu↔Rwu)∧∀u(Vu↔Rvu)∧W≈V)→w=v)∧∃W(∀v(Wv↔∃u(Rvu))∧N(W,mj))];formkanunusedvariable.

27

Whereas‘Cat(...)’mayberegardedasanatomicpredicate,‘Prime-less-than-6(...)’abbreviates

acomplexformulaconstructedusingthearithmeticalpredicatesdefinedinfootnote16.

28

Sofarwehaveonlybeenconcernedwiththearithmeticoffinitecardinals.Butitisworthnoting

thatasimilartransformationcouldbeappliedtothelanguageofinfinitecardinalarithmetic.

22

29

HereandinwhatfollowsIassumethat,asaprecautionagainstvariableclashes,φcontainsno

variablesfortheform󰀁Zi󰀂,󰀁Ri󰀂or󰀁Si󰀂.

30

Infact,theresultisslightlymoregeneral.Supposeφ(mi1,...,min)isaformulaofLandlet

ψ(Zi1,...,Zin)beTr(φ(mi1,...,min));suppose,moreover,thatthereareatmostfinitelymanyF1s,atmostfinitelymanyF2s,...,andatmostfinitelymanyFns.Thenφ(mi1,...,min)istruewhenmi1isthenumberoftheF1s,mi2isthenumberoftheF2s,...,andministhenumberoftheFnsjustincaseψ(Zi1,...,Zin)istruewhentheZi1saretheF1s,theZi2saretheF2s,...,andtheZinsaretheFns.

31

SeeBoolos(1984)andBoolos(1985a).

Formoreonmodalstrategies,seepartIIofBurgessandRosen(1997).Hodesdiscussesamodal

32

strategyinsectionIIIofHodes(1984).

33

SeeWright(1997),sectionI,andHale(1997).

Foramoredetailedtreatmentofthisissueseemy....Itisworthnotingthatthecompleteness

34

ofthesecond-orderDedekind-Peanoaxiomsyieldsasimilarresultforthecaseofpuresecond-orderarithmetic,andthatthequasi-categoricityresultinMcGee(1997)yieldsasimilarresultforthecaseofpureset-theory.

35

UnlikenominalizationintermsofRamseysentences,Trrespectsthelogicalconnectivesandquan-

tifiers:

•Tr(󰀁¬φ󰀂)=‘¬’󰀔Tr(󰀁φ󰀂),

•Tr(󰀁φ∧ψ󰀂)=Tr(󰀁φ󰀂)󰀔‘∧’󰀔Tr(󰀁ψ󰀂),•Tr(󰀁∃miφ󰀂)=󰀁∃fZi󰀂󰀔Tr(󰀁φ󰀂),•Tr(󰀁∃Miφ󰀂)=󰀁∃Ri󰀂󰀔Tr(󰀁φ󰀂).

36

Thecasewhereψhasnofreearithmeticalvariablesdistinctfrommi,andthecasewhereψdoes

notcontainmifreerequiretrivialdifferencesinterminology.Weignorethemforthesakeofbrevity.

37

ThecasewhereψhasnofreearithmeticalvariablesdistinctfromMj,andthecasewhereψdoes

notcontainMjfreerequiretrivialdifferencesinterminology.Weignorethemforthesakeofbrevity.

23

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25