John MacFarlane - Frege, Kant and the Logic in Logicism

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1、Frege,Kant,andtheLogicinLogicism∗JohnMacFarlane†DraftofFebruary1,20021TheproblemLetmestartwithawell-knownstory.Kantheldthatlogicandconceptualanalysisalonecannotaccountforourknowledgeofarithmetic:“howeverwemightturnandtwistourconcepts,wecouldnever,bythemereanalysis

2、ofthem,andwithouttheaidofintuition,discoverwhatisthesum[7+5]”(KrV:B16).FregetookhimselftohaveshownthatKantwaswrongaboutthis.AccordingtoFrege’slogicistthesis,everyarithmeticalconceptcanbedefinedinpurelylogicalterms,andeverytheoremofarithmeticcanbeprovedusingonlytheb

3、asiclawsoflogic.HenceKantwaswrongtothinkthatourgraspofarithmeticalconceptsandourknowledgeofarithmeticaltruthdependonanextralogicalsource—thepureintuitionoftime(1884:§89,§109).Arithmetic,properlyunderstood,isjustapartoflogic.NevermindwhetherFregewasrightaboutthis.I

4、wanttoaddressadifferentquestion:doesFrege’spositiononarithmeticreallycontradictKant’s?IdonotdenythatFregeendorsed(F)ArithmeticisreducibletologicorthatKantendorsed∗Forcommentsonearlierversionsofthispaper,IamgratefultoaudiencesatUTAustin,UCBerkeley,UCLA,NYU,andPrinc

5、eton,andtoBobBrandom,JoeCamp,SteveEngstrom,AnjaJauernig,ØysteinLinnebo,DorotheaLotter,DanielleMacbeth,LionelShapiro,HansSluga,andtwoanonymousreferees.†DepartmentofPhilosophy,UniversityofCalifornia,Berkeley.E-mail:jgm@uclink.berkeley.edu.1(K)Arithmeticisnotreducibl

6、etologic.1But(F)and(K)arecontradictoriesonlyif‘logic’hasthesamesenseinboth.Anditisnotatallclearthatitdoes.First,theresourcesFregerecognizesaslogicalfaroutstripthoseofKant’slogic(Aris-toteliantermlogicwithasimpletheoryofdisjunctiveandhypotheticalpropositionsaddedon

7、).ThemostdramaticdifferenceisthatFrege’slogicallowsustodefineconceptsusingnestedquantifiers,whileKant’sislimitedtorepresentinginclusionrelations.2Forexample,usingFregeanlogic(inmodernnotation)wecansaythatarelationRisadenseorderingjustincase(D)(∀x)(∀y)(Rxy⊃(∃z)(Rxz&R

8、zy)).But(asFriedman1992hasemphasized)wecannotexpressthisconditionusingthere-sourcesofKant’slogic.3ForKant,theonlywaytorepresentdensenessistomodelitonthe