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1、WarrenGoldfarbRussell'sReasonsforRamificationiRussellintroducedaformoframificationinhis1906paper"On'Insolubilia'andTheirSolutionbySymbolicLogic."1Thereitisappliedtopropositions.Ex-tendedandsomewhatmodified,ramificationisthecentralcomponentofthethe-oryoftypesasitispresentedin"MathematicalLogi
2、casBasedontheTheoryofTypes"in1908andinPrincipiaMathematicalThatis,Russelldidnotseparatethetheoryoforders,whichembodiestheramificationofprepositionalfunctions,fromthetheoryoftypes.AdisentanglementofthesetwonotionswasfirsturgedbyRamseyin1925,whenheformulatedasimpletheoryoftypes.3Ramificationco
3、uldthen,andonlythen,beseenasasuperpositionofatheoryofordersonabasictype-theoreticstructure.Foroversixtyyearsnow,ramificationhaselicitedanintriguingbipolarreac-tion.ThosewhohavemostsharedRussell'smotivationsinthephilosophyofmathematicshavethoughtitconfusedormisguided(thusRamsey,andalsoCar-nap
4、,Godel,andQuine).4YetmanywhosebasicoutlookisrathermoredistantfromRussell'shavefoundthenotionalluring,havecontinuedtostudyitsnatureandconsequences(althoughofteninsettingsremovedfromthetheoryoftypes),andhavearguedforitsimportanceinfoundationalstudies.5Ramificationofadomainofabstractentitiesist
5、heresultofrequiringthatlegitimatespecificationsofsuchentitiesbepredicative.Brieflyput,aspecifica-tionispredicativeifitcontainsnoquantifierthatrangesoverauniversetowhichtheentityspecifiedbelongs.6(Obviously,wespeakofspecificationsinaninter-pretedlanguage.)Thepredicativityrequirementallowsaspe
6、cificationtolicenseanexistenceclaimonlyiftheentitywhoseexistenceisinferredliesoutsidetheuniversesofthequantificationsinthespecification.Thus,therequirementwillyieldahierarchyofentities:thoseatanygivenlevelofthehierarchyarejustthosethatarespecifiableusingquantifiersthatrangeonlyoverlowerlevel
7、softhehier-archy.Ramificationisjustthisdivisionofentitiesintolevels.Thereisaparticularphilosophicalcastthat,itseems,hastobeputonthena-24RUSSELL'SREASONSFORRAMIFICATION25tureoftheentitiesunderdiscussioninorderforthepredicativityrequirement,andhencef
