Vagueness, Kant and Topology A Study of Formal Epi

Vagueness,KantandTopologyAStudyofFormalEpistemologyTobepublishedinJournalofPhilosophicalLogicGiovanniBonioloSilvioValentiniDip.diFilosofiaDip.diMatematicaUniversit`adiPadovaUniversit`adiPadovaPiazzaCapitaniato,3viaG.Belzoni,7I–35131Padova,ItalyI–35131Padova,Italye-mail:[e-mail]-mail:[e-mail]roposeanapproachtovaguenesscharacterisedbytwofeatures.Thefirstoneisphilosophical:wemovealongaKantianpathemphasizingtheknowingsubject’sconceptualapparatus.Thesecondoneisformal:tofacevagueness,andourphilosophicalviewonit,weproposetousetopologyandformaltopology.WeshowthattheKantianandthetopologicalfeaturesjoinedtogetherallowusanatypical,butpromising,wayofconsideringvagueness.Keywords:concept,constitution,explication,formaltopology,topology,transcen-dentalism,vagueness.1IntroductionThetopicofvagueconceptsisanoldvexataquaestio.Indeed,itsoriginmaybefoundinthesoritesparadox,whichseemstohavebeenfirstformulatedbyEu-bulidesofMiletustocontrastStoics’logic.Sincethen,vagueconceptshavebeencalledalsoinotherways:‘openconcepts’,‘incompleteconcepts’,‘porouscon-cepts’[Waismann1945],‘polytheticconcepts’[Needham1975],etc.Actually,accordingtosomescholars,therearedifferencesbehindthesenames;butallthiswouldleadtotheconclusionthattheconceptof‘vagueconcept’isitselfrathervague.However,alsothankstotherichWilliamson’stextwhichprovidesbothatheoreticalinsightandahistoricalaccountofvagueness[Williamson1994],intheseyearssuchatopichasbecomethebattlefieldofanalways-increasingnumberofcontenders(cf.[KeefeandSmith1996,Keefe2000,Merricks2001,GraffandWilliamson2001]).Primafacie,somepeoplesuggestadereinterpretationofvagueness,ac-cordingtowhichvaguenessisintheworld(theonticapproach);otherpeopleproposeadedictointerpretationofvagueness,accordingtowhichvaguenessis1inourwayofspeakingabouttheworld(thesemanticapproach);stillothersspurforadeignorantiainterpretationofvagueness,accordingtowhichvaguenessdealswithourlackofknowledge(theepistemicapproach).Inwhatfollows,weproposetomovealongadifferentinterpretativepath,namely,thetranscendentalinterpretationofvaguenness.Evenifitdealssome-howwiththeworld,somehowwiththelanguage,andsomehowwithourig-norance,itdifferssubstantiallyfromtheotherapproachessinceitisendorsedwithaKantianepistemology.Asfarasweknow,inthecontemporarydebateonvaguenessthislineofthoughthasnotbeenconsideredatall,butwethinkthatitleadstopromisingandusefulresultsas,forexample,whenwetacklethesoritesparadoxinsection3.2.Wedonotlimitourselvestoindicateadifferentwayofconsideringvagueness.Weshowthatitsnaturalhomerestsintoatopologicalformalismwhichreallyseemstobeabletoofferapreciseandrigorousexplication,`alaCarnap(cf.[Carnap1950]),ofourKantian,ortranscendental,perspective.SincetheKantianpointofviewisquiteeccentricwithrespecttothedebateonvagueness,ourfirsttaskwillconcernitsnon-formalpresentation(see§2).WhenwewilldiscusssuchaKantianapproach,wewillshowalsohowitdiffersfromtheontic,thesemantic,andtheepistemicpaths.Thenwewillmovetoitsformalisationbymeansoftopologyandformaltopology(see§3and§4).Toconclude,wewishtoemphasisethatwhatwearegoingtoproposehereatanabstractandgenerallevelmightbeapplied,atalessabstractlevel,torealproblems.Forexample,wethinkthatour,bothphilosophicalandmathemati-cal,proposalcanbeusefullyappliedtothedesignofdatabasesand,ingeneral,tomanycasesofknowledgerepresentation.2ThetranscendentalapproachtovaguenessAssaid,weproposeaneccentricapproachtovagueness,wheretheeccentricityhastobeunderstoodwithrespecttothemoreusualapproaches,thatis,theontic,theepistemic,andthesemanticones.Inparticular,wewillsuggestatranscendentalapproach.Weusethewordtranscendental,thatis,atechnicalKantianterm,tomeantheconditionsforthepossibilityofknowledge:“Ientitletranscendentalallknowledgewhichisoccupiednotsomuchwithobjectsaswiththemodeofourknowledgeofobjects”(see[Kant17811-17872],p.59,A11-12/B25).ThisinvolvesthatweassumetheKantian“Copernicanrevolution”accordingtowhichtheprimaryandmainelementofthecognitiveprocessistheknowingsubjectwithhisconceptualapparatusallowinghimtocognitivelygrasptheworldandliveinit.Moreover,beingaKantianmeansacceptingthewell-knowncriticismstotheapodicticyoftheoriginalKant’scategorialsetup,andthereforeacceptingitsliberalisation.Hereitisnotimportantthekindofliberalisationaccepted.Whatisrelevantisthatwestartfromtheideathattheknowingsubjectconstitutestheworldinacognitivelysignificantway.Asitshouldbequiteknown,constitutingtheworlddoesnotmeanatallconstructingtheworld,asKanthimselfhighlightedinthe21787secondeditionofhisKritikderreinenVernunft.TheKantianpositionisnotanidealisticpositionbutarealisticpositionwithrespecttotheworld.So,outoftheknowingsubject’smind,thereistheworld.But,ifwewanttorenderitcognitivelysignificant,wemustimposeourconceptualapparatusonit.Conceptsdonotconstructobjectsatall,insteadconceptsconstituteobjectsbygivingthemcognitivesignificance,andthusinsertingthemintoaclass(cf.[Cassirer1910]).Thisoccursinanaturalway,sincethegivenconstitutionispossibleexactlybecausetheknowingsubjecthasthesuitableconcepts,thatis,sincehehasconceptually