3 The Heap 英文版 悖論 邏輯學(xué)教材

TheParadoxesofEubulidesofMiletus(4thcenturyb.c.)3.TheHeap

(a.k.a.TheBaldMan,a.k.a.TheSoritesParadox)TheParadoxConsiderthisobvioustruth:additionof1kernelofwheatistooinsignificanttoturnwhatisnotaheapofwheatintoaheapofwheatand,similarly,subtractionof1kernelofwheatistooinsignificanttoturnwhatisaheapofwheatintowhatisnotaheapofwheat.TheobvioustruthhastheparadoxicalconsequencesshownbythefollowingSoriticalargument:1grainofwheatisnotaheap.If1grainofwheatisnotaheap,then2grainsofwheatarenotaheap.[bytheobvioustruth]If2grainsofwheatarenotaheap,then3grainsofwheatarenotaheap.[bytheobvioustruth]10,000.If9,999grainsofwheatarenotaheap,then10,000grainsarenotaheap. [bytheobvioustruth]Therefore,10,000grainsofwheatarenotaheap.Theargumentisvalid.(Ineffect,itjustuses10,000applicationsofthetruth-preservinginferencerulemodusponens.)And,itspremisesallseemtobetrue:(1)isclearlytrue;(2)istruebytheobvioustruththatadditionof1littlekernelcan’tturnanon-heapintoaheap;and(3)(10,000)aretrueforthesamereason.But,theconclusionisobviouslyfalse.So,theparadoxhereis,again,aseeminglysoundargumentwithafalseconclusion.TheHeartoftheMatterTheheartoftheHeapparadoxiscapturedinthreeassertionseachofwhichseemstrue:smallchangesdon’tmakeforadifferenceintheapplicationofsoriticalpredicateslike‘...isaheap’or‘...isbald.’largechangesdomakeforadifferenceintheapplicationofsuchpredicates,alargechangeisnothingmorethanalargenumberofsmallchanges.ButastheSoriticalargumentseemstoshow,thetriad(a)-(c)isinconsistent,i.e.,allofitsmemberscannotbetruetogether.FormalizationLet’ssymbolizetheSoriticalargument:Fa1Fa1nFa2Fa2nFa3...n.Fan-1nFan?.?Fan‘F’isthepredicateletterforthesoriticalpredicate—inthiscase‘...isnotaheap’一and‘a(chǎn)」，‘a(chǎn)2’，etc.areindividualconstantsdenotingtheobjectstowhichthepredicateapplies,e.g.,1kernelofwheat,acollectionof2kernelsofwheat,etc.ThegeneralconditionsforaSoritesargumentare:theseriesa「anisorderedconsecutively(e.g.byincreasingnumberofkernels),‘F’satisfiesthefollowingconditions:itistrueofa1itisfalseofaneachadjacentpairintheseries,a.anda.,aresimilarenoughsoasnottodifferinrespectof‘F.’ JJ+[Thus,wecanconstructamoreemotionallycompellingSoritesargumentwith‘F’interpretedas‘isnotaperson’andtheseriesa「andenotingthestagesofembryonicdevelopmentfromzygotetoinfant.]Ul.VaguenessThesourceoftheSoritesparadoxisthesemanticphenomenonofvagueness:aword’sorconcepfshavingagrayareaofapplication,i.e.,borderlineorindeterminatecases,circumstancesinwhichthewordorconceptneitherclearlyappliesnorclearlyfailstoapply.Ifeverywordorconceptwereaspreciseas,say,rationalnumber,whichhasnoborderlinecases,thentherewouldbenoSoritesparadox.Thatis,iftherewereasharpboundaryfor‘.isnotaheap’一anumberksuchthatkgrainsclearlywerenotaheapbutk+1grainsclearlywereaheap—thenoneofthepremisesintheSoritesargumentwouldbefalse.Specifically,thek+1stpremise—FaknFak+1—wouldbefalse,for'Faj(‘kgrainsofwheatisnotaheap’)wouldbetruebut‘Fak+1’(‘k+1grainsofwheatisnotaheap’)wouldbefalse.Remember,aconditionalwithatrueantecedentandafalseconsequentisfalse.So,we’dnolongerhaveasoundargumentwithafalseconclusion.

Remember,aconditionalwithatrueantecedentandafalseconsequentisfalse.But,‘...isnotaheap,’‘...isbald,’etc.certainlydon’tseemtohavesuchsharpboundaries.So,weseemstuckwiththeparadox.ReactionsNaturalLanguageisn’tLogicalOneproposalistodenythatformallogiccharacterizesnaturallanguage—atleastuntilitsvaguepredicatesarereplacedwithpreciseones—forthesymboliclanguagesofformallogicaredeliberatelyprecisewhilenaturallanguagesarelousywithvagueness.ThiswasFrege’sattitudeaswellasRussell’s.Idon’tcareforthisapproach.For,argumentswhosepremisesandconclusionsareexpressedinnaturallanguagecanbevalidornot.Indeed,naturallanguageargumentsconstructedwithvaguepredicatescanbevalidornot.Andvalidityiscertainlywithinthescopeofformallogic.IgnoranceofSharpMeaningsAnotherproposalistoclaimthat,despiteappearances,oneofthepremisesinSoriticalargumentsisreallyfalse.Itfollowsthattherereallyisasharpboundaryforeverysoriticalpredicatelike‘...isaheap’一somespecificnumberwhichsharplydistinguishesheapsfromnon-heaps.Todefendthisidea,letmerecasttheformoftheSoriticalargumentfromaseriesofconditionalpremisestoasingleuniversalpremise:Fa1(VFa1(Vn)(FanFaJn^^nI1??⑶Fa10,000Foranynumbern,ifthemanwithnhairsisbald,thenthemanwithn+1hairsisbald.Therefore,(3)themanwith10,000hairsonhisheadisbald.Theargumentisvalid.Since(1)isundeniablytrueand(3)isundeniablyfalse,avoidingtheparadoxrequiresthatpremise(2)befalse.Therefore:Hence:(馳?(馳?(Fa“nFa“+)Thus:[byQN][bymaterialimplicationandDeMorgan’slaw]Thatis,theredoesexistaparticularnumber,callitk,suchthatthemanwithkhairsonhisheadisbaldbutthemanwithk+1hairsisnotbald.It’sjustthatwedon’t(orcan’t)knowwhatkis.Inotherwords,therereallyarenovaguewordsorconcepts,andwhatseemslikevaguenessisnotasemanticphenomenonbutrathertheresultofourignoranceabouttheprecisemeaningsofourownwordsandconcepts.Idon’tlikethisreplyeither.Languageisourcreation,andthereisn’tmoretoitthanweputinit.So,itisveryimplausiblethatthemeaningsofourtermsshouldbesofarbeyondourknowledge.FuzzySetTheoryandFuzzyLogicThereisanintuitiveconnectionbetweenanobjecfsmembershipinasetandthetruthofasentencethatpredicatesapropertyoftheobject,viz.ifabelongstothesetofF-things—ae{x|xisF}—thenthesentence‘Fa’istrue.Intuitively,then,itseemsthatanyobjectiseitherinoroutofagivenset.Accordingtofuzzysettheory,ontheotherhand,therearean(uncountably)infinitenumberofdegreesofsetmembership,asmanydegreesastherearerealnumbersin[0,1].So,acanbecompletelynon-F,...,alittleF,...,alittlemoreF,...,prettyF,...,quiteF,...,veryF,..,orfullyF.And,therearecorrespondinglymanydegreesoftruth:ifaetodegreen{x|xisF},then‘Fa’istruetodegreen.Forexample,asamanlosesmoreandmorehair,hebecomesmoreandmorebaldandtheassertion‘Sisbald’becomesmoreandmoretrue.Intheory,theconceptofdegreesoftruthimpliesdegreesofvalidity,and,withfurtherdevelopment,thefuzzytheoristhopestoshowthattheconclusionofaSoriticalargumentdoesnotfollowfromitspremiseswithasufficientdegreeofvaliditytobeparadoxical.Iwon’tpursuethedevelopmentofthissolutionbecauseIalreadyfinditenormouslyimplausiblethatourconceptoftruthshouldbesounbelievablyrefinedastoincludeanon-denumerablyinfinitenumberofdistincttruth-values.Truth-ValueGaps.Intheclassicallogicthatwe’vestudied,therearetwo,andonlytwo,truth-values:thetrueandthefalse.Fuzzylogicisanexampleofnon-classical(ordeviant)logicofthesortcalledmany-valued.Alogicoftruth-valuegapsisadifferentsortofnon-classicallogicaltheory.Forexample,if‘71kernelsisnotaheap’isneitherclearlytruenorclearlyfalse,thenithasnotruth-value一thesentenceisatruth-valuegap.Thispromisestoresolvetheparadox,foriftherearepremisesinaSoriticalargumentthatgetnotruth-value,thentheargumentisnolongersound.However,itisnotobviousthattruth-valuegapswillresolvetheparadox.Imaginethatweadoptagappylogicaltheorywheresomesentenceswithvaguepredicates