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1、THEPROBLEMOFARTIFICIALPRECISIONINTHEORIESOFVAGUENESS:ANOTEONTHEROLEOFMAXIMALˆCONSISTENCYVINCENZOMARRAAbstract.Theproblemofartificialprecisionisamajorobjectiontoanytheoryofvaguenessbasedonrealnumbersasdegreesoftruth.Supposeyouarewillingtoadmitthat,undersuffici
2、entlyspecifiedcircumstances,apredicationof“isred”receivesaunique,exactnumberfromtherealunitinterval[0,1].Youshouldthenbecommittedtoexplainwhatisitthatdeterminesthatvalue,settlingforinstancethatmycoatisredtodegree0.322ratherthan0.321.InthisnoteIrevisitthepro
3、blemintheimportantcaseofLukasiewiczinfinite-valuedpropositionallogicthatbringstotheforegroundtherˆoleofmaximallyconsistenttheories.Iarguethattheproblemofartificialprecision,ascommonlyconceivedofintheliterature,actuallyconflatestwodistinctproblemsofaverydiffere
4、ntnature.1ThemonadicpredicateP(x):=“xisprime”,interpretedoverthesetofnaturalnumbersx>1,is(absolutely)precise:itsextensionisthesetofprimenumbers;itsanti-extensionisthesetofcompositenumbers;eachnumbereitherbelongstotheextensionofPortoitsanti-extension,butnot
5、toboth;andinprinciplethereisnoissueastowhetheragivennumberbeprimeorcomposite—thoughinpracticeitmaybeimpossibletoascertainwhichisthecaseforanastronomicinstanceofx.Bycontrast,themonadicpredicateR(x):=“xisred”,interpretedoverthesetofallobjects,is(tosomeextent
6、)vague:itsextensionoughttobethesetofallredobjects;itsanti-extensionoughttobethesetofallnon-redobjects;butitmaynotbeclear,eveninprinciple,justwhichobjectsdoqualifyasred,andwhichasnon-red—thinkofapeculiartintattheborderlinebetweenredandpink.arXiv:1306.4369v1
7、[math.LO]18Jun2013Istherealogicofvaguepredicates—or,forthatmatter,ofvaguepropositions?Anydefiniteanswer,atpresent,islikelytobecontentious.Thephilosophicalliteratureonvaguenessisrelativelylarge;accountsofthemaincompetingtheoriesmaybefoundin[Wil96,Kee00,Sha06
8、,Smi08].OneclusteroftheoriesisbasedontheassumptionthatanyinstantiationofavaguepredicateRbyaconstantcwhosedenotatumlies,intuitively,attheborderlinebetweentheextensionofRanditscomplement,isonlytruetoadegree.Thu
