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1、THEAXIOMATICSTRUCTUREOFEMPIRICALCONTENTCHRISTOPHERP.CHAMBERS,FEDERICOECHENIQUE,ANDERANSHMAYAAbstract.Inthispaper,weprovideaformalframeworkforstudy-ingtheempiricalcontentofagiventheory.Wede nethefalsi -ableclosureofatheorytobetheleastweakeningofthetheorythatmakesonlyfalsi ableclaims.Thefalsi able
2、closureisournotionofempiricalcontent.Weprovethattheempiricalcontentofatheorycanbeexactlycapturedbyacertainkindofaxiom-atization,onethatusesaxiomswhichareuniversalnegationsofconjunctionsofatomicformulas.Ourresultsestablishanexplicitconnectionbetweenthedataoneassumesonemayobserve,andtheempiricalco
3、ntentofthetheory.Wepresentapplicationstostandardrevealedpreferencetheory,andtorecenttheoriesofmultipleselvesfrombehavioraleconomics.Date:July10,2010.ChambersandEcheniquearealiatedwiththeDivisionoftheHumanitiesandSocialSciences,CaliforniaInstituteofTechnology,PasadenaCA91125.Shmayaisaliatedwith
4、theKelloggSchoolofManagementatNorthwesternUniversity,2001SheridanRoad,Evanston,IL60208-2001.WethankNageebAli,YaronAzrieli,BettinaKlaus,TomaszStrzalecki,andLeeatYarivforcomments.ChambersandEcheniqueacknowledgesupportfromtheNSFthroughgrantSES-0751980.Emails:chambers@hss.caltech.edu(Chambers),fede@
5、caltech.edu(Echenique),ande-shmaya@kellogg.northwestern.edu(Shmaya).1Contents1.Introduction11.1.Theconcepts.11.2.Falsi ability,empiricalcontentandfalsi ableclosure.31.3.Axiomatizations.41.4.Partialobservability.61.5.Illustrationoffalsi ableclosure.61.6.Relativetheories.71.7.Jointhypotheses.71.8.
6、Applications.71.9.Previousliterature81.10.Outline102.Theoriesandstructures103.Falsi ableClosure:Semantics124.Syntax164.1.Jointhypotheses204.2.Relativenotions204.3.Aresultonaxiomatizationsusingunobservables235.RelationtoTarski235.1.Datasetsvs.substructures235.2.Tarski'sresultonrelationalsystems24
7、5.3.ThetheoremofLos-Tarski266.Applications276.1.Application:Multipleselvespreferences276.2.Application:Afriat'stheorem337.Othernotionsofrefutability358.Conclusion38AppendixA.Thedualoffalsi ablecompleteness40AppendixB.Basicd
