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1、BasicSetTheoryLX502-SemanticsISeptember11,20081.MotivationWhenyoustartreadingthesenotes,thefirstthingyoushouldbeaskingyourselvesis“WhatisSetTheoryandwhyisitrelevant?”ThoughPropositionalLogicwillproveausefultooltodescribecertainaspectsofmeaning,likethereasoningin(1),
2、itisabluntinstrument.(1)a.JacquessanganddancedTherefore:Jacquessangb.IfJacqueskissesSmurfette,shewillslapJacquesJacqueskissedSmurfetteTherefore:SmurfetteslapsJacquesPropositionalLogicwon’thelpuswiththeargumentin(2),whichiswhywewillextendittoPredicateLogic.(2)Everyma
3、nismortalAristotleisamanTherefore:Aristotleismortal.Ourintuitionstellusthatifthefirstandsecondsentencearetrue,thenthethirdnecessarilyfollows.Thereasoninggoeslikethis.Supposethefirstsentenceistrue:thecollectionofmenispartofthecollectionofmortalthings.Nowsupposethesec
4、ondsentenceisalsotrue:Aristotleisaman.Thefirstsentencesaysthatifathingisinthecollectionofmen,itisalsointhecollectionofmortalthings.SoitfollowsthatAristotleisamortalthing(orsimplymortal),whichiswhatthethirdsentencestates.Thisisrepresentedpictoriallyin(3).(3)Thecollec
5、tionofthingsintheworldthataremortalFidoFredSueAristotleBobThecollectionofthingsintheworldthataremen.(Sentence1tellsustheyareamongthethingsthataremortal.)ThesemanticsofPredicateLogicisdefinedintermsofSetTheory.12.SetsandSetMembershipAbunchofcarrots,aclassroomfullofst
6、udents,aherdofelephants:theseareallexamplesofsetsofthings.Intuitively,youcanthinkofasetasanabstractcollectionofobjects,whichmaycorrespondtothingsinthe1worldortoconcepts,etc.Asetissometimesalsocalledacollection.Theobjectsthatarecollectedinasetarecalleditsmembersorele
7、ments.So,acarrot,astudent,andanelephant,aremembersofthesetsdescribedabove.ASetmayalsocontainothersetsasitsmembers.Ioffernodefinitionofwhatasetisbeyondtheintuitivenotiondescribedabove.Instead,Iamgoingtoshowyouwhatcanbedonewithsets.ThisisatypicalapproachtoSetTheory,i.
8、e.,setsaretreatedasprimitivesofthetheoryandarenotdefinableinmorebasicterms.Iadoptthenotationin(4)forconvenience.(4)a.Capitallettersreprese
