15ArgumentsValidityandContradictionValidity13 Firstapproximation Whenanargumentisvalidthetruthofthepremissesguaranteesthetruthoftheconclusion10 Anargumentisvalidifitcant bethecasethatallofth ID: 401594
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Outline(1)Introductory(2)Validity(3)CourseOverview(4)SetsandRelations 1.5Arguments,Validity,andContradictionValidity1/3 Firstapproximation. Whenanargumentisvalid,thetruthofthepremissesguaranteesthetruthoftheconclusion.10 Anargumentisvalidifit`can't 'bethecasethatallofthepremissesaretrueandtheconclusionisfalse. Validitydoesnot dependoncontingentfacts. Validitydoesnot dependonlawsofnature. Validitydoesnot dependonthemeaningsofsubject-speci cexpressions. Validitydependspurelyonthe`form'oftheargument. 1.5Arguments,Validity,andContradictionExamples Argument1 Notvalid Zenoisatortoise.Therefore,Zenoistoothless. Thetruthofthepremissdoesnotprovideasu cientlystrongguaranteeofthetruthoftheconclusion Argument2 Valid Zenoisatortoise.Alltortoisesaretoothless.Therefore,Zenoistoothless. 1.5Arguments,Validity,andContradictionValidity2/3 Characterisation(p.19) Anargumentislogicallyvalidifandonlyif:thereisno interpretationunderwhich:(i)thepremissesarealltrue,and(ii)theconclusionisfalse. 1.5Arguments,Validity,andContradictionArgument2revisited Argument2 Valid Zenoisatortoise.Alltortoisesaretoothless.Therefore,Zenoistoothless. Argument2a Valid TheresaMayisaConservative.AllConservativesareLiberalDemocrats.Therefore,TheresaMayisaLiberalDemocrat. Argument2b Valid Radonisanoblegas.Allnoblegasesarechemicalelements.Therefore,Radonisachemicalelement. Note:argument2aisavalidargumentwithafalseconclusion. 1.5Arguments,Validity,andContradictionSubject-speci cversuslogicalexpressions Examples:logicalterms all,every,some,no.not,and,or,unless,if,onlyif,ifandonlyif. Examples:subject-speci cterms Zeno,TheresaMay,France,TheNorthSea,Radon,soap,bread,GDP,logicalpositivism,...tortoise,toothless,Conservative,nobelgas,philosopher,chemicalelement,...loves,owns,reactswith,votedfor,... OverviewCourseoverview 1:Validity;IntroductiontoSetsandRelations2:SyntaxandSemanticsofPropositionalLogic3:FormalizationinPropositionalLogic4:TheSyntaxofPredicateLogic 5:TheSemanticsofPredicateLogic6:NaturalDeduction7:FormalizationinPredicateLogic8:IdentityandDe niteDescriptions 1.1SetsSets2/2 Factaboutsets Setsareidenticalifandonlyiftheyhavethesameelements. Example Thefollowingsetsareallidentical: fLennon,McCartney,Harrison,Ringog fRingo,Lennon,Harrison,McCartneyg fRingo,Ringo,Ringo,Lennon,Harrison,McCartneyg fx:xisaBeatleg fx:xsangleadvocalsonanAbbeyRoadtrackg 1.2BinaryrelationsWorkedexampleWritedownthefollowingrelationasasetoforderedpairs.Drawitsarrowdiagram.40 TherelationofbeingcountriesinGBsharingaborder fhEngland,Scotlandi;hScotland,Englandi;hEngland,Walesi;hWales,Englishig Scotland Wales-- Englandll JJ 1.2BinaryrelationsPropertiesofrelations1/3 De nition(p.9) AbinaryrelationRisre exiveonasetSi: foralldinS:thepairhd;diisanelementofR.45 Informally:everymemberofSbearsRtoitself. Example Re exiveonthesetofhumanbeings Therelationofbeingthesameheightas Example Notre exiveonthisset Therelationofbeingtallerthan Example Notre exiveonf1;2;3g fh1;1i;h2;2i;h1;3ig Re exiveonf1;2g 1.2BinaryrelationsPropertiesofrelations2/3 De nition(p.9) AbinaryrelationRissymmetriconsetSi: foralld;einS:ifhd;ei2Rthenhe;di2R. Informally:anymemberofSbearsRtoasecondonlyifthesecondbearsRbacktothe rst. Example Symmetriconthesetofhumanbeings Therelationofbeingasiblingof Example Notsymmetriconthisset Therelationofbeingabrotherof 1.2BinaryrelationsPropertiesofrelations3/3 De nition AbinaryrelationRistransitiveonSi: foralld;e;finS:ifhd;ei2Randhe;fi2R,thenalsohd;fi2R Informally:ifanymemberofSbearsRtoasecond,andthesecondalsobearsRtoathird,the rstbearsRtothethird. Example Transitiveonthesetofhumanbeings Therelationofbeingtallerthan Example Nottransitiveonthisset Therelationofnothavingthesameheight(1cm) 1.3FunctionsFunctions De nition(p.14) AbinaryrelationFisafunctioniforalld;e;f: ifhd;ei2Fandhd;fi2Fthene=f. Informally,everythingstandsinFtoatmostonething. Example Thefunctionthatsquarespositiveintegers.fh1;1i;h2;4i;h3;9i;:::gfhx;yi:y=x2;forxapositiveintegerg50 ExampleAstraightforwardandelementaryexample(a)Whatisabinaryrelation?(b)ConsidertherelationRofsharingexactlyoneparent:R=fhd;ei:dandeshareexactlyoneoftheirparentsgDeterminewhetherRis:(i)re exiveonthesetofhumanbeings(ii)symmetriconthesetofhumanbeings(iii)transitiveonthesetofhumanbeingsExplainyouranswers. ExampleAstraightforwardandelementaryexample(b)R=fhd;ei:dandeshareexactlyoneoftheirparentsg(i)IsRre exiveonthesetofhumanbeings? No. Isharetwoparentswithmyself,notone. (ii)IsRsymmetriconthesetofhumanbeings? Yes. Ifhumanbeingsdandeshareexactlyoneparent,clearlyeanddtheverysamepeopleshareexactlyoneparenttoo. (iii)IsRtransitiveonthesetofhumanbeings? No. Forexample,mymaternalhalf-sisterRachelandIshareexactlyoneparent,andmeandmypaternalhalf-sisterDebbyshareexactlyoneparent,butRachelandDebbysharenoparents.