bEx1071 8xCubexLargex9xCubexLargex3 TetcCubec4 Tetc5 8xCubexLargexTheargumenthasthetruthfunctionalform1 ABC2 DA3 EB4 E5 DThisisnottautologicallyvalidby ID: 442712
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(b)(Ex10.7)1 9x(Cube(x)^Large(x))!(Cube(c)!Large(c))2 8x:(Cube(x)^Large(x))$:9x(Cube(x)^Large(x))3 Tet(c)!:Cube(c)4 Tet(c)5 8x:(Cube(x)^Large(x))Theargumenthasthetruth-functionalform:1 A!(B!C)2 D$:A3 E!:B4 E5 DThisisnottautologicallyvalid:bypremise4Eisthecase,andsobypremise3Bisnotthecase.ThenB!Cistrue.However,thismeanspremise1canbetruewhetherAistrueorfalse.Supposeitistrue.ThenDisfalse,bypremise2.Butthentheconclusionisfalse.Howevertheargumentislogicallyvalid.Weassignthenamectoalargecubeifthereisone.Butpremises3and4tellusthatcdoesnotnameacube.Thusnothingisalargecube,andtheconclusionmustbetrue.2.Decidewhethertheconclusionofeachofthefollowingargumentsis:(a)atautologicalconsequenceofthepremises;(b)aFOconsequenceofthepremisesbutnotatautologicalconsequenceofthepremises;(c)alogicalconsequenceofthepremisesbutnotaFOconsequenceofthepremises.Ineachcasewriteoutasmanyofthefollowingasareneededtojustifyyourargument:(a)thetruthfunctionalformoftheargument,(b)theargumentwiththepredicatesreplacedbymeaninglesslettersorwords,(c)arstordercounterexample.2 1 P(a)2 Q(b)3 :(a=b)TheconclusionisnotaFOconsequenceofthepremises.ForsupposethatP(x)meansthatxissmallandQ(x)meansthatxisred.Thenwecouldhaveasmallredobjectnamedbothaandb,makingthepremisestrueandtheconclusionfalse.(c)(Ex10.16)1 Cube(a)2 :Cube(a)3 :(a=b)Thetruth-functionalformoftheargumentis:1 A2 :A3 :BThisisatautologicalvalidity.(Why?)(d)(Ex10.18)1 8z(Small(z)$Cube(z))2 Cube(d)3 Small(d)Thetruth-functionalformoftheargumentis:1 A2 B3 CThisisnottautologicallyvalid.Afterreplacingpredicatestheargumentis:4