ConsequenceWecanalsodenedifferenttypesofconsequenceSentenceQisanXconsequenceofsentencesP1PnifftherearenoXpossibleworldswhereP1PnaretrueandQisfalseForXtautologicallogicaletcSpecialcase ID: 190636
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SentenceequivalenceRecalltwodenitionsfromlastclass:1.AsentenceisanXpossiblesentenceifitistrueinsomeXpossibleworld.Cube(a)isTWpossiblesentence.2.AsentenceisanXnecessityifitistrueinallXpossibleworlds.a=aisalogicalnecessity.3.TwosentencesareXequivalentiftheyhavethesametruthvalueinallXworlds.ThesentencesAand::Aaretautologicallyequivalent.ClassnotesOct11,2002p.1TautologicalequivalenceTwosentencescanbeshowntobetautologicallyequivalentusingtruthtables:ABCA_(B^C)(A_B)^(A_C)TTTTTTTTTTFTFTTTTFTTFTTTTFFTFTTTFTTTTTTTFTFFFTFFFFTFFFFTFFFFFFFFClassnotesOct11,2002p.2Truthtablescanalsoshowthattwosentencesarenottautologicallyequivalent:a=bb=aa=ba=b^b=aTTTTTFTFFTFFFFFFTheaboveshowsthata=banda=b^b=aarenottautologicallyequivalent.ClassnotesOct11,2002p.3TautologicalvslogicalequivalenceIftwosentencesaretautologicallyequivalent,theyarelogicallyequivalent.Thus,A_(B^C)islogicallyequivalentto(A^C)_(A^B),andnotjusttautequivalent.Butthereversedoesnothold:LogicalEquivalence6)TautologicalEquivalencea=bislogicallyequivalenttoa=b^b=a,but,aswesaw,nottautologicallyequivalent.ClassnotesOct11,2002p.4 ConsequenceWecanalsodenedifferenttypesofconsequence:SentenceQisanXconsequenceofsentencesP1;:::;PnifftherearenoXpossibleworldswhereP1;:::;PnaretrueandQisfalse.ForX=tautological,logical,etc.Specialcase:SentenceQisatautologicallyconsequenceofsentencesP1;:::;PniffthereisnorowofatruthtablewhereP1;:::;PnaretrueandQisfalse.ClassnotesOct11,2002p.5Consequence,equivalence,necessityAnargumentwithpremisesP1;:::;PnandconclusionQisvalidjustincaseQisaconsequenceofP1;:::;Pn.ForsentencesAandB,ifAimpliesBandBimpliesA,AandBareequivalent.Forproof,assumeopposite.ThenoneTandotherF.Ifasentenceisnecessary(avalidity),itisaconsequenceofanyothersentence.Ifasentenceisimpossible,anyothersentenceisaconsequenceofit.ClassnotesOct11,2002p.6TautologicalconsequenceexamplesAisaconsequenceofA^B:ABAA^BTTTTTFTFFTFFFFFFNoatomicsentencetautologicallyimpliesanotherone:Green(grass)Tall(jim)Green(grass)Tall(jim)TTTTTFTFFTFTFFFFClassnotesOct11,2002p.7Tautologicalconsequenceexamples2A^BisatautologicalconsequenceofAandB_:A,butnoteitherAorBseparately:ABAB_:AA^BTTTTTTFTFFFTFTFFFFTFClassnotesOct11,2002p.8 TautologicalvslogicalconsequenceTautologicalconsequence)logicalconsequence.(Why?)SonowweknowthatAandB_:AlogicallyimplyA^B.However,logicalconsequence6)tautologicalconsequencea=bisalogicalconsequenceofb=a,butnotatautologicalconsequence.Likewise,TautologicalconsequenceimpliesTWconsequence,butnotviceversa.ClassnotesOct11,2002p.9TautConFitchtheprogramallowsyoutousetheruleTautConwhenthesentencefollowsfromthepreviousones.1A2B_:A3A^BTautCon:1,2Thiscansaveyoutheworkofmakingatruthtableandcheckingallthecontents.ClassnotesOct11,2002p.10AnaCon,FOConThecomputeralsohasthemorepowerfulrulesAnaConandFOCon.1a=b2Cube(a)3Cube(b)FOCon:1,2ReplacingFOConwithTautCondoesn'twork:a=bCube(a)Cube(b)TTFClassnotesOct11,2002p.11TautConvsFOConvsAnaConWe'lllearnmoreabouttheselater.Fornow:Aworldistautologicallypossibleiffthesententialconnectives(_,^,:)workthesameway.AworldisFO(FirstOrder)possibleifaboveandequality(=)workthesameway.AworldisAnalyticallypossibleifaboveandTWpredicates(Larger,etc.)workthesameway.Forpracticalreasons,BetweenandAdjoinsdon'thavetoworkthesameway.ClassnotesOct11,2002p.12 Variousconexamples1.TautCon:A_BimpliesB_A2.FOCon:a=bimpliesb=a3.AnaCon:Larger(b;a)impliesSmaller(a;b)TautCon)FOCon)AnaCon.AnaConisthestrongest,buttakesthelongest.ClassnotesOct11,2002p.13SubstitutionprincipleIfPandQareequivalent,thensoareS(P)andS(Q).Examples:B,::B,soA^(::B_C),A^(B_C).:(A_B),:A^:B,so:(A_B)_C,(:A^:B)_C.a=a,b=b,soCube(a)^a=a,Cube(a)^b=b.ClassnotesOct11,2002p.14NegationnormalformAsentenceisinnegationnormalformifanynegationsapplyonlytoatomicsentences.ExamplesofsentencesinNNF::Green(grass)^(:a=a^Larger(a;b))A_BExamplesofsentencesnotinNNF::(A^B)::Cube(a)A^(B_::C)ClassnotesOct11,2002p.15TransformingsentencesintoNNF1.MovenegationsasfarinaspossibleusingDeMorganequivalences::(A_B),:A^:B:(A^B),:A_:B2.Canceloutdoublenegationsusingequivalenceof::AandA.Example:1.:(A^:B)2.:A_::BDeM2above3.:A_B:elimClassnotesOct11,2002p.16 Distributionlawsfor^,_A^(B_C),(A^B)_(A^C)A_(B^C),(A_B)^(A_C)Comparethedistributionlawforand+:A(B+C),(AB)+(AC)However,+doesn'tdistributeover.LikeDeMorgan'slaws,wecanusethistosimplifycom-plexsentences.ClassnotesOct11,2002p.17DisjunctionNormalFormMoredenitions:Asentenceisaliteraliffitisatomicorthenegationofanatomicsentence.Grass(green),:a=aAsentenceisindisjunctivenormalformiffit'sinnegationnormalform,andanyconjunctionsareinsidethedisjunctions.Grass(green):A_(B^:C):a=aClassnotesOct11,2002p.18MovingsentencesintoDNF1.PutsentenceintoNNF.2.UsedistributionruleA^(B_C),(A^B)_(A^C)Example:1.:((A^:B)_C)2.:(A^:B)^:CDeMorgan3.(:A_::B)^:CDeMorganagain4.(:A_B)^:C:elim,nowinNNF5.(:A^:C)_(B^:C)^distributionClassnotesOct11,2002p.19ConjunctionNormalFormAsentenceisinconjunctionnormalformifit'sinnegationnormalformandthedisjunctionsareinsidetheconjunctions.SentencesinCNF:A,:a=b,A_B,(A_:B)^(:C_D)SentencesnotinCNF:::A,(B^C)_A.TheCNFprocedureissimilartotheDNFprocedure:1.PutsentenceintoNNF.2.UsedistributionruleA_(B^C),(A_B)^(A_C).ClassnotesOct11,2002p.20