资料课件讲义英文版课件2 2

定义设p,q为二命题，复合命题“如果p,则q”称作p与q的蕴涵式，记作pq，并称p是蕴涵式的前项，q为蕴涵式的后项.称作蕴涵联结词，并规定，pq为假当且仅当p为真q为假.,2.2CONDITIONALSTATEMENTS条件命题,Implication蕴含(1)Ifpandqarestatements.thecompoundstatementifpthenq,denotedpq,iscalledaconditionalstatement(条件命题),orimplication(蕴含).Thestatementpiscalledtheantecedent(前项)orhypothesis(假设),andthestatementqiscalledtheconsequent(后项)orconclusion(推论).Theconnective“if.Then”isdenotedbythesymbol“”.,Example1:Writetheimplicationpqforeachofthefollowing.(a)p:Iamhungry.q:Iwilleat.(b)p:Itissnowing.q:3+5=8Solution(a)lfIamhungry,thenIwilleat.(b)Ifitissnowing,then3+5=8.,If2+3=4thenIamasuperman.p:2+3=4;q:Iamasuperman;pqWhatisthehypothesis?Whatistheconclusion?Whatisthetruthvalueofthisproposition?Afalsehypothesisimpliesanyconclusion.Anyimplystatementwithafalsehypothesisisalwaystrue!蕴含悖论,注意：1.在自然语言中,“ifpthenq”往往意味着p,q有某种内在的因果关系;但在数理逻辑中“pq”中的p,q不一定存在什么内在联系；2.在数学中“ifpthenq”表示的是前项p为真，则后项q为真的推理关系，但在数理逻辑中，当前项p为假时，仍有“pq”为真.,“pisasufficientconditionforq”;“p是q的充分条件”;“qisanecessaryconditionforp”.“q是p的必要条件”“q,ifp”;“如果p，那么q”;“pimpliesq”;“p推出q”;,Someotherexpressionsofconditionalstatementpq,If2+2=5.thenIamthekingofEngland.p:2+2=5;q:IamthekingofEngland.pqWhatisthetruthvalueofthissentence?,当且仅当p为真q为假时，才有pq为假.当前项p为假时，仍有“pq”为真.,(2)Ifpqisanimplication,thentheconverse(逆)ofpqistheimplicationqp,andthecontrapositive(逆否命题)ofpqistheimplicationqp.,Example2:Givetheconverseandthecontrapositiveoftheimplication“Ifitisraining,thenIgetwet.”Solution:Wehavep:Itisraining;andq:Igetwet.Theconverseisqp:IfIgetWet,thenitisraining.Thecontrapositiveisqp:Ifldonotgetwet,thenitisnotraining.,2.Equivalence等价Ifpandqarestatements,thecompoundstatementpifandonlyifq,denotedbypq,iscalledanequivalence(等价)orbiconditional(双条件命题).Theconnectiveifandonlyifisdenotedbythesymbol“”.Theequivalencepqcanalsobestatedaspisanecessaryandsufficientconditionforq.,Thetruthvaluesofpqaregivenintheleft.Observethatpqistrueonlywhenbothpandqaretrueorwhenbothpandqarefalse.,howaboutpqand(pq)(qp)?(pq)(pq)(qp)isalwaystrue!Computeitstruthtable.So:pq(pq)(qp),Logicallyequivalent,pq,TTTFFTFF,TFTT,pq,q,p,FFTT,FTFT,qp,TFTT,(pq)(qp),TTTT,Example4:Maketruetableof(pq)(qp):,ASpecialstatement,Astatementthatistrueforallpossiblevaluesofitspropositionalvariablesiscalledatautology(重言式).Astatementthatisalwaysfalseiscalledacontradiction(矛盾)oranabsurdity(谬论),andastatementthatcanbeeithertrueorfalse,dependingonthetruthvaluesofitspropositionalvariables,iscalledacontingency(不定式).,Theorem1:PropertiesofLogicalOperations,CommutativeProperties1.pqqp2.pqqpAssociativeProperties3.p(qr)(pq)r4.p(qr)(pq)rDistributiveProperties5.p(qr)(pq)(pr)6.p(qr)(pq)(pr)IdempotentProperties7.ppp8.qqqPropertiesofNegation9.(p)p10.(pq)pq11.(pq)pqDeMorganslaws,CommutativeProperties1.pqqp2.pqqpAssociativeProperties3.p(qr)(pq)r4.p(qr)(pq)rDistributiveProperties5.p(qr)(pq)(pr)6.p(qr)(pq)(pr)IdempotentProperties7.ppp8.qqqPropertiesofNegation9.(p)p10.(pq)pq11.(pq)pqDeMorganslaws,Theimplicationoperationalsohasanumberofimportantproperties.,Theorem2:1.(pq)(p)q)2.(pq)(qp)3.(pq)(pq)(qp)4.(pq)(pq)5.(pq)(pq)(qp),(xP(x)xP(x)(xP(x)x(P(x)x(P(x)Q(x)xP(x)xQ(x)xP(x)xQ(x)x(P(x)Q(x)x(P(x)Q(x)xP(x)xQ(x)x(P(x)Q(x)xP(x)xQ(x)(xP(x)(xQ(x)x(P(x)Q(x)是一个重言式.x(P(x)Q(x)xP(x)xQ(x)是一个重言式.,Forquantifications,truthtableisinvalid,sosomeaxiomsareneede