ABSTRACT
Logicaldeductionisaprocedureforestablishingthataproposition(the conclusion)isalogicalconsequenceofotherpropositions(thepremises)--thatifthepremisesarealltrue,theconclusionmustalsobetrue. Itconsistsinderivingthelogicalconsequenceclaimthatwewanttoestablishfromother"moresimple"logicalconsequenceclaims.Thus,for example,wecanestablishthatapropositionpisalogicalconsequence ofasetofpropositionsSbyshowingthatpisalogicalconsequenceofa propositionqwhichis,inturn,alogicalconsequenceofS,orthatShas asalogicalconsequenceadisjunctiveproposition,eachofwhosedisjunctshaspasalogicalconsequence.Thesemoresimpleclaimscanin turnbederivedfromotherlogicalconsequenceclaims,butthisprocess musteventuallystop,withlogicalconsequenceclaimswhosecorrectness istakenforgranted.Hence,toestablishalogicalconsequenceclaimby logicaldeductionweneedtousetwokindsoftool.First,weneedacatalogueoflogicalconsequenceclaimsfromwhichweattempttoderivethe claimthatwewanttoestablish.Second,weneedacollectionofprinciplestounderwritethederivation.Weshallrefertothesetoolsasdeductiverules.Weshallclassifydeductiverulesascategoricalandhypothetical,dependingonwhethertheyexpressasufficientconditionfor logicalconsequenceorasufficientconditionforalogicalconsequence claimtofollowfromotherlogicalconsequenceclaims.Weshallreferto
acollectionofdeductiverulesasadeductivesystem,andtotheprocess ofestablishingalogicalconsequenceclaimwithadeductivesystemas deducing(inthesystem)theconclusionfromthepremises.Deducinga conclusioncfromasetofpremisesPinadeductivesystemissupposed toshowthat,inacceptingthelogicalconsequenceclaimsexpressedby therulesofthesystem,wewouldbecommittedtoacceptingalsothatcis alogicalconsequenceofP.Ifadeductivesystemistoservethispurpose,itsruleswillhavetobespecifiedinsuchawaythattodeducec fromPinthesystemwedon'tneedtomakeanyfurtherassumptions aboutlogicalconsequence.Oneofthetraditionalaspirationsoflogic wastheprovisionofadeductivesystemwhich,foreverysetofpremises P,wouldallowustodeducefromPallitslogicalconsequencesand nothingelse,i.e.asysteminwhichwecoulddeduceapropositioncfrom PifandonlyifcisalogicalconsequenceofP.Ourmaingoalinthis chapterandthenextistoshowthat,withrespecttotheinstancesoflogicalconsequencegeneratedbythestructureoffirst-orderpropositions, thisaspirationcanbefulfilled.
