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1、外文文献译文SyntaxandsemanticsAformallanguageusuallyrequiresasetofformationrules—i.e.,acompletespecificationofthekindsofexpressionsthatshallcountaswell-formedformulas(sentencesormeaningfulexpressions),applicablemechanically,inthesensethatamachinecouldcheckwhetheracandidatesatisfiestherequirements.Thisspec
2、ificationusuallycontainsthreeparts:(1)alistofprimitivesymbols(basicunits)givenmechanically,(2)certaincombinationsofthesesymbols,singledoutmechanicallyasformingthesimple(atomic)sentences,and(3)asetofinductiveclauses—inductiveinasmuchastheystipulatethatnaturalcombinationsofgivensentencesformedbysuchlo
3、gicalconnectivesasthedisjunction“or,”whichissymbolized“∨”;“not,”symbolized“∼”;and“forall,”symbolized“(∀),”areagainsentences.[“(∀)”iscalledaquantifier,asisalso“thereissome,”symbolized“(∃)”.]Sincethesespecificationsareconcernedonlywithsymbolsandtheircombinationsandnotwithmeanings,theyinvolveonlythesyn
4、taxofthelanguage.Aninterpretationofaformallanguageisdeterminedbyformulatinganinterpretationoftheatomicsentencesofthelanguagewithregardtoadomainofobjects.,bystipulatingwhichobjectsofthedomainaredenotedbywhichconstantsofthelanguageandwhichrelationsandfunctionsaredenotedbywhichpredicatelettersandfuncti
5、onsymbols.Thetruth-value(whether“true”or“false”)ofeverysentenceisthusdeterminedaccordingtothestandardinterpretationoflogicalconnectives.Forexample,p·qistrueifandonlyifpandqaretrue.(Here,thedotmeanstheconjunction“and,”notthemultiplicationoperation“times.”)Thus,givenanyinterpretationofaformallanguage,
6、aformalconceptoftruthisobtained.Truth,meaning,anddenotationaresemanticconcepts.If,inaddition,aformalsysteminaformallanguageisintroduced,certainsyntacticconceptsarisenamely,axioms,rulesofinference,andtheorems.Certainsentencesaresingledoutasaxioms.Theseare(thebasic)theorems.Eachruleofinferenceisanindu
7、ctiveclause,statingthat,ifcertainsentencesaretheorems,thenanothersentencerelatedtotheminasuitablewayisalsoatheorem.Ifpand“eithernot-porq”(∼p∨q)aretheorems,forexample,thenqisatheorem.Ingeneral,atheorem
