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2、ntaxandsemanticsAformallanguageusuallyrequiresasetofformationrules—i.e.,acompletespecificationofthekindsofexpressionsthatshallcountaswell-formedformulas(sentencesormeaningfulexpressions),applicablemechanically,inthesensethatamachinecouldcheckwhetheracand
3、idatesatisfiestherequirements.Thisspecificationusuallycontainsthreeparts:(1)alistofprimitivesymbols(basicunits)givenmechanically,(2)certaincombinationsofthesesymbols,singledoutmechanicallyasformingthesimple(atomic)sentences,and(3)asetofinductiveclauses—i
4、nductiveinasmuchastheystipulatethatnaturalcombinationsofgivensentencesformedbysuchlogicalconnectivesasthedisjunction“or,”whichissymbolized“∨”;“not,”symbolized“∼”;and“forall,”symbolized“(∀),”areagainsentences.[“(∀)”iscalledaquantifier,asisalso“thereissome
5、,”symbolized“(∃)”.]Sincethesespecificationsareconcernedonlywithsymbolsandtheircombinationsandnotwithmeanings,theyinvolveonlythesyntaxofthelanguage.Aninterpretationofaformallanguageisdeterminedbyformulatinganinterpretationoftheatomicsentencesofthelanguage
6、withregardtoadomainofobjects.,bystipulatingwhichobjectsofthedomainaredenotedbywhichconstantsofthelanguageandwhichrelationsandfunctionsaredenotedbywhichpredicatelettersandfunctionsymbols.Thetruth-value(whether“true”or“false”)ofeverysentenceisthusdetermine
7、daccordingtothestandardinterpretationoflogicalconnectives.Forexample,p·qistrueifandonlyifpandqaretrue.(Here,thedotmeanstheconjunction“and,”notthemultiplicationoperation“times.”)Thus,givenanyinterpretationofaformallanguage,aformalconceptoftruthisobtained.
8、Truth,meaning,anddenotationaresemanticconcepts.If,inaddition,aformalsysteminaformallanguageisintroduced,certainsyntacticconceptsarisenamely,axioms,rulesofinference,andtheorems.Certainsentencesaresingledoutasaxioms.Theseare
