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1、KRISTERSEGERBERGTWO-DIMENSIONALMODALLOGIC*I.INTRODUCTIONThispaperaroseinresponsetoaproblemposed,inconversation,byLennartAqvist.WorkingwithordinaryKripke-typesemanticsforpropositionallogic,Aqvistwantedtoaddtohislanguagetwonewoperators0and0’ofanunusualkind.Whatmadethemunu
2、sualwasthatformulasinwhichtheyoccurredcouldnotbeevaluatedinamodelatonepoint,asisusuallydone:suchformulaswouldhavetobeevaluatedattwopointssimultaneously.Supposewearedealingwithacertainmodel,andletuswrite,withreferencetothatmodel,W(A,X,y)forthetentativenotion“TheformulaAi
3、strueatthepointxwithrespecttothepointy”.ThesemanticalconditionsAqvistwantedtoimposeon0and0’werethese:WOA,x,Y)iffWA,Y,Y);W(O’A,x,y)iffW(A,y,x).Ontheotherhand,foraformulaBnotcontaining0or0’wewouldhaveW(B,x,y)iffW(B,x,z).0and0’arespecialcasesofanoperatorCthatsatisfies,fors
4、omefunctionsfandg,theconditionWCA,x,Y)iffWA,f(x,~1,g(x,~4).Suchoperatorsprovidesimpleinstancesofwhatmaybetermed‘two-dimensional’operators.Morecomplicatedexamplesareeasilyfound,forinstance,W(C’A,x,y)iffx=y;W(C”(A,B),x,y)iffforallz,ifW(A,x,z)thenWB,z,Y).Aresuchoperatorsof
5、anyphilosophicalinterest?IthasbeenarguedthatJournalofPhilosophicalLogic2(1973)77-96.AllRightsReservedCopyrightQ1973byD.ReidelPublishingCompany,Dordrecht-Holland78KRISTERSEGERBERGatleastsomenewoperatorsofthatvaguelydelineatedcategoryareneededifonewantstogiveanaccount,wit
6、hinapost-Kripkeanframework,ofcertainaspectsofEnglish.Thatthesiswillnotbedefendedhere,butweshallofferoneexampletoillustratetheallegedneed.ConsiderthesentenceIthasalwaysbeenthecasethattherewillbeaseabattletomorrow.Hereweshalltakeitforgrantedthattoassertthissentenceisnotto
7、assertthatitwasalwaystrueinthepastthattherewouldbeaseabattleonthefollowingday(andhence,sincetimehasnobeginning,thateverydayaseabattlehastakenplace).Accordingtothereadingwerecognize,thewordtomorrowreferstothedayafterthedaythesentenceisuttered;soinordertodecidewhetherthes
8、entenceisutteredtruthfullyitisim-portanttoknowwhenitisuttered.Infact,unlessthesentenceisregardedasutteredatace
