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1、NotreDameJournalofFormalLogicVolume42,Number4,2001ACloserLookatSomeSubintuitionisticLogicsSergioCelaniandRamonJansanaAbstractInthepresentpaperwestudysystematicallyseveralconsequencere-lationsontheusuallanguageofpropositionalintuitionisticlogicthatcanbedenedsemanticallybyusingKripkeframesan
2、dthesamedeningtruthcondi-tionsfortheconnectivesasinintuitionisticlogicbutwithoutimposingsomeoftheconditionsontheKripkeframesthatarerequiredintheintuitionisticcase.Thelogicssoobtainedarecalledsubintuitionisticlogicsintheliterature.Wede-partfromtheperspectiveofconsideringalogicjustasasetofth
3、eoremsandalsodepartfromtheperspectivetakenbyRestallinthatweconsiderstandardKripkemodelsinsteadofmodelswithabasepoint.WestudytherelationsbetweensubintuitionisticlogicsandmodallogicsgivenbythetranslationconsideredbyDošen.Moreover,weclassifythelogicsobtainedaccordingtothehierarchyconsideredinA
4、bstractAlgebraicLogic.1IntroductionTheexpression“subintuitionisticlogic”isusedintheliterature(Restall[10],Wans-ing[18])torefertopropositionallogicsinthepropositionallanguagewithconnec-tives^;_;!andpropositionalconstant?thataredenedsemanticallybyusingKripkeframesandthesamedeningtruthcondit
5、ionsfortheconnectivesasinin-tuitionisticlogicbutwithoutimposingsomeoftheconditionsontheKripkeframesthatarerequiredintheintuitionisticcase.Forinstance,onecandispensewiththereexivityoftherelationRoftheKripkeframesorwiththetransitivityorwiththepersistence(heredity)conditionthatrequiresthatthe
6、valuationsassigntothevari-ablessetsofpointswhichareclosedunderR.Intherstsectionof[18],WansingarguesveryconvincinglyinfavoroftakingseriouslytheseweakeningsoftheusualconditionsimposedonKripkeframesforintuitionisticlogic.HisproposalistoconsidertherelationofaframeasdescribingReceivedJune23,200
7、0;acceptedSeptember17,2002;printedAugust29,20032001MathematicsSubjectClassication:Primary,03B20,03B6D,03B45;Secondary,03699Keywords:subintuitionisticlogic,modallogic,intuitionisticlogic,protoalgebraiclogics,equivalentiallogics,algebraizablelogicsc2003Un
