DEFINING BASIC KINDS OF PROPERTIES

1DEFININGBASICKINDSOFPROPERTIESJiříRaclavskýAbstract:ThispaperfollowsinPavelTichý’sconceptofdistinguishingbetweentrivial(i.e.constant)andnon-trivialproperties.ThisclassificationhasbeenextendedbyPavelCmorejwhodistinguishedtwokindsofnon-trivialproperties,namelypurelyempiricalandpartlyessential(whicharepartlyempirical)properties(partlyessentialpropertyisessentialforcertainindividual(s),butthatitisnotforother(s)).Thepresentstudyprovidesrigorousformaldefinitionsoftrivial/non-trivial,essential/non-essential,andpurelyempirical/partlyessential/purelyessentialpropertieswithrespecttothepossiblepartialityoftheseproperties.Inaddition,adefinitionoftriviallyvoidpropertiescompletingtheclassificationofthekindsofpropertiestoformaquadrupletisproposed.Furthermore,theintroductionoftheconceptofaccidentalpropertiesgivesrisetoanother(yetnotequivalent)quadruplet:purelyaccidental/partlyessential(i.e.partlyaccidental)/purelyessential/voidproperties.PavelTichý,alogicianwhodevelopedaspecificintensionallogic,namelytransparentintensionallogic(hereafterTIL),usedhislogicnotonlyforalogico-semanticanalysisofnaturallanguagebuthealsoappliedittoproblemsinseveralphilosophicalareas.Thefieldweareconcernedwithinthispapercanbecalled‘thelogicofproperties’.Intensionallogicmaybeviewedasaconceptualtoolhelpingtodevelopintensionalmetaphysics.Accordingtoitsmainassumption,anindividualcaninstantiatevariousproperties.Propertiesaremodelledasfunctionsfrompossibleworldsandhaveclassesofindividualsastheirvalues(extensions).Sincepossessingofapropertybyanindividualiswhatconstitutesafact,apreciseconceptualhandlingofpropertiesisofgreatimportanceforthecoreofmetaphysics.Tichýhimselfusedsomeclassificationsofpropertiesofhisownbuthedidnotwriteasystematicpaperonit(hiskeyclassificationemergedfromhiscommentsonthedoctrineofbareindividuals).PavelCmorej,afollowerofTichý,paidcloseattentiontopropertiesanddefinedotherkindsofthem.Naturally,propertiescanbeclassifiedaccordingtovariouscriteriabutonlysomeofthesehavebeendiscussedbythefollowersofTichý.Thetopicisthereforestillopenforfurtherdevelopment.Theaimofthispaperistosummarizeandextendup-to-dateknowledgeofthisareaandofferexactdefinitionsofthekindsofproperties(withrespecttopartiality),whichisanecessarystepforfurtherinvestigations.Theconceptualconstructionisbasedonafewsimplelogicalconceptslikeidentity,connectivesandquantifiers.Thegradualinvolvementinthequestionhelpsustoguaranteetheadequacyofproposeddefinitions.Thewholeconstructionculminatesinthedefinitionsofthreerelatedcategoriesofproperties(whicharefurtherdivided):trivial/non-trivialproperties(i.e.2propertieswithconstantorwithnon-constantrangeofextensions),essential/non-essentialproperties(i.e.propertieswhichare,orwhicharenot,necessarilypossessedbyacertainindividual),andpurelyempirical/partlyessential/purelyessential/triviallyvoidproperties(i.e.propertieswithentirelycontingentextensions,propertiespossessedbysomeindividualsnecessarilybutbyotherscontingently,propertiesinstantiatedbyanindividualnecessarily,propertiesnotinstantiablebyanyindividual).SpecialattentionispaidtopartlyessentialpropertieswhichhavealreadybeeninvestigatedbyCmorej.1Thetwokindsofvoidpropertiesandaccidental/non-accidentalpropertiesrepresentanewcontributionextendingCmorej’sclassification.Theydosonotonlyincompletingtheoriginallytripledivisionintotheabovementionedquadrupletofkindsofpropertiesbecausethereisalsoanotherquadrupletconceivable.Thisquadrupletdividespropertiesintopurelyaccidental/partlyessential(i.e.partlyaccidental)/purelyessential/voidproperties(nownon-triviallyvoidpropertiesarenotcoveredbythefirstcategoryasintheprecedingquadruple,butbythelastone).TRANSPARENTINTENSIONALLOGICNaturallanguagebymeansofwhichweformulateourtheorieswasnotoriginallydesignedforsuchartificialservice.Itwasspontaneouslydevelopedbyordinarypeopleforeasycommunicationoffactswithoutgoingintosubtletiesweneedtosortout.Thedangerofnaturallanguageambiguityforphilosophers’theorieswasstressedbyFrancisBacon.Subsequently,GeorgWilhelmLeibnizpromotedtheideaof‘conceptualscript’,aninstrumentdesignedtohelptheoreticianstosolvetheirabstractquestions.ThismovementwasjoinedbyGottlobFrege,whodevelopedthefirstmodern‘shorthand’writing.Nevertheless,Frege’soriginalproposalhastobemodifiedbecausepredicatelogiccannotsufficientlyhandleitemswithinrealisticempiricalframework,whichisinvestigatedbyintensionalmetaphysics.BelowweareusingthenotationoftransparentintensionallogicdevelopedbyTichýattheverybeginningof1970th(see[Tichý2004],[Tichý1988]).TILcanbecharacterizedasahigherorderintensionallogicwithatemporalparameter.Itishigherorderlogicforitquantifiesnotonlyoverindividualsbutalsooverpropertiesofindividualsoroverpropertiesofproperties,etc.Predicatelogiccanbeviewedasacalculuswhose(model-theoretic)interpretationisbasedonobjectsbelongingtothecategoryofindividuals(markedι),ortruth-values(markedο).TILalsoadoptsatomiccategoriesofpossibleworlds(ω)andrealnumbers1TheauthorishappytoexpressherehiswarmestcongratulationstoPavelCmorejontheoccasionofhis70thbirthdayandispleasedtodedicatethisp