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Yasusi HASIMOTO

Schoolof InformationScience,

Japan Advanced Instituteof Science and Technology

January 12,2001

Abstract

Inthis thesis,weinvestigatemodallogicssemanticallyby usingbothalgebraicsemanticsandgeneral

Kripketypesemantics. Wewill discussseveral topics onmodallogic. Thoughthetopic varies, thereis

auniqueunderlyingmotif throughthewholethesis,i.e. the dualitybetween algebrasandframes.

Kripketypesemanticsformodallogicshasmadeagreatsuccessintheseyears. Thisismainlyduetothe

factthatKripketypesemanticsoersusintuitivelycomprehensibleandeasilymanageable,mathematical

models formodallogics. Ontheotherhand,while algebraicstructures lackthese properties, theyhave

one quiteimportant merit which Kripketype semantics lacks. That is, every modal logic is complete

withrespectto algebraicsemantics.

Tosupplementthisdefect,KripketypesemanticsbasedongeneralKripkeframeswasintroduced. This

semanticsbridgesbetweenoriginalKripketypesemanticsandalgebraicsemantics. Infact,bytheStone

duality,wehaveanicecorrespondencebetweengeneralKripkeframesandalgebras.Throughthisduality,

itbecomespossibleto getimportantresultsongeneralKripkeframesfrom resultsonthecorresponding

classofalgebras,whichareobtainedbyusingthefruitsofuniversalalgebra.

Thersttopicofourthesisispseudo-Euclideanlogics. Forxednon-negativeintegersmandn,letE

k

bethelogicwhich isobtainedfrom thesmallestnormal(classical) modal logicKbyadding theaxiom

3 k

!2 m

3 n

,wherek0. WewillgiveacompleteanswertothequestionwhenE

k E

k 0

holds.

Second, we discuss intuitionistic modal logics. For Kripke type semantics, we discuss nite model

property ofintuitionisticmodallogics by ltration method. Foralgebraicsemantics,wehavesucceeded

togiveadescriptionofsubdirectlyirreduciblealgebrasforvariouskindsofmodalHeyting algebras. By

usingthedualitytheory,thisresultcanbetranslatedintoaresultonadescriptionofirreducible(nite)

Kripkeframes.

Finally,weintroduceanewtypeofproductsofmodallogics,callednormalproducts. Normalproducts

resemble productsfamiliar to researcherof measure theoryand topology,and are dened asageneral-

ization ofproducts of algebras of sets. Ourproducts of modallogics canbe dened either bymeansof

normalproducts of general frames, orbymeansofnormal productsof modal algebras. Sinceournotion

ofproductsisbasedhighly onthedualitytheory,it hassuch anicepropertyasfollows; theproductof

twogeneral frames is isomorphic to thedual of the product of the correspondingdual algebras. This

broughtus adesired eect that the denition of the normalproduct of modal logics L

1 and L

2 is not

aectedbythechoiceofclassesofgeneralframes(or,modalalgebras)whichdetermineL

1 andL

2 . Note

thatthis isnotthecaseforusualproductsof modallogics.

The notionofnormalproducts isquitenaturalfrom theviewpoint ofdualitytheory. Thereforethis

enablesus toextendthenotionofproductstootherlogicslikeintuitionisticmodallogics.

Key Words: modal logics, intuitionistic modal logics, duality between algebras and

frames,subdirectly irreduciblealgebras, products of modallogics