Notre Dame Journal of Formal Logic

Kripke Completeness of Infinitary Predicate Multimodal Logics

Yoshihito Tanaka


Kripke completeness of some infinitary predicate modal logics is presented. More precisely, we prove that if a normal modal logic $ \bf L$ above $ \bf K$ is $ \cal {D}$-persistent and universal, the infinitary and predicate extension of $ \bf L$ with BF$\scriptstyle \omega_{{1}}$ and BF is Kripke complete, where BF$\scriptstyle \omega_{{1}}$ and BF denote the formulas $ \bigwedge_{{{i\in\omega}}}^{{}}$ $ \Box$ pi $ \supset$ $ \Box$ $ \bigwedge_{{{i\in\omega}}}^{{}}$pi and $ \forall x \Box \varphi \supset \Box \forall x \varphi$, respectively. The results include the completeness of extensions of standard modal logics such as $ \bf K$, and its extensions by the schemata T, B, 4, 5, D, and their combinations. The proof is obtained by extending the correspondence between the representation of modal algebras and the completeness of propositional modal logic to infinite.

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Notre Dame J. Formal Logic, Volume 40, Number 3 (1999), 326-340.

First available in Project Euclid: 28 May 2002

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Primary: 03B45: Modal logic (including the logic of norms) {For knowledge and belief, see 03B42; for temporal logic, see 03B44; for provability logic, see also 03F45}


Tanaka, Yoshihito. Kripke Completeness of Infinitary Predicate Multimodal Logics. Notre Dame J. Formal Logic 40 (1999), no. 3, 326--340. doi:10.1305/ndjfl/1022615613.

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