Bezhanishvili, Guram; Zakharyaschev, Michael Logics over MIPC. (English) Zbl 0940.03022 RIMS Kokyuroku 1021, 86-95 (1997). The paper considers propositional intuitionistic modal logics which are normal extensions of the calculus MIPC introduced by Prior and Bull. The authors use Ono’s Kripke-type semantics for MIPC – an Ono frame is a set \(W\) of possible worlds with two accessibility relations: “intuitionistic” partial order \(R\) and “modal” quasi-order \(Q\) containing \(R\), such that \([ (xQy) \Rightarrow \exists z (xRz, yQz, zQy) ]\). Some completeness and FMP results are stated without proofs. The authors refer to a combination of the standard canonical model technique and the filtration method for superintuitionistic logics and for classical modal logics. In particular, any logic of finite \(R\)-depth enjoys FMP (and hence is complete w.r.t. Ono frames); on the other hand, there exists a continuum of Ono-incomplete logics of any finite \(Q\)-depth and of any finite \(Q\)-width. The problem of Ono-completeness of any logic of finite \(R\)-width (an analogue of Fine’s result on normal extensions of K4) remains open. Reviewer: D.Skvortsov (Moskva) Cited in 1 Document MSC: 03B45 Modal logic (including the logic of norms) Keywords:intuitionistic modal logic; completeness results; finite model property; Kripke-type semantics; Ono frame; possible worlds PDFBibTeX XMLCite \textit{G. Bezhanishvili} and \textit{M. Zakharyaschev}, RIMS Kokyuroku 1021, 86--95 (1997; Zbl 0940.03022)