×

Logics over MIPC. (English) Zbl 0940.03022

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.

MSC:

03B45 Modal logic (including the logic of norms)
PDFBibTeX XMLCite