Connected modal logics
Bearing in mind the fact that the real line \(\mathbb R\), and hence every subalgebra of the closure algebra \(\mathbb R^+=(\mathcal P(\mathbb R),\mathbf{cl})\), where \(\mathcal P(\mathbb R)\) and \(\mathbf{cl}\) denote the powerset of \(\mathbb R\) and the closure operator over subsets of \(\mathbb R\), respectively, are connected, the authors introduce the concept of a connected logic over Lewis' modal system S4 and show that each connected logic with the finite model property (fmp) is the logic corresponding to a subalgebra of \(\mathbb R^+\). This result is a generalization of the famous McKinsey-Tarski theorem (see [\textit{J. C. C. McKinsey} and \textit{A. Tarski}, ``The algebra of topology'', Ann. Math. (2) 45, 141--191 (1944; Zbl 0060.06206)]). They also show that each logic over S4.1 is a connected logic, and that there are continuum many connected logics with the fmp and continuum many connected logics without the fmp. An immediate consequence of this result and the Blok-Esakia correspondence between modal and superintuitionistic logics is that each superintuitionistic logic is a connected logic and that each superintuitionistic logic with the fmp corresponds to a subalgebra of the Heyting algebra of all open subsets of \(\mathbb R\).
scientific article; zbMATH DE number 5886771
scientific article