Decidability by Filtrations for graded normal logics (graded modalities V)

1. I n t r o d u c t i o n Graded normal logics ( G N L s ) are the extensions of modal normal logics to a language with graded modalities. The interpretation in usual Kripke models of a formula <>hA (n < w), whose main operator is a graded possibility, is there are more than n accessible worlds where A is true. When graded modalities were introduced (in [7], independently rediscovering a former idea of [8]), the purpose was to offer axiomatizations for the graded versions of the normal modal logics, and then to show completeness, compactness and decidability theorems for the fifteen main G N L s between K ° and $5 °. Completeness and compactness were fully proved in several steps ([8], [7], [5], [6], [2]), while the decidability had only partial answers in [7], [9], [1], [10], [11]. We prove decidability for all of the main G N L s by a suitable version of the notion of filtration, completing the basic investigation of graded modalities. Usually, when talking about filtrations one has in mind to start from a given model, then to obtain a quotient model with respect to a certain kind of set of formulas, and finally to suitably arrange the accessibility relation ([12], [13], [14], [3]). On the contrary, we reduce a model to a finite one by combining generated models with usual filtration techniques and with controls on the grades of modalities, respecting in a quite natural way the properties of the accessibility relation. Finally, as a corollary, since graded modalities really extend the usual ones, our notion of filtration can be used also for the usual normal modal logics (by restricting the attention to the formulas that contain as modal operators only n0 and <>0), avoiding the problems that usual filtrations offer to respect the properties of the accessibility relation ([3]). Presented by J an Zygmunt ; Received March 25, 1992; Revised February 10, 1993 Studia Logica 53: 61-73, 1994. © 1994 Kluwer Academic Publishers. Printed in the Netherlands.