Decidability of admissibility:On a problem by friedman and its solution by rybakov

Rybakov (1984) proved that the admissible rules of IPC are decidable. We give a proof of the same theorem, using the same core idea, but couched in the many notions that have been developed in the mean time. In particular, we illustrate how the argument can be interpreted as using refinements of the notions of exactness and extendibility

On finite model property for admissible rules

WOS: 000083646500008Our investigation is concerned with the finite model property (fmp) with respect to admissible rules. We establish general sufficient conditions for absence of fmp w.r.t. admissibility which are applicable to modal logics containing K4: Theorem 3.1 says that no logic lambda containing K4 with the co-cover property and of width > 2 has fmp w.r.t. admissibility. Surprisingly many, if not to say all, important modal logics of width > 2 are within the scope of this theorem - K4 itself, S4, GL, K4.1, K4.2, S4.1, S4.2, GL.2, etc. Thus the situation is completely opposite to the case of the ordinary fmp - the absolute majority of important logics have fmp, but not with respect to admissibility. As regards logics of width I 2, there exists a Bone for fmp w.r.t. admissibility. It is shown (Theorem 4.3) that all modal logics lambda of width I 2 extending S4 which are not sub-logics of three special tabular logics (which is equipotent to all these lambda extend a certain subframe logic defined over S4 by omission of four special frames) have fmp w.r.t. admissibility