Criteria for admissibility of inference rules. Modal and intermediate logics with the branching property

The main result of this paper is the following theorem: each modal logic extendingK4 having the branching property belowm and the effective m-drop point property is decidable with respect to admissibility. A similar result is obtained for intermediate intuitionistic logics with the branching property belowm and the strong effective m-drop point property. Thus, general algorithmic criteria which allow to recognize the admissibility of inference rules for modal and intermediate logics of the above kind are found. These criteria are applicable to most modal logics for which decidability with respect to admissibility is known and to many others, for instance, to the modal logicsK4,K4.1,K4.2,K4.3,S4.1,S4.2,GL.2; to all smallest and greatest counterparts of intermediate Gabbay-De-Jong logicsD_n; to all intermediate Gabbay-De-Jong logicsD_n; to all finitely axiomatizable modal and intermediate logics of finite depth etc. Semantic criteria for recognizing admissibility for these logics are offered as well.The results of this paper were obtained by the author during a stay at the Free University of Berlin with support of the Alexander von Humboldt Foundation in 1992 – 1993.Presented byWolfgang Rauntenberg