An Inheritance Criterion for the Admissible Inference Rules of K4

We give a criterion for inheritance of the admissible inference rules of the modal logic K4 by the finitely approximable modal logics extending K4. We give a negative solution to the question of inheritance of the admissible rules of K4 by tabular logics. We exhibit a series of examples of modal logics which inherit or fail to inherit the admissible inference rules of K4.