Rejecting inadmissible rules in reduced normal forms in S4

Several methods for checking admissibility of rules in the modal logic $S4$ are presented in [1], [15]. These methods determine admissibility of rules in $S4$, but they don't determine or give substitutions rejecting inadmissible rules. In this paper, we investigate some relations between one of the above methods, based on the reduced normal form rules, and sets of substitutions which reject them. We also generalize the method in [1], [15] for one rule to admissibility of a set of rules.