  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.
