Independent bases for admissible rules in pretable logics

Independent bases of admissible inference rules are studied; namely, we treat inference rules in pretable modal logics over S4, and in pretable superintuitionistic logics. The Maksimova-Esakia-Meskhi theorem holds that there exist exactly five pretable S4-logics and precisely three pretable superintuitionistic ones. We argue that all pretable modal logics and all pretable super-intuitionistic logics have independent bases for admissible inference rules.