 corresponding to Rose’s counterexample, to \(\mathbf{IPC}\) also has the disjunction property. Iemhoff [2005] proved that \(\mathbf{IPC}\) is the only intermediate propositional logic with the disjunction property which is closed under all the Visser rules. Iemhoff and Metcalfe [2009] developed a formal calculus for generalized admissibility for \(\mathbf{IPC}\) and some intermediate logics. Goudsmit [2015] is a thorough study of the admissible rules of intermediate logics, with a comprehensive bibliography.