Intermediate Logics and Visser's Rules

Visser’s rules form a basis for the admissible rules of IPC. Here we show that this result can be generalized to arbitrary intermediate logics: Visser’s rules form a basis for the admissible rules of any intermediate logic L for which they are admissible. This implies that if Visser’s rules are derivable for L then L has no non-derivable admissible rules. We also provide a necessary and sufficient condition for the admissibility of Visser’s rules. We apply these results to some specific intermediate logics, and obtain that Visser’s rules form a basis for the admissible rules of e.g. De Morgan logic, and that Dummett’s logic and the propositional Gödel logics do not have non-derivable admissible rules.