Intermediate Logics and Visser's Rules

Abstract

Visser's rules form a basis for the admissible rules of ${\sf IPC}$. 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 ${\sf L}$ for which they are admissible. This implies that if Visser's rules are derivable for ${\sf L}$ then ${\sf L}$ has no nonderivable 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, for example, De Morgan logic, and that Dummett's logic and the propositional Gödel logics do not have nonderivable admissible rules.