Intermediate logics and Visser's rules
A set \(R\) of admissible rules forms a basis of admissible rules for a logic \(L\) if all the admissible rules of \(L\) can be derived from \(R\). It was proved by the author [J. Symb. Log. 66, 281--294 (2001; Zbl 0986.03013)] that the so-called Visser's rules form a basis of admissible rules for the intuitionistic propositional calculus \textbf{IPC}. This paper discusses bases of admissible rules for other intermediate logics. The paper shows that if all Visser's rules are admissible for an intermediate logic \(L\), then Visser's rules form a basis for \(L\). This implies that if all Visser's rules are derivable in \(L\), then all the admissible rules of \(L\) are derivable. As a result, the author obtains that the Dummett logic \textbf{LC} of linear Kripke frames does not have non-derivable admissible rules. The paper also gives a criterion for admissibility of Visser's rules. Namely, it is shown that Visser's rules are admissible (and therefore, by the result mentioned above, form a basis) for an intermediate logic \(L\) iff \(L\) has the offspring property (which is a special kind of an extension property). As a corollary, the author derives that Visser's rules are admissible for the intermediate logic \textbf{KC} of all directed Kripke frames, and that there exist logics with the disjunction property for which not all Visser's rules are admissible.
scientific article; zbMATH DE number 2186751
scientific article