On structural completeness of implicational logics

Abstract

We consider the notion of structural completeness with respect to arbitrary (finitary and/or infinitary) inferential rules. Our main task is to characterize structurally complete intermediate logics. We prove that the structurally complete extension of any pure implicational in termediate logic C can be given as an extension of C with a certain family of schematically denned infinitary rules; the same rules are used for each C. The cardinality of the family is continuum and, in the case of (the pure implicational fragment of) intuitionistic logic, the family cannot be reduced to a countable one. It means that the structurally complete extension of the intuitionistic logic is not countably axiomatizable by schematic rules.