A(nother) characterization of intuitionistic propositional logic

Abstract

In Iemhoff (J. Symbolic Logic, to appear) we gave a countable basis V for the admissible rules of IPC . Here, we show that there is no proper superintuitionistic logic with the disjunction property for which all rules in V are admissible. This shows that, relative to the disjunction property, IPC is maximal with respect to its set of admissible rules. This characterization of IPC is optimal in the sense that no finite subset of V suffices. In fact, it is shown that for any finite subset X of V , for one of the proper superintuitionistic logics D n constructed by De Jongh and Gabbay (J. Symbolic Logic 39 (1974)), all the rules in X are admissible. Moreover, the logic D n in question is even characterized by X: it is the maximal superintuitionistic logic containing D n with the disjunction property for which all rules in X are admissible. Finally, the characterization of IPC is proved to be effective by showing that it is effectively reducible to an effective characterization of IPC in terms of the Kleene slash by De Jongh (Kino et al. eds., Intuitionism and Proof Theory, North-Holland, Amsterdam, 1970).