# PP-2000-06: A(nother) characterization of Intuitionistic Propositional Logic

PP-2000-06: Iemhoff, Rosalie (2000) A(nother) characterization of Intuitionistic Propositional Logic. [Report]

In \cite{iemhoff} we gave a countable basis $\cal V$ 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 $\cal V$ 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 $\cal V$ suffices. In fact, it is shown that for any finite subset $X$ of $\cal V$, for one of the proper superintuitionistic logics $D_n$ constructed by De Jongh and Gabbay (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 (1970).