A(nother) characterization of Intuitionistic Propositional Logic
Rosalie Iemhoff
Abstract:
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).