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A(nother) characterization of intuitionistic propositional logic. (English) Zbl 0988.03045
Summary: In a previous paper [the author, J. Symb. Log. 66, 281-294 (2001; Zbl 0986.03013)], we gave a countable basis \({\mathcal V}\) 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 \({\mathcal V}\) 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 \({\mathcal V}\) suffices. In fact, it is shown that for any finite subset \(X\) of \({\mathcal V}\), for one of the proper superintuitionistic logics \(D_n\) constructed by D. de Jongh and D. M. Gabbay [J. Symb. Log. 39, 67-78 (1974; Zbl 0289.02032)], 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 D. de Jongh [in: A. Kino et al. (eds.), Intuitionism and Proof Theory, Proc. Summer Conf. Buffalo/NY 1968, 211-217 (1970; Zbl 0225.02020)].

MSC:
03B55 Intermediate logics
03B20 Subsystems of classical logic (including intuitionistic logic)
03F50 Metamathematics of constructive systems
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