A(nother) characterization of intuitionistic propositional logic.

*(English)*Zbl 0988.03045Summary: 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 |

##### Keywords:

intermediate logics; intuitionistic logic; superintuitionistic logic; disjunction property; admissible rules; Kleene slash
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\textit{R. Iemhoff}, Ann. Pure Appl. Logic 113, No. 1--3, 161--173 (2002; Zbl 0988.03045)

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##### References:

[1] | D.H.J. de Jongh, A characterization of the intuitionistic propositional calculus, in: A. Kino, J. Myhill, R.E. Vesley (Eds.), Intuitionism and Proof Theory, North-Holland, Amsterdam, 1970. · Zbl 0225.02020 |

[2] | D.H.J. de Jongh, A class of intuitionistic connectives, in: J. Barwise, H.J. Keisler, K. Kunen (Eds.), The Kleene Symposium, North-Holland Publishing Company, Amsterdam, 1980. · Zbl 0479.03015 |

[3] | Gabbay, D.M.; de Jongh, D.H.J., A sequence of decidable finitely axiomatizable intermediate logics with the disjunction property, J. symbolic logic, 39, 67-78, (1974) · Zbl 0289.02032 |

[4] | Iemhoff, R., On the admissible rules of intuitionistic propositional logic, J. symbolic logic, 66, 281-294, (2001) · Zbl 0986.03013 |

[5] | Kleene, S.C., Disjunction and existence under implication in elementary intuitionistic formalisms, J. symbolic logic, 27, 11-18, (1962) · Zbl 0112.24502 |

[6] | Rybakov, V.V., Admissibility of logical inference rules, (1997), Elsevier Amsterdam · Zbl 0872.03002 |

[7] | C.A. Smoryński, Applications of Kripke Models, in: A.S. Troelstra (Ed.), Mathematical Investigations of Intuitionistic Arithmetic and Analysis, Springer, Berlin, 1973. |

[8] | Troelstra, A.S.; van Dalen, D., Constructivism in mathematics, Vol. I, (1988), North-Holland Amsterdam · Zbl 0653.03040 |

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