A(nother) characterization of intuitionistic propositional logic

✍ Scribed by Rosalie Iemhoff

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 123 KB
• Volume: 113
• Category: Article
• ISSN: 0168-0072
• DOI: 10.1016/s0168-0072(01)00056-2

✦ Synopsis

In Iemho (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 ÿnite subset of V su ces. In fact, it is shown that for any ÿnite subset X of V, for one of the proper superintuitionistic logics Dn constructed by De Jongh and Gabbay (J. Symbolic Logic 39 (1974)), all the rules in X are admissible. Moreover, the logic Dn in question is even characterized by X : it is the maximal superintuitionistic logic containing Dn with the disjunction property for which all rules in X are admissible. Finally, the characterization of IPC is proved to be e ective by showing that it is e ectively reducible to an e ective characterization of IPC in terms of the Kleene slash by De Jongh (Kino et al. eds.,