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Admissible bases via stable canonical rules. (English) Zbl 1397.03016
The dichotomy property means that a canonical multi-conclusion rule is either admissible or equivalent to an assumption-free rule. The authors establish the same property for stable multi-conclusion canonical rules for the intuitionistic propositional calculus (IPC), and the modal systems K4 and S4 (see [V. V. Rybakov, Admissibility of logical inference rules. Amsterdam: Elsevier (1997; Zbl 0872.03002)]). An immediate consequence of the main result is decidability of the admissibility problem for IPC, K4 and S4, while the admissibility for the basic modal logic K remains a long standing open problem.

MSC:
03B20 Subsystems of classical logic (including intuitionistic logic)
03B25 Decidability of theories and sets of sentences
03B45 Modal logic (including the logic of norms)
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