Basis of globally admissible rules for logic $S4$

V. V. Rimatskiy

The Bulletin of Irkutsk State University. Series Mathematics, Tome 50 (2024), pp. 152-169

Résumé

Setting the basic rules of inference is fundamental to logic. The most general variant of possible inference rules are admissible inference rules:in logic $L$, a rule of inference is admissible if the set of theorems $L$ is closed with respect to this rule. The study of admissible inference rules was stimulated by the formulation of problems about decidability by admissibility (Friedman) and the presence of a finite basis of admissible rules (Kuznetsov) in Int logic. In the early 2000s, for most basic non-classical logics and some tabular logics, the Fridman-Kuznetsov problem was solved by describing an explicit basis for admissible rules. The next stage in the study of admissible inference rules for non-classical logics can be considered the concept of a globally admissible inference rule. Globally admissible rules in the logic $L$ are those inference rules that are admissible simultaneously in all (with finite model property) extensions of the given logic. Such rules develop and generalize the concept of an admissible inference rule. The presented work is devoted to the study of bases for globally admissible rules of logic $S4$. An algorithm for constructing a set of inference rules in a reduced form was described, forming the basis for globally admissible inference rules in $S4$ logic.

Détail

Keywords: modal logic, frame and model Kripke, admissible and globally admissible inference rule.

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