¹Siberian Federal University, Krasnoyarsk, Russian Federation

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.

Rimatskiy V. V. Basis of Globally Admissible Rules for Logic S4. The Bulletin of Irkutsk State University. Series Mathematics, 2024, vol. 50, pp. 152–169. (in Russian)

https://doi.org/10.26516/1997-7670.2024.50.152

1. Rimatskiy V.V. Globally Admissible Inference Rules. The Bulletin of Irkutsk State University. Series Mathematics, 2022, vol. 42, pp. 138–160. https://doi.org/10.26516/1997-7670.2022.42.138 (in Russian)
2. Rimatskiy V.V., Kiyatkin V.R. Independent bases for admissible rules of pretabular modal logic and its extensions. Siberian Electronic Mathematical Reports, 2013, vol. 10, pp. 79–89.(in Russian) http://semr.math.nsc.ru
3. Rimatskiy V.V. An explixit basis for WCP-globally admissible inference rules. Algebra and Logic. 2023. vol. 62, no. 2. С. 149 – 165. https://doi.org/10.1007/s10469-024-09733-6
4. Rybakov V.V. Basis for admissible inference rules of logic S4 and Int. Algebra and Logic, 1985, vol. 24, no. 1, pp. 55–68.
5. Fridman H. One hundred and two problems in mathematical logic. Journal of Symbolic Logic, 1975, vol. 40, no. 3, pp. 113–130.
6. Iemhoff R. A(nother) characterization of Intuitionistic Propositional Logic. Annals of Pure and Applied Logic, 2001, vol. 113, no. 1-3, pp. 161–173. https://doi.org/10.1016/S0168-0072(01)00056-2
7. Iemhoff R. On the admissible rules of Intuitionistic Propositional Logic. Journal of Symbolic Logic, 2001, vol. 66, no. 2. pp. 281–294.
8. Jeˇr´abek E., Admissible rules of modal logics. Journal of Logic and Computation. 2005, vol. 15, no. 4, pp. 411–431.
9. Jeˇr´abek E. , Independent bases of admissible rules. Logic Journal of the IGPL, 2005, vol. 16, no. 3, pp. 249–267.
10. Lorenzen P. Einfung in Operative Logik und Mathematik. Berlin, Gottingen, Heidelberg, 1955.
11. Rimatskiy V.V. Description of modal logics which enjoy co-cover property. Siberian Electronic Mathematical Reports, vol. 19, is. 1, pp. 316–325.
12. Rybakov V.V., Construction of an Explicit Basis for Rules admissible in Modal system S4. Mathematical Logic Quarterly, 2001, vol. 147, no. 2, pp. 441–451.
13. Rybakov V.V., Terziler M., Remazki V.V. Basis in Semi-Redused Form for the Admissible Rules of the Intuitionistc Logic IPC. Mathematical Logic Quarterly, 2001, vol. 46, no. 2. pp. 207–218.
14. Rybakov V.V. Admissibility of logical inference rules, Studies in Logic and the Foundations of Mathematics. New-York, Amsterdam, Elsevier Sci. Publ., 1997, vol. 136, 611 p.
15. Rybakov V.V., Rimatski V.V. A note on Globally admissible inference rules for modal and superintuitionistic logics. Bulletin of the Section of Logic, 2005, vol. 34, no. 2, pp. 1–7.