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«THE BULLETIN OF IRKUTSK STATE UNIVERSITY». SERIES «MATHEMATICS»
«IZVESTIYA IRKUTSKOGO GOSUDARSTVENNOGO UNIVERSITETA». SERIYA «MATEMATIKA»
ISSN 1997-7670 (Print)
ISSN 2541-8785 (Online)

List of issues > Series «Mathematics». 2025. Vol 54

Decidability of Global Admissibility of Inference Rules in Logic 𝑆4

Author(s)

Vitaliy V. Rimatskiy1

1 Siberian Federal University, Krasnoyarsk, Russian Federation

Abstract
In the early 2000s, the key questions of the theory of admissible rules (decidability by admissibility, the presence of a basis) were resolved for most basic non-classical logics. The question arose about the direction of development of this theory. One of the directions of further study of admissible rules became globally admissible inference rules, i.e. rules admissible in all (finitely approximable) extensions of a given logic or in some class of extensions. For them, the problem of decidability, the presence of a finite or explicit basis, etc. also arises. In the presented work the problem of decidability of globally admissible rules of logic 𝑆4 is investigated. For rules, the model for which satisfies some natural properties, the necessary and sufficient condition of global admissibility in logic 𝑆4 (𝐺𝑟𝑧) is obtained. The specified properties of the model 𝑀(𝑟; 𝑋) do not depend on the choice of the given logic, which allowed to apply the technique of truth of the rule on the n-characteristic model. Based on the obtained description, an algorithm for checking the global admissibility of an arbitrary rule in a reduced form is proposed. Thus, the problem of global admissibility in logic 𝑆4 (𝐺𝑟𝑧) is decidable.
About the Authors
Vitaliy V. Rimatskiy, Cand. Sci. (Phys.-Math), Assoc. Prof., Siberian Federal University, Krasnoyarsk, 660041, Russian Federation, Gemmeny@rambler.ru
For citation
Rimatskiy V. V. Decidability of Global Admissibility of Inference Rules in Logic 𝑆4. The Bulletin of Irkutsk State University. Series Mathematics, 2025, vol. 54, pp. 129–142. (in Russian) https://doi.org/10.26516/1997-7670.2025.54.129
Keywords
modal logic, frame and model Kripke, admissible and globally admissible inference rule
UDC
510.643; 517.11
MSC
03F25, 03B35
DOI
https://doi.org/10.26516/1997-7670.2025.54.129
References
  1. Mints G.E. Inference of admissible rules. Journal of Soviet mathematic, 1976, vol. 6, no. 4, pp. 417–421. (in Russian) 
  2. Rimatskiy V.V. Globally Admissible Inference Rules. The Bulletin of Irkutsk State University. Series Mathematics, 2022, vol. 42, pp. 138–160. (in Russian) https://doi.org/10.26516/1997-7670.2022.42.138
  3. 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. 
  4. Rimatskiy V.V. An explixit basis for WCP-globally admissible inference rules. Algebra and Logic. 2023. vol. 62, no. 2, pp. 149–165. https://doi.org/10.33048/alglog.2023.62.204
  5. 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
  6. Rybakov V.V. Basis for admissible inference rules of logic 𝑆4 and 𝐼𝑛𝑡. Algebra and Logic, 1985, vol. 24, no. 1, pp. 55–68. 
  7. Fridman H. One hundred and two problems in mathematical logic. Journal of Symbolic Logic, 1975, vol. 40, no. 3, pp. 113–130. 
  8. 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
  9. Iemhoff R. On the admissible rules of Intuitionistic Propositional Logic. Journal of Symbolic Logic, 2001, vol. 66, no. 2. pp. 281–294. 
  10. Jeˇr´abek E., Admissible rules of modal logics. Journal of Logic and Computation. 2005, vol. 15, no. 4. pp. 411–431. 
  11. Jeˇr´abek E. , Independent bases of admissible rules. Logic Journal of the IGPL, 2005, vol. 16, no. 3, pp. 249–267. 
  12. Lorenzen P. Einfung in Operative Logik und Mathematik. Berlin, Gottingen, Heidelberg, 1955. 
  13. Port J., The deducibilities of S5, Journal of Phylosophical Logic, 1981, vol. 10, no. 1, pp. 409–422. 
  14. Rimatskiy V.V. Description of modal logics which enjoy co-cover property. Siberian Electronic Mathematical Reports, 2022, vol. 19, iss. 1, p. 316-325. 
  15. 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. 
  16. 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. 
  17. 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. 
  18. 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.

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