On the quantified version of the Belnap–Dunn modal logic

Cover Page

Cite item

Full Text

Open Access Open Access
Restricted Access Access granted
Restricted Access Subscription Access

Abstract

We develop a quantified version of the propositional modal logic BK from an article by Odintsov and Wansing, which is based on the (non-modal) Belnap–Dunn system; we denote this version by QBK. First, by using the canonical model method we prove that QBK, as well as some important extensions of it, is strongly complete with respect to a suitable possible world semantics. Then we define translations (in the spirit of Gödel–McKinsey–Tarski) that faithfully embed the quantified versions of Nelson's constructive logics into suitable extensions of QBK. In conclusion, we discuss interpolation properties for QBK-extensions.

About the authors

Alexander Vitalevich Grefenshtein

Steklov Mathematical Institute of Russian Academy of Sciences

Author for correspondence.
Email: katze.tail@gmail.com

without scientific degree, no status

Stanislav Olegovich Speranski

Steklov Mathematical Institute of Russian Academy of Sciences

Email: katze.tail@gmail.com
ORCID iD: 0000-0001-6386-5632
Scopus Author ID: 55532074400
ResearcherId: L-2043-2016
Candidate of physico-mathematical sciences, no status

References

  1. S. C. Kleene, “On the interpretation of intuitionistic number theory”, J. Symb. Log., 10:4 (1945), 109–124
  2. A. S. Troelstra, D. van Dalen, Constructivism in mathematics, v. I, Stud. Logic Found. Math., 121, North-Holland Publishing Co., Amsterdam, 1988, xx+342+XIV pp.
  3. D. Nelson, “Recursive functions and intuitionistic number theory”, Trans. Amer. Math. Soc., 61 (1947), 307–368
  4. D. Nelson, “Constructible falsity”, J. Symb. Log., 14:1 (1949), 16–26
  5. А. А. Mapков, “Конструктивная логика”, В ст.: “Заседания математического семинара ЛОМИ”, УМН, 5:3(37) (1950), 187–188
  6. A. Almukdad, D. Nelson, “Constructible falsity and inexact predicates”, J. Symb. Log., 49:1 (1984), 231–233
  7. S. P. Odintsov, Constructive negations and paraconsistency, Trends Log. Stud. Log. Libr., Springer, New York, 2008, vi+240 pp.
  8. С. К. Клини, Введение в метаматематику, ИЛ, М., 1957, 526 с.
  9. N. D. Belnap, Jr., “A useful four-valued logic”, Modern uses of multiple-valued logic (Indiana Univ., Bloomington, Ind., 1975), Episteme, 2, D. Reidel Publishing Co., Dordrecht–Boston, MA, 1977, 5–37
  10. J. M. Dunn, “Intuitive semantics for first-degree entailments and ‘coupled trees’ ”, Philos. Stud., 29:3 (1976), 149–168
  11. S. P. Odintsov, H. Wansing, “Modal logics with Belnapian truth values”, J. Appl. Non-Class. Log., 20:3 (2010), 279–301
  12. S. P. Odintsov, E. I. Latkin, “BK-lattices. Algebraic semantics for Belnapian modal logics”, Studia Logica, 100:1-2 (2012), 319–338
  13. S. P. Odintsov, S. O. Speranski, “The lattice of Belnapian modal logics: special extensions and counterparts”, Log. Log. Philos., 25:1 (2016), 3–33
  14. S. P. Odintsov, S. O. Speranski, “Belnap–Dunn modal logics: truth constants vs. truth values”, Rev. Symb. Log., 13:2 (2020), 416–435
  15. D. M. Gabbay, V. B. Shehtman, D. P. Skvortsov, Quantification in nonclassical logic, v. 1, Stud. Logic Found. Math., 153, Elsevier B. V., Amsterdam, 2009, xxiv+615 pp.
  16. С. О. Сперанский, “О модальной логике бирешeток и еe расширениях”, Алгебра и логика, 60:6 (2021), 612–635
  17. S. P. Odintsov, H. Wansing, “Disentangling FDE-based paraconsistent modal logics”, Studia Logica, 105:1 (2017), 1221–1254
  18. K. Sano, H. Omori, “An expansion of first-order Belnap–Dunn logic”, Log. J. IGPL, 22:3 (2014), 458–481
  19. Y. Gurevich, “Intuitionistic logic with strong negation”, Studia Logica, 36:1-2 (1977), 49–59
  20. S. P. Odintsov, H. Wansing, “Inconsistency-tolerant description logic: motivation and basic systems”, Trends in logic, 50 years of Studia Logica, Trends Log. Stud. Log. Libr., 21, Kluwer Acad. Publ., Dordrecht, 2003, 301–335
  21. D. M. Gabbay, L. Maksimova, Interpolation and definability. Modal and intuitionistic logics, Oxford Logic Guides, 46, The Clarendon Press, Oxford Univ. Press, Oxford, 2005, xiv+508 pp.

Supplementary files

Supplementary Files
Action
1. JATS XML
Download

Copyright (c) 2024 Грефенштейн А.V., Сперанский С.O.

Согласие на обработку персональных данных

 

Используя сайт https://journals.rcsi.science, я (далее – «Пользователь» или «Субъект персональных данных») даю согласие на обработку персональных данных на этом сайте (текст Согласия) и на обработку персональных данных с помощью сервиса «Яндекс.Метрика» (текст Согласия).