Geodesic
On the quantified version of the Belnap–Dunn modal logic
Sbornik. Mathematics, Tome 215 (2024) no. 3, pp. 323-354 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We develop a quantified version of the propositional modal logic $\mathsf{BK}$ from an article by Odintsov and Wansing, which is based on the (non-modal) Belnap–Dunn system; we denote this version by $\mathsf{QBK}$. First, by using the canonical model method we prove that $\mathsf{QBK}$, 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 $\mathsf{QBK}$. In conclusion, we discuss interpolation properties for $\mathsf{QBK}$-extensions. Bibliography: 21 titles.
Keywords: modal logic, constructive logic, strong negation, possible world semantics
Mots-clés : quantification. 
@article{SM_2024_215_3_a2,
     author = {A. V. Grefenshtein and S. O. Speranski},
     title = {On the quantified version of the {Belnap{\textendash}Dunn} modal logic},
     journal = {Sbornik. Mathematics},
     pages = {323--354},
     year = {2024},
     volume = {215},
     number = {3},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2024_215_3_a2/}
}
TY  - JOUR
AU  - A. V. Grefenshtein
AU  - S. O. Speranski
TI  - On the quantified version of the Belnap–Dunn modal logic
JO  - Sbornik. Mathematics
PY  - 2024
SP  - 323
EP  - 354
VL  - 215
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/SM_2024_215_3_a2/
LA  - en
ID  - SM_2024_215_3_a2
ER  - 
%0 Journal Article
%A A. V. Grefenshtein
%A S. O. Speranski
%T On the quantified version of the Belnap–Dunn modal logic
%J Sbornik. Mathematics
%D 2024
%P 323-354
%V 215
%N 3
%U http://geodesic.mathdoc.fr/item/SM_2024_215_3_a2/
%G en
%F SM_2024_215_3_a2
A. V. Grefenshtein; S. O. Speranski. On the quantified version of the Belnap–Dunn modal logic. Sbornik. Mathematics, Tome 215 (2024) no. 3, pp. 323-354. http://geodesic.mathdoc.fr/item/SM_2024_215_3_a2/

[1] S. C. Kleene, “On the interpretation of intuitionistic number theory”, J. Symb. Log., 10:4 (1945), 109–124 | DOI | MR | Zbl

[2] A. S. Troelstra and D. van Dalen, Constructivism in mathematics, v. I, Stud. Logic Found. Math., 121, North-Holland Publishing Co., Amsterdam, 1988, xx+342+XIV pp. | MR | Zbl

[3] D. Nelson, “Recursive functions and intuitionistic number theory”, Trans. Amer. Math. Soc., 61 (1947), 307–368 | DOI | MR | Zbl

[4] D. Nelson, “Constructible falsity”, J. Symb. Log., 14:1 (1949), 16–26 | DOI | MR | Zbl

[5] A. A. Markov, “Constructive logic”, in “Meetings of the Mathematical Seminar LOMI”, Uspekhi Mat. Nauk, 5:3(37) (1950), 187–188 (Russian)

[6] A. Almukdad and D. Nelson, “Constructible falsity and inexact predicates”, J. Symb. Log., 49:1 (1984), 231–233 | DOI | MR | Zbl

[7] S. P. Odintsov, Constructive negations and paraconsistency, Trends Log. Stud. Log. Libr., Springer, New York, 2008, vi+240 pp. | DOI | MR | Zbl

[8] S. C. Kleene, Introduction to metamathematics, D. Van Nostrand Co., Inc., New York, 1952, x+550 pp. | MR | Zbl

[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 | MR | Zbl

[10] J. M. Dunn, “Intuitive semantics for first-degree entailments and ‘coupled trees’ ”, Philos. Stud., 29:3 (1976), 149–168 | DOI | MR | Zbl

[11] S. P. Odintsov and H. Wansing, “Modal logics with Belnapian truth values”, J. Appl. Non-Class. Log., 20:3 (2010), 279–301 | DOI | MR | Zbl

[12] S. P. Odintsov and E. I. Latkin, “BK-lattices. Algebraic semantics for Belnapian modal logics”, Studia Logica, 100:1–2 (2012), 319–338 | DOI | MR | Zbl

[13] S. P. Odintsov and S. O. Speranski, “The lattice of Belnapian modal logics: special extensions and counterparts”, Log. Log. Philos., 25:1 (2016), 3–33 | DOI | MR | Zbl

[14] S. P. Odintsov and S. O. Speranski, “Belnap–Dunn modal logics: truth constants vs. truth values”, Rev. Symb. Log., 13:2 (2020), 416–435 | DOI | MR | Zbl

[15] D. M. Gabbay, V. B. Shehtman and D. P. Skvortsov, Quantification in nonclassical logic, v. 1, Stud. Logic Found. Math., 153, Elsevier B. V., Amsterdam, 2009, xxiv+615 pp. | MR | Zbl

[16] S. O. Speranski, “Modal bilattice logic and its extensions”, Algebra and Logic, 60:6 (2022), 407–424 | DOI | MR | Zbl

[17] S. P. Odintsov and H. Wansing, “Disentangling FDE-based paraconsistent modal logics”, Studia Logica, 105:1 (2017), 1221–1254 | DOI | MR | Zbl

[18] K. Sano and H. Omori, “An expansion of first-order Belnap–Dunn logic”, Log. J. IGPL, 22:3 (2014), 458–481 | DOI | MR | Zbl

[19] Y. Gurevich, “Intuitionistic logic with strong negation”, Studia Logica, 36:1–2 (1977), 49–59 | DOI | MR | Zbl

[20] S. P. Odintsov and 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 | DOI | MR | Zbl

[21] D. M. Gabbay and L. Maksimova, Interpolation and definability. Modal and intuitionistic logics, Oxford Logic Guides, 46, The Clarendon Press, Oxford Univ. Press, Oxford, 2005, xiv+508 pp. | DOI | MR | Zbl