The lattice of Belnapian modal logics: Special extensions and counterparts

Sergei P. Odintsov, Stanislav O. Speranski



Let K be the least normal modal logic and BK its Belnapian version, which enriches K with ‘strong negation’. We carry out a systematic study of the lattice of logics containing BK based on:

• introducing the classes (or rather sublattices) of so-called explosive, complete and classical Belnapian modal logics;

• assigning to every normal modal logic three special conservative extensions in these classes;

• associating with every Belnapian modal logic its explosive, complete and classical counterparts.

We investigate the relationships between special extensions and counterparts, provide certain handy characterisations and suggest a useful decomposition of the lattice of logics containing BK.


algebraic logic; paraconsistent logic; many-valued modal logic; strong negation

Full Text:



Almukdad, A., and D. Nelson, “Constructible falsity and inexact predicates”, Journal of Symbolic Logic, 49, 1 (1984): 231–233. DOI: 10.2307/2274105

Arieli, O., and A. Avron, “Reasoning with logical bilattices”, Journal of Logic, Language and Information, 5, 1 (1996): 25–63. DOI: 10.1007/BF00215626

Belnap, N., “How a computer should think”, pages 30–56 in G. Ryle (ed.), Contemporary Aspects of Philosophy, Oriel Press, Stockfield, 1977.

Belnap, N., “A useful four-valued logic”, pages 8–37 in J.M. Dunn, and G. Epstein (eds), Modern Uses of Multiple-Valued Logic, D. Reidel, Dordrecht, 1977.

Berman, J., “Distributive lattices with an additional unary operation”, Aequationes Mathematicae, 16, 1–2 (1977): 165–171. DOI: 10.1007/BF01836429

Blok, W.J., and D. Pigozzi, Algebraizable Logics, Memoirs of the American Mathematical Society 396, AMS, Providence, 1989. DOI: 10.1090/memo/0396

Brady, R.T., “Completeness proofs for the systems RM3 and BN4”, Logique et Analyse, 25, 97 (1982): 9–32.

Dunn, J.M., “Intuitive semantics for first-degree entailments and ‘coupled trees”’, Philosophical Studies, 29, 3 (1976): 149–168. DOI: 10.1007/BF00373152

Fidel, M.M., “An algebraic study of a propositional system of Nelson”, pages 99–117 in Proceedings of the First Brazilian Conference on Mathematical Logic (Campinas, 1977), Lecture Notes in Pure and Applied Mathematics 39, M. Dekker, New York, 1978.

Gabbay, D.M., and L. Maksimova, Interpolation and Definability: Modal and Intuitionistic Logics, Clarendon Press, Oxford, 2005. DOI: 10.1093/acprof:oso/9780198511748.001.0001

Goble, L., “Paraconsistent modal logic”, Logique et Analyse, 49, 193 (2006): 3–29.

Gurevich, Yu., “Intuitionistic logic with strong negation”, Studia Logica, 36, 1–2 (1977): 49–59. DOI: 10.1007/BF02121114

Jung, A., and U. Rivieccio, “Kripke semantics for modal bilattice logic”, pages 438–447 in Extended Abstracts of the 28th Annual ACM/IEEE Symposium on Logic in Computer Science, IEEE Computer Society Press, 2013.

Kracht, M., “On extensions of intermediate logics by strong negation”, Journal of Philosophical Logic, 27, 1 (1998): 49–73. DOI: 10.1023/A:1004222213212

Kracht, M., Tools and Techniques in Modal Logic, Elsevier, Amsterdam, 1999.

Maksimova, L.L., “Craig’s theorem in superintuitionistic logics and amalgamable varieties of pseudo-Boolean algebras”, Algebra and Logic, 16, 6 (1977): 427–455. DOI: 10.1007/BF01670006

Meyer, R.K., S. Giambrone, and R.T. Brady, “Where gamma fails”, Studia Logica, 43, 3 (1984): 247–256. DOI: 10.1007/BF02429841

Nelson, D., “Constructible falsity”, Journal of Symbolic Logic, 14, 1 (1949): 16–26. DOI: 10.2307/2268973

Odintsov, S.P., “Algebraic semantics for paraconsistent Nelson’s logic”, Journal of Logic and Computation, 13, 4 (2003): 453–468. DOI: 10.1093/logcom/13.4.453

Odintsov, S.P., “On the representation of N4-lattices’, Studia Logica, 76, 3 (2004): 385–405. DOI: 10.1023/B:STUD.0000032104.14199.08

Odintsov, S.P., “The class of extensions of Nelson’s paraconsistent logic”, Studia Logica, 80, 2–3 (2005): 291–320. DOI: 10.1007/s11225-005-8472-9

Odintsov, S.P., Constructive Negations and Paraconsistency, Springer, Dordrecht, 2008. DOI: 10.1007/978-1-4020-6867-6

Odintsov, S.P., “On axiomatizing Shramko-Wansing’s logic”, Studia Logica, 91, 3 (2009): 407–428. DOI: 10.1007/s11225-009-9181-6

Odintsov, S.P., and E.I. Latkin, “BK-lattices. Algebraic semantics for Belnapian modal logics”, Studia Logica, 100, 1–2 (2012): 319–338. DOI: 10.1007/s11225-012-9380-4

Odintsov, S.P., and H. Wansing, “Modal logics with Belnapian truth values”, Journal of Applied Non-Classical Logics, 20, 3 (2010): 279–301. DOI: 10.3166/jancl.20.279-304

Priest, G., An Introduction to Non-Classical Logic: From If to Is, Cambridge University Press, Cambridge, 2008.

Priest, G., “Many-valued modal logics: a simple approach”, Review of Symbolic Logic, 1, 2 (2008): 190–203. DOI: 10.1017/S1755020308080179

Ramos, F.M., and V.L. Fernández, “Twist-structures semantics for the logics of the hierarchy I n P k ”, Journal of Applied Non-Classical Logics, 19, 2 (2009): 183–209. DOI: 10.3166/jancl.19.183-209

Rasiowa, H., “N-lattices and constructive logic with strong negation”, Fundamenta Mathematicae, 46, 1 (1958): 61–80.

Sendlewski, A., “Nelson algebras through Heyting ones: I”, Studia Logica, 49, 1 (1990): 105–126. DOI: 10.1007/BF00401557

Slaney, J., “Relevant logic and paraconsistency”, pages 270–293 in L. Bertossi, A. Hunter, and T. Schaub (eds.), Inconsistency Tolerance, Lecture Notes in Computer Science 3300, Springer, Berlin, 2004. DOI: 10.1007/978-3-540-30597-2_9

Speranski, S.O., “On Belnapian modal algebras: representations, homomorphisms, congruences, and so on”, Siberian Electronic Mathematical Reports, 10 (2013): 517–534. DOI: 10.17377/semi.2013.10.040

Speranski, S.O., “On connections between BK-extensions and K-extensions”, pages 86–90 in Extended Abstracts of the Ninth Advances in Modal Logic (Copenhagen, 2012), 2012.

Urquhart, A., “Distributive lattices with a dual homomorphic operation”, Studia Logica, 38, 2 (1979): 201–209. DOI: 10.1007/BF00370442

Vakarelov, D., “Notes on N-lattices and constructive logic with strong negation”, Studia Logica, 36, 1–2 (1977): 109–125. DOI: 10.1007/BF02121118

Vorob’ev, N.N., “A constructive propositional logic with strong negation” (in Russian), Doklady Akademii Nauk SSSR, 85, 3 (1952): 465–468.

ISSN: 1425-3305 (print version)

ISSN: 2300-9802 (electronic version)

Partnerzy platformy czasopism