Previously, Došen and Božić introduced four independent intuitionistic modal logics, one for each of four types of modal operators—necessity N, possibility P, impossibility Im, and unnecessity Un. These logics are denoted HKM, where M ∈ {N, P, Un, Im}. Interest in treating the four types of modal operators separately is associated with just the fact that these cannot be reduced to each other over intuitionistic logic. Here we study extensions of logics HKM that have normal companions. It turns out that all extensions of the logics HKN and HKUn possess normal companions. For the extensions of HKP and HKIm, we obtain a criterion for the existence of normal companions, which is postulated as the presence of some modal law of double negation. Also we show how adding of this law influences expressive capacities of a logic. Of particular interest is the result saying that extensions of HKP and HKIm have normal companions only if they are definitionally equivalent to those of HKN and HKUn respectively. This result is one more example of the differences in behavior of the four types of modal operators over intuitionistic logic.
Similar content being viewed by others
Notes
It is worth noting separately that the name is possibly not quite ‘fortunate,’ since a considerable portion of those results had been published by Maksimova [11] before this was done by Blok and Esakia.
References
F.Wolter and M. Zakharyaschev, “Intuitionistic modal logic,” in Synthese Libr., 280, Kluwer, Dordrecht (1999), pp. 227-238.
V. H. Sotirov, “Modal theories with intuitionistic logic,” in Proc. Conf. Math. Logic, (Sofia, 1980), Bulgarian Academy of Sciences (1984), pp. 139-171.
M. Božić and K. Došen, “Models for normal intuitionistic logics,” Stud. Log., 43, 217-245 (1984).
K. Došen, “Negative modal operators in intuitionistic logic,” Publ. Inst. Math., Nouv. Sér., 35(49), 3-14 (1984).
K. Došen, “Models for stronger normal intuitionistic modal logics,” Stud. Log., 44, No. 1, 39-70 (1985).
S. Drobyshevich, “On classical behavior of intuitionistic modalities,” Log. Log. Philos., 24, No. 1, 79-104 (2015).
S. A. Drobyshevich and S. P. Odintsov, “The finite model property for negative modalities,” Sib. Electr. Math. Rep., 10, 1-21 (2013); http://semr.math.nsc.ru/v10/p1-21.pdf
F. Wolter and M. Zakharyaschev, “The relation between intuitionistic and classical modal logics,” Algebra and Logic, 36, No. 2, 73-92 (1997).
F. Wolter, “Superintuitionistic companions of classical modal logics,” Stud. Log., 58, No. 2, 229-259 (1997).
F. Wolter and M. Zakharyaschev, “On the Blok–Esakia theorem,” in Outst. Contrib. Log., 4, Springer, Dordrecht (2014), pp. 99-118.
L. L. Maksimova and V. V. Rybakov, “A lattice of normal modal logics,” Algebra snd Logic, 13, No. 2, 105-122 (1974).
S. Burris and H. P. Sankappanavar, A Course in Universal Algebra, Grad. Texts Math., 78, Springer-Verlag, New York (1981).
H. Rasiowa and R. Sikorski, The Mathematics of Metamathematics, Monografien Matematyczne, Warsaw (1963).
A. Chagrov and M. Zakharyaschev, Modal Logic, Oxford Logic Guides, 35, Clarendon Press, Oxford (1997).
R. Goldblatt, Mathematics of Modality, CSLI Lecture Notes, 43, Stanford University, Stanford, CA (1993).
E. Pacuit, Neighborhood Semantics for Modal Logic, Short Textb. Log., Springer, Cham (2017).
Author information
Authors and Affiliations
Corresponding author
Additional information
S. A. Drobyshevich Supported by Mathematical Center in Akademgorodok, Agreement with RF Ministry of Education and Science No. 075-15-2022-281.
Translated from Algebra i Logika, Vol. 61, No. 6, pp. 659-686, November-December, 2022. Russian https://doi.org/10.33048/alglog.2022.61.601.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Drobyshevich, S.A. Normal Companions of Intuitionistic Modal Logics. Algebra Logic 61, 445–465 (2023). https://doi.org/10.1007/s10469-023-09712-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10469-023-09712-3