Abstract
In the context of modal logics one standardly considers two modal operators: possibility (\(\Diamond \)) and necessity (\(\Box \)) [see for example Chellas (Modal logic. An introduction, Cambridge University Press, Cambridge, 1980)]. If the classical negation is present these operators can be treated as inter-definable. However, negative modalities (\(\Diamond \lnot \)) and (\(\Box \lnot \)) are also considered in the literature [see for example Béziau (Log Log Philos 15:99–111, 2006. https://doi.org/10.12775/LLP.2006.006); Došen (Publ L’Inst Math, Nouv Sér 35(49):3–14, 1984); Gödel, in: Feferman (ed.), Collected works, vol 1, Publications 1929–1936, Oxford University Press, New York, 1986, p. 300; Lewis and Langford (Symbolic logic, Dover Publications Inc., New York, 1959, p. 497)]. Both of them can be treated as negations. In Béziau (Log Log Philos 15:99–111, 2006. https://doi.org/10.12775/LLP.2006.006) a logic \(\mathbf{Z}\) has been defined on the basis of the modal logic \(\mathbf{S5}\). \(\mathbf{Z}\) is proposed as a solution of so-called Jaśkowski’s problem [see also Jaśkowski (Stud Soc Sci Torun 5:57–77, 1948)]. The only negation considered in the language of \(\mathbf{Z}\) is ‘it is not necessary’. It appears that logic \(\mathbf{Z}\) and \(\mathbf{S5}\) inter-definable. This initial correspondence result between \(\mathbf{S5}\) and \(\mathbf{Z}\) has been generalised for the case of normal logics, in particular soundness-completeness results were obtained [see Marcos (Log Anal 48(189–192):279–300, 2005); Mruczek-Nasieniewska and Nasieniewski (Bull Sect Log 34(4):229–248, 2005)]. In Mruczek-Nasieniewska and Nasieniewski (Log Univ 12:207–219, 2018. https://doi.org/10.1007/s11787-018-0184-9) it has been proved that there is a correspondence between \(\mathbf{Z}\)-like logics and regular extensions of the smallest deontic logic. To obtain this result both negative modalities were used. This result has been strengthened in Mruczek-Nasieniewska and Nasieniewski (Bull Sect Log 46(3–4):261–280, 2017) since on the basis of classical positive logic it is enough to solely use \(\Box \lnot \) to equivalently express both positive modalities and negation. Here we strengthen results given in Mruczek-Nasieniewska and Nasieniewski (Log Univ 12:207–219, 2018. https://doi.org/10.1007/s11787-018-0184-9) by showing correspondence for the smallest regular logic. In particular we give a syntactic formulation of a logic that corresponds to the smallest regular logic. As a result we characterise all logics that arise from regular logics. From this follows via respective translations a characterisation of a class of logics corresponding to some quasi-regular logics where \(\mathbf{S2}^{\mathbf{0}}\) is the smallest element. Moreover, if a given quasi-regular logic is characterised by some class of models, the same class can be used to semantically characterise the logic obtained by our translation.
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The authors are grateful to the anonymous referees for their valuable comments. The authors of this work benefited from support provided by Polish National Science Centre (NCN), Grant number 2016/23/B/HS1/00344.
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Mruczek-Nasieniewska, K., Nasieniewski, M. On Correspondence of Standard Modalities and Negative Ones on the Basis of Regular and Quasi-regular Logics. Stud Logica 108, 1087–1123 (2020). https://doi.org/10.1007/s11225-019-09890-1
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DOI: https://doi.org/10.1007/s11225-019-09890-1