On Correspondence of Standard Modalities and Negative Ones on the Basis of Regular and Quasi-regular Logics

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.