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

Studia Logica 108 (5):1087-1123 (2020)

 Authors Marek Nasieniewski Nicolaus Copernicus University Abstract In the context of modal logics one standardly considers two modal operators: possibility ) and necessity ) [see for example Chellas ]. If the classical negation is present these operators can be treated as inter-definable. However, negative modalities ) and ) are also considered in the literature [see for example Béziau ; Došen :3–14, 1984); Gödel, in: Feferman, Collected works, vol 1, Publications 1929–1936, Oxford University Press, New York, 1986, p. 300; Lewis and Langford ]. Both of them can be treated as negations. In Béziau a logic \ has been defined on the basis of the modal logic \. \ is proposed as a solution of so-called Jaśkowski’s problem [see also Jaśkowski ]. The only negation considered in the language of \ is ‘it is not necessary’. It appears that logic \ and \ inter-definable. This initial correspondence result between \ and \ has been generalised for the case of normal logics, in particular soundness-completeness results were obtained [see Marcos :279–300, 2005); Mruczek-Nasieniewska and Nasieniewski :229–248, 2005)]. In Mruczek-Nasieniewska and Nasieniewski it has been proved that there is a correspondence between \-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 :261–280, 2017) since on the basis of classical positive logic it is enough to solely use \ to equivalently express both positive modalities and negation. Here we strengthen results given in Mruczek-Nasieniewska and Nasieniewski 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 \ 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. Keywords No keywords specified (fix it) Categories (categorize this paper) ISBN(s) DOI 10.1007/s11225-019-09890-1 Options Mark as duplicate Export citation Request removal from index Translate to english

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## References found in this work BETA

Modal Logic: An Introduction.Brian F. Chellas - 1980 - Cambridge University Press.
An Essay in Classical Modal Logic.Krister Segerberg - 1971 - Uppsala, Filosofiska Föreningen Och Filosofiska Institutionen Vid Uppsala Universitet.
Algebraic Semantics for Modal Logics II.E. J. Lemmon - 1966 - Journal of Symbolic Logic 31 (2):191-218.

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