Abstract
This paper studies modal logics whose extensions all have the finite model property, those whose extensions are all recursively axiomatizable, and those whose extensions are all finitely axiomatizable. We call such logics FMP-ensuring, RA-ensuring and FA-ensuring respectively, and prove necessary and sufficient conditions of such logics in \(\mathsf {NExtK4.3}\). Two infinite descending chains \(\{{\textbf{S}}_{k}\}_{k\in \omega }\) and \(\{{\textbf{S}} _{k}^{*}\}_{k\in \omega }\) of logics are presented, in terms of which the necessary and sufficient conditions are formulated as follows: A logic in \(\mathsf {NExtK4.3}\) is FMP-ensuring iff it extends \({\textbf{S}}_{k}\) for some \(k\in \omega \), it is RA-ensuring iff it extends \({\textbf{S}}_{k}^{*}\) for some \(k\in \omega \), and it is FA-ensuring iff it is finitely axiomatizable and extends \({\textbf{S}}_{k}^{*}\) for some \(k\in \omega \).
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Acknowledgements
I would like to give thanks to Yan Zhang for a couple of discussions on related topics, and to Frank Wolter and Yan Zhang for providing some references. I would also like to give thanks to the two anonymous referees provided by this journal for their comments and suggestions to improve the paper.
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Presented by Yde Venema; Received March 5, 2022.
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Xu, M. FMP-Ensuring Logics, RA-Ensuring Logics and FA-Ensuring Logics in \(\text {NExtK4.3}\). Stud Logica 111, 899–946 (2023). https://doi.org/10.1007/s11225-023-10046-5
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DOI: https://doi.org/10.1007/s11225-023-10046-5