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 \(k\in \omega \).

本文研究了外延全部具有有限模型性质的模态逻辑、外延全部递归公理化的模态逻辑以及外延全部有限公理化的模态逻辑。我们分别将这种逻辑称为FMP-确保、RA-确保和FA-确保，并在\(\mathsf {NExtK4.3}\)中证明这种逻辑的充分必要条件。两条无限下降链\(\{{\textbf{S}}_{k}\}_{k\in \omega }\)和\(\{{\textbf{S}} _{k}^{*提出了}\}_{k\in \omega }\)个逻辑，根据这些逻辑，充分必要条件表述如下：\(\mathsf {NExtK4.3}\)中的逻辑是 FMP 确保的当且仅当它扩展了\({\textbf{S}}_{k}\)一些\(k\in \omega \)，它是 RA 确保当且仅当它扩展\({\textbf{S}}_{k}^{*}\)对于某些\(k\in \omega \)，并且它是 FA 确保的当且仅当它是有限公理化的并且对于某些\(k\in \omega \)扩展\({\textbf{S}}_{k}^{*}\)。