FMP-Ensuring Logics, RA-Ensuring Logics and FA-Ensuring Logics in NExtK4.3

This paper studies modal logics whose extensions all have the finite model property, those whose extensions are all recursively axiomatizable, and those whose exten-sions 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 NExtK4.3. Two infinite descending chains {S-k}(k??) and {S-k(*)}(k??) of logics are presented, in terms of which the necessary and sufficient conditions are formulated as follows: A logic in NExtK4.3 is FMP-ensuring iff it extends Sk for some k ? ?, it is RA-ensuring iff it extends S-k(*) for some k ? ?, and it is FA-ensuring iff it is finitely axiomatizable and extends S*k for some k ? ?.