Residual Finiteness for Admissible Inference Rules

We look into methods which make it possible to determine whether or not the modal logics under examination are residually finite w. r. t. admissible inference rules. A general condition is specified which states that modal logics over $K4$ are not residually finite w.ṙ.ṫ. admissibility. It is shown that all modal logics $\lambda$ over $K4$ of width strictly more than 2 which have the co-covering property fail to be residually finite w. r. t. admissible inference rules; in particular, such are $K4$, $GL$, $K4.1$, $K4.2$, $S4.1$, $S4.2$, and $GL.2$. It is proved that all logics $\lambda$ over $S4$ of width at most 2, which are not sublogics of three special table logics, possess the property of being residually finite w. r. t. admissibility. A number of open questions are set up.