Geodesic
Residual Finiteness for Admissible Inference Rules
Algebra i logika, Tome 40 (2001) no. 5, pp. 593-618.

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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.
Keywords: modal logic, residual finiteness for admissible inference rules. 
@article{AL_2001_40_5_a6,
     author = {V. V. Rybakov and V. R. Kiyatkin and T. Oner},
     title = {Residual {Finiteness} for {Admissible} {Inference} {Rules}},
     journal = {Algebra i logika},
     pages = {593--618},
     publisher = {mathdoc},
     volume = {40},
     number = {5},
     year = {2001},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/AL_2001_40_5_a6/}
}
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V. V. Rybakov; V. R. Kiyatkin; T. Oner. Residual Finiteness for Admissible Inference Rules. Algebra i logika, Tome 40 (2001) no. 5, pp. 593-618. http://geodesic.mathdoc.fr/item/AL_2001_40_5_a6/