Geodesic
Admissible inference rules and semantic property of modal logics
The Bulletin of Irkutsk State University. Series Mathematics, Tome 37 (2021), pp. 104-117

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Firstly semantic property of nonstandart logics were described by formulas which are peculiar to studied a models in general, and do not take to consideration a variable conditions and a changing assumptions. Evidently the notion of inference rule generalizes the notion of formulas and brings us more flexibility and more expressive power to model human reasoning and computing. In 2000-2010 a few results on describing of explicit bases for admissible inference rules for nonstandard logics (S4, K4, H etc.) appeared. The key property of these logics was weak co-cover property. Beside the improvement of deductive power in logic, an admissible rule are able to describe some semantic property of given logic. We describe a semantic property of modal logics in term of admissibility of given set of inference rules. We prove that modal logic over logic $GL$ enjoys weak co-cover property iff all given rules are admissible for logic.
Keywords: modal logic, frame and model Kripke, admissible inference rule, weak co-cover property. 
@article{IIGUM_2021_37_a7,
     author = {V. V. Rimatskiy},
     title = {Admissible inference rules and semantic property of modal logics},
     journal = {The Bulletin of Irkutsk State University. Series Mathematics},
     pages = {104--117},
     publisher = {mathdoc},
     volume = {37},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IIGUM_2021_37_a7/}
}
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V. V. Rimatskiy. Admissible inference rules and semantic property of modal logics. The Bulletin of Irkutsk State University. Series Mathematics, Tome 37 (2021), pp. 104-117. http://geodesic.mathdoc.fr/item/IIGUM_2021_37_a7/