Admissible Inference Rules and Semantic Property of Modal Logics

• Main Author: V.V. Rimatskiy
• Format: Article
• Language: English
• Published: Irkutsk State University 2021-09-01
• Series: Известия Иркутского государственного университета: Серия "Математика"
• Subjects: modal logic; frame and model kripke; admissible inference rule; weak co-cover property
• Online Access: http://mathizv.isu.ru/en/article/file?id=1388
• ISSN: 1997-7670; 2541-8785

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.