Simplified Kripke-Style Semantics for Some Normal Modal Logics

Pietruszczak :163–171, 2009. https://doi.org/10.12775/LLP.2009.013) proved that the normal logics \, \ ), \ are determined by suitable classes of simplified Kripke frames of the form \, where \. In this paper, we extend this result. Firstly, we show that a modal logic is determined by a class composed of simplified frames if and only if it is a normal extension of \. Furthermore, a modal logic is a normal extension of \ ; \; \) if and only if it is determined by a set consisting of finite simplified frames ; such frames with \ or \; such frames with \). Secondly, for all normal extensions of \, \, \ and \, in particular for extensions obtained by adding the so-called “verum” axiom, Segerberg’s formulas and/or their T-versions, we prove certain versions of Nagle’s Fact :319–328, 1981. https://doi.org/10.2307/2273624) ). Thirdly, we show that these extensions are determined by certain classes of finite simplified frames generated by finite subsets of the set \ of natural numbers. In the case of extensions with Segerberg’s formulas and/or their T-versions these classes are generated by certain finite subsets of \.