Simplified Kripke-Style Semantics for Some Normal Modal Logics
Andrzej Pietruszczak, Mateusz Klonowski, Yaroslav Petrukhin · Studia Logica · 2019
Pietruszczak (Bull Sect Log 38(3/4):163–171, 2009 . https://doi.org/10.12775/LLP.2009.013 ) proved that the normal logics \(\mathrm {K45}\) , \(\mathrm {KB4}\) ( \(=\mathrm {KB5}\) ), \(\mathrm {KD45}\) are determined by suitable classes of simplified Kripke frames of the form \(\langle W,A\rangle \) , where \(A\subseteq W\) . 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 \(\mathrm {K45}\) . Furthermore, a modal logic is a normal extension of \(\mathrm {K45}\) (resp. \(\mathrm {KD45}\) ; \(\mathrm {KB4}\) ; \(\mathrm {S5}\) ) if and only if it is determined by a set consisting of finite simplified frames (resp. such frames with \(A e \varnothing \) ; such frames with \(A=W\) or \(A=\varnothing \) ; such frames with \(A=W\) ). Secondly, for all normal extensions of \(\mathrm {K45}\) , \(\mathrm {KB4}\) , \(\mathrm {KD45}\) and \(\mathrm {S5}\) , 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 (J Symbol Log 46(2):319–328, 1981 . https://doi.org/10.2307/2273624 ) (which concerned normal extensions of \(\mathrm {K}5\) ). Thirdly, we show that these extensions are determined by certain classes of finite simplified frames generated by finite subsets of the set \(\mathbb {N}\) 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 \(\mathbb {N}\) .