Simplified Kripke-Style Semantics for Some Normal Modal Logics

Abstract

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 $A\ne \varnothing$; 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 $\mathbb {N}$.