Simplified Kripke-Style Semantics for Some Normal Modal Logics

摘要

Pietruszczak (Bull Sect Log 38(3/4):163-171, 2009. https://doi.org/10.12775/ LLP.2009.013) proved that the normal logics K45, KB4 (= KB5), KD45 are determined by suitable classes of simplified Kripke frames of the form < W, A >, where A subset of 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 K45. Furthermore, a modal logic is a normal extension of K45 (resp. KD45; KB4; S5) if and only if it is determined by a set consisting of finite simplified frames (resp. such frames with A not equal O; such frames with A = W or A = O; such frames with A = W). Secondly, for all normal extensions of K45, KB4, KD45 and 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 K5). Thirdly, we show that these extensions are determined by certain classes of finite simplified frames generated by finite subsets of the set 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 N.