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 K 45 , KB 4 (= KB 5), KD 45 are determined by suitable classes of simplified Kripke frames of the form ⟨ W, A⟩ , where A⊆ 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 K 45. Furthermore, a modal logic is a normal extension of K 45 (resp. KD 45 ; KB 4 ; S 5) if and only if it is determined by a set consisting of finite simplified frames (resp. such frames with A≠ ∅; such frames with A= W or A= ∅; such frames with A= W). Secondly, for all normal extensions of K 45 , KB 4 , KD 45 and S 5 , 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 K 5). 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.