Simplified Kripke-Style Semantics for Some Normal Modal Logics
Studia Logica 108 (3):451-476 (2020)
Abstract
Pietruszczak (Bull Sect Log 38(3/4):163–171, 2009) 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⊆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≠∅; such frames with A=W or A=∅; 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) (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.Author Profiles
DOI
10.1007/s11225-019-09849-2
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Citations of this work
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References found in this work
The decidability of normal k5 logics.Michael C. Nagle - 1981 - Journal of Symbolic Logic 46 (2):319-328.
Simplified Kripke style semantics for some very weak modal logics.Andrzej Pietruszczak - 2009 - Logic and Logical Philosophy 18 (3-4):271-296.
A Method of Generating Modal Logics Defining Jaśkowski’s Discussive Logic D2.Marek Nasieniewski & Andrzej Pietruszczak - 2011 - Studia Logica 97 (1):161-182.
On Theses Without Iterated Modalities of Modal Logics Between C1 and S5. Part 1.Andrzej Pietruszczak - 2017 - Bulletin of the Section of Logic 46 (1/2).
On Theses without Iterated Modalities of Modal Logics Between C1 and S5. Part 2.Andrzej Pietruszczak - 2017 - Bulletin of the Section of Logic 46 (3/4).