Pietruszczak (Bull Sect Log 38(3/4):163–171, 2009 . https://doi.org/10.12775/LLP.2009.013 ) proved that the normal logics $$\mathrm {K45}$$ K 45 , $$\mathrm {KB4}$$ KB 4 ( $$=\mathrm {KB5}$$ = KB 5 ), $$\mathrm {KD45}$$ KD 45 are determined by suitable classes of simplified Kripke frames of the form $$\langle W,A\rangle $$ ⟨ W , A ⟩ , where $$A\subseteq W$$ 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 $$\mathrm {K45}$$ K 45 . Furthermore, a modal logic is a normal extension of $$\mathrm {K45}$$ K 45 (resp. $$\mathrm {KD45}$$ KD 45 ; $$\mathrm {KB4}$$ KB 4 ; $$\mathrm {S5}$$ S 5 ) if and only if it is determined by a set consisting of finite simplified frames (resp. such frames with $$A\ne \varnothing $$ A ≠ ∅ ; such frames with $$A=W$$ A = W or $$A=\varnothing $$ A = ∅ ; such frames with $$A=W$$ A = W ). Secondly, for all normal extensions of $$\mathrm {K45}$$ K 45 , $$\mathrm {KB4}$$ KB 4 , $$\mathrm {KD45}$$ KD 45 and $$\mathrm {S5}$$ 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 $$\mathrm {K}5$$ K 5 ). Thirdly, we show that these extensions are determined by certain classes of finite simplified frames generated by finite subsets of the set $$\mathbb {N}$$ 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 $$\mathbb {N}$$ N .

中文翻译：

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一些正规模态逻辑的简化 Kripke 式语义

Pietruszczak (Bull Sect Log 38(3/4):163–171, 2009 . https://doi.org/10.12775/LLP.2009.013 ) 证明了正常逻辑 $$\mathrm {K45}$$ K 45 , $ $\mathrm {KB4}$$ KB 4 ( $$=\mathrm {KB5}$$ = KB 5 ), $$\mathrm {KD45}$$ KD 45 由形式 $ 的简化 Kripke 框架的合适类别确定$\langle W,A\rangle $$ ⟨ W , A ⟩ ，其中 $$A\subseteq W$$ A ⊆ W 。在本文中，我们扩展了这一结果。首先，我们证明模态逻辑是由一个由简化框架组成的类决定的，当且仅当它是 $$\mathrm {K45}$$ K 45 的正常扩展。此外，模态逻辑是 $$\mathrm {K45}$$ K 45 (resp. $$\mathrm {KD45}$$ KD 45 ; $$\mathrm {KB4}$$ KB 4 ; $$ \mathrm {S5}$$ S 5 ) 当且仅当它由一个由有限简化框架组成的集合确定（相应的框架具有 $$A\ne \varnothing $$ A ≠ ∅ ; 这样的帧 $$A=W$$ A = W 或 $$A=\varnothing $$ A = ∅ ; 这样的帧 $$A=W$$ A = W ）。其次，对于 $$\mathrm {K45}$$ K 45 、 $$\mathrm {KB4}$$ KB 4 、 $$\mathrm {KD45}$$ KD 45 和 $$\mathrm {S5} 的所有正常扩展$$ S 5 ，特别是对于通过添加所谓的“真实”公理、Segerberg 公式和/或其 T 版本获得的扩展，我们证明了 Nagle 事实的某些版本（J Symbol Log 46(2):319–328 , 1981 . https://doi.org/10.2307/2273624 )（涉及 $$\mathrm {K}5$$ K 5 的正常扩展）。第三，我们表明这些扩展是由某些类别的有限简化框架决定的，这些有限简化框架由自然数集合 $$\mathbb {N}$$ N 的有限子集生成。