The aim of this paper is to study a particular family of non-deterministic semantics for modal logics that has eight truth-values. These eight-valued semantics can be traced back to Omori and Skurt (2016), where a particular member of this family was used to characterize the normal modal logic K. The truth-values in these semantics convey information about a proposition’s truth/falsity, whether the proposition is necessary/not necessary, and whether it is possible/not possible. Each of these triples is represented by a unique value. In this paper we will study which modal logics can be obtained by changing the interpretation of the \(\Box \) modality, assuming that the interpretation of other connectives stays constant. We will show what axioms are responsible for a particular interpretations of \(\Box \). Furthermore, we will study subsets of these axioms. We show that some of the combinations of the axioms are equivalent to well-known modal axioms. We apply the level-valuation technique to all of the systems to regain the closure under the rule of necessitation. We also point out that some of the resulting logics are not sublogics of S5 and comment briefly on the corresponding frame conditions that are forced by these axioms. Ultimately, we sketch a proof of meta-completeness for all of these systems.

本文的目的是研究具有八个真值的模态逻辑的非确定性语义的特定族。这些八值语义可以追溯到 Omori 和 Skurt (2016)，其中该家族的一个特定成员被用来表征正常模态逻辑K。这些语义中的真值传达了有关命题的真/假、命题是否必要/不必要以及是否可能/不可能的信息。每个三元组都由一个唯一的值表示。在本文中，我们将研究通过改变\(\Box \)模态的解释可以获得哪些模态逻辑，假设其他连接词的解释保持不变。我们将展示哪些公理负责\(\Box \)的特定解释。此外，我们将研究这些公理的子集。我们证明了一些公理的组合等价于众所周知的模态公理。我们将水平评估技术应用于所有系统，以根据必要性规则重新获得关闭。我们还指出，一些结果逻辑不是S5的子逻辑，并简要评论了这些公理所强制的相应框架条件。最终，我们为所有这些系统绘制了元完整性的证明。