Dugundji proved in 1940 that most parts of standard modal systems cannot be characterized by a single finite deterministic matrix. In the eighties, Ivlev proposed a semantics of four-valued non-deterministic matrices (which he called quasi-matrices), in order to characterize a hierarchy of weak modal logics without the necessitation rule. In a previous paper, we extended some systems of Ivlev’s hierarchy, also proposing weaker six-valued systems in which the (T) axiom was replaced by the deontic (D) axiom. In this paper, we propose even weaker systems, by eliminating both axioms, which are characterized by eight-valued non-deterministic matrices. In addition, we prove completeness for those new systems. It is natural to ask if a characterization by finite ordinary (deterministic) logical matrices would be possible for all those Ivlev-like systems. We will show that finite deterministic matrices do not characterize any of them.

中文翻译：

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具有非确定性语义的模态逻辑：第一部分—命题案例

Dugundji在1940年证明，标准模态系统的大多数部分都不能用单个有限确定性矩阵来表征。八十年代，Ivlev提出了四值非确定性矩阵（他称为准矩阵）的语义，以刻画不需要模态规则的弱模态逻辑的层次结构。在过去的研究中，我们扩展Ivlev型层次结构的某些系统中，也提出了较弱的六值体系，其中（T）公理由道义取代（d）公理。在本文中，我们通过消除两个以八值非确定性矩阵为特征的公理来提出甚至更弱的系统。此外，我们证明了这些新系统的完整性。自然会问，对于所有那些类似Ivlev的系统，用有限的普通（确定性）逻辑矩阵进行表征是否可能。我们将证明有限确定性矩阵不代表任何矩阵。