Modal Semantics without Possible Worlds:A Study of Hierarchical Non-deterministic Many-valuedSemantics

这个项目的目的是系统地研究和发展非确定性多值分层语义，或卡恩斯语义，用于各种正态和非正态模态逻辑。我们将致力于回答这样的问题：这种语义是否可以作为模态逻辑中已建立的Kripke语义的合适替代。然而，在很长一段时间内，将模态逻辑作为多值逻辑呈现似乎是一个死亡的研究计划，非确定性矩阵的重新发现现在已经改变了这一图景。正如预期的那样，模态逻辑的非确定性多值语义比普通矩阵语义要复杂得多。然而，正如最近所显示的那样，当涉及到对原始可能性算子的公理化时，非确定性多值语义明显比Kripke的可能世界语义更具表现力。因此，多值方法不仅避免了对可能世界或语义评价的类似指标的任何承诺，而且还承诺提供比标准方法更正式的优势。特别地，我们的目标是在不同的公理化和语言及其各自的非确定性语义之间建立一种对应理论，类似于克里普克可能世界语义的对应理论。此外，我们将使用额外的模态运算符扩展模态语言，例如现实运算符，并计划根据卡恩斯的语义研究模态逻辑的建设性变体。最后，我们将讨论情态概念的哲学含义，它不是用可能世界来表达的。

The aim of this project is the systematical investigation and development of non-deterministic many-valued hierarchical semantics, or Kearns' semantics, for various normal and non-normal modal logics. We will aim at answering the questions whether such semantics can be a suitable alternative to the established Kripke semantics for modal logics. Whereas for a long time presenting modal logics as many-valued logics seemed to be a dead research programme, the rediscovery of non-deterministic matrices has now changed the picture. As was to be expected, the non-deterministic many-valued semantics for modal logics is considerably more complex than the ordinary matrix semantics. However, as has been shown very recently, the non-deterministic many-valued semantics is significantly more expressive than Kripke's possible worlds semantics when it comes to axiomatizing a primitive possibility operator. The many-valued approach thus not only avoids any commitment to possible worlds or similar indices of semantic evaluation, but also promises to offer some formal advantage over the standard approach. In particular, we aim at a correspondence theory, similar to the correspondence theory of Kripke's possible worlds semantics, between different axiomatizations and, languages and their respective non-deterministic semantics. Furthermore, we will extend the modal language with additional modal operators, such as an actuality operator and plan on investigating constructive variants of modal logics in terms of Kearns’ semantics. Finally, we will discuss the philosophical implications of a concept of modality, which is not expressed in terms of possible worlds.