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

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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.