Author: Leung, Kam Sing
The study of modal logic often starts with that of unary operators applied to sentences, denoting some notions of necessity or possibility. However, we adopt a more general approach in this dissertation. We begin with object languages that possess multi-ary modal operators, and interpret them in relational semantics, neighbourhood semantics and algebraic semantics. Some topics on this subject have been investigated by logicians for some time, and we present a survey of their results. But there remain areas to be explored, and we examine them in order to gain more knowledge of our territory. More specifically, we propose polyadic modal axioms that correspond to seriality, reflexivity, symmetry, transitivity and euclideanness of multi-ary relations, and prove soundness and completeness of normal systems based on these axioms. We also put forward polyadic classical systems determined by classes of neighbourhood frames of finite types such as superset-closed frames, quasi-filtroids and filtroids. Equivalences between categories of modal algebras and categories of relational frames and neighbourhood frames are demonstrated. Furthermore some of the systems studied in this dissertation are shown to be translationally equivalent. While the first part of our study is purely formal, we take a different route in the second part. The multi-ary modal operators, previously interpreted in classes of mathematical structures, are given meanings in ordinary discourse. We read them as modalities in normative thinking, for instance, as the ``ought'' when we say ``you ought to visit your parents, or at least call them if you cannot visit them''. A series of polyadic modal logics, called systems of deontic residuation, are proposed. They represent real-life situations involving, for example, normative conflicts and contrary-to-duty obligations better than traditional deontic logics based on unary modal operators do.
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