Model Theoretical Aspects of Normal Polyadic Modal Logic: An Exposition

Studies in Logic, Vol. 12, No. 3 (2019): 79–101 PII: 1674-3202(2019)-03-0079-23

Jixin Liu

Abstract. In this paper, we give an exposition on the model theoretical aspects of normal polyadic modal logic (PML), which is a modal logic with n-ary modalities generalizing the basic normal modal logic . Compared to the basic normal modal logic , PML is much less studied. Basic results about PML scattered in the literature are often stated without proofs, except in certain algebraic setting, as they are considered as straightforward generalizations of the results of . Besides the missing details, the very limited available expositions are errorprone even in well-known textbooks and papers, since the generalization to the polyadic setting from the monadic one is sometimes non-trivial, which requires different techniques. Therefore, we think there is a need for a detailed exposition of the basic model theoretical results of PML proved in the modal logic setting, to provide a unified reference for further studies of PML, and this is the goal of the paper. In this paper, we review the definition of filtration and ultrafilter extension for polyadic language and give proofs for some basic theorems including the saturation theorem of ultrafilter extension in a purely model theoretical way other than algebraic one. Then we give a clarification on proving van-Benthem characterization theorem of PML in order to exhibit differences in the proof from the monadic cases. Finally, we also give a model theoretical proof for the Craig Interpolation Theorem of PML while the theorem was treated as a corollary of some algebraic results in the literature.