Jixin Liu

This paper connects the following three apparently unrelated topics: an epistemic framework fighting logical omniscience, a class of generalized graphs without the arities of relations, and a family of non-normal modal logics rejecting the aggregative axiom. Through neighborhood frames as their meeting point, we show that, among many completeness results obtained in this paper, the limit of a family of weakly aggregative logics is both exactly the modal logic of hypergraphs and also the epistemic logic of local reasoning with veracity and positive introspection. The logics studied are shown to be decidable based on a filtration construction.

This paper connects the following four topics: a class of generalized graphs whose relations do not have fixed arities called hypergraphs, a family of non-normal modal logics rejecting the aggregative axiom, an epistemic framework fighting logical omniscience, and the classical group knowledge modality of ‘someone knows’. Through neighborhood frames as their meeting point, we show that, among many completeness results obtained in this paper, the limit of a family of weakly aggregative logics is both exactly the modal logic of hypergraphs and also the epistemic logic of local reasoning with veracity and positive introspection, and upon adding a single combinatorial axiom, it is also the logic of ‘someone knows’ for a fixed finite number of positively introspective agents. At the core of all these completeness results is a new canonical neighborhood model construction for monotone modal logics that is capable of dealing with all these diverse cases. We also provide an axiomatization for the logic of all non-n-colorable hypergraphs based on a filtration argument that also shows the decidability of the logics of hypergraphs we study.

Weakly Aggregative Modal Logic ) is a collection of disguised polyadic modal logics with n-ary modalities whose arguments are all the same. \ has interesting applications on epistemic logic, deontic logic, and the logic of belief. In this paper, we study some basic model theoretical aspects of \. Specifically, we first give a van Benthem–Rosen characterization theorem of \ based on an intuitive notion of bisimulation. Then, in contrast to many well known normal or non-normal modal logics, we show that each basic \ system \ lacks Craig interpolation. Finally, by model theoretical techniques, we show that an extension of \ does have Craig interpolation, as an example of amending the interpolation problem of \.