We extend Dunn’s treatment of various forms of negation developed in the context of his theory of generalized Galois logics (known as gaggle theory), by dropping the assumption of distribution. We also study modal operators of possibility and impossibility in a non-distributive context and in standard Kripke semantics, thus improving significantly over existing approaches developed in the last decade or so on the semantics of modalities when distribution of conjunction over disjunction and conversely is dropped. We prove representation and completeness theorems for the related logical calculi in appropriate Kripke frames. Without distribution, the points of the frame (we call them information sites) appear as possessing incomplete only information, supporting the truth of a disjunction without necessarily supporting the truth of either or Our approach is based on and extends past results we have obtained on the (topological) representation (and Stone type duality) of non-distributive lattices with additional operators.
Hartonas, C. (2016). Reasoning with Incomplete Information in Generalized Galois Logics Without Distribution: The Case of Negation and Modal Operators. In Outstanding Contributions to Logic (Vol. 8, pp. 279–312). Springer. https://doi.org/10.1007/978-3-319-29300-4_14