Sahlqvist Correspondence for Instantial Neighbourhood Logic

Studies in Logic, Vol. 14, No. 3 (2021): 1–23  PII: 1674-3202(2021)-03-0001-23

Zhiguang Zhao

Abstract.

In the present paper, we investigate the Sahlqvist-type correspondence theory for instantial neighbourhood logic (INL), which can talk about existential information about the neighbourhoods of a given world and is a mixture between relational semantics and neighbourhood semantics. The increased expressivity and its ability to talk about certain relational patterns of the neighbourhood function makes it possible to ask what kind of properties can this language define on the frame level, whether the “Sahlqvist” fragment of instantial neighbourhood logic could be larger than the rather small KW-fragment. (H. Hansen, 2003) We have two proofs of the correspondence results, the first proof is obtained by using standard translation and minimal valuation techniques directly, the second proof follows M. Gehrke et al. (2005) and H. Hansen (2003), where we use bimodal translation method to reduce the correspondence problem in instantial neighbourhood logic to normal bimodal logics in classical Kripke semantics. We give some remarks and future directions at the end of the paper.