Generalised Epstein Semantics Characterization, Interpolation, Undefinability, and (In)Completeness

Abstract

This paper is a continuation of investigations concerning Epstein semantics from (Krawczyk 2022). The main tool of the present paper is the model-theoretic S-set construction introduced in (Krawczyk 2022). We use it to prove several results: 1) that each Epstein model has uncountably many equivalent Epstein models, 2) that the logic of generalised Epstein models is the S-set invariant fragment of CPL (analogon of the celebrated van Benthem characterization theorem for modal logic), 3) that several sets of Epstein models are undefinable, 4) that logics of undefinable sets of relations can be finitely axiomatised. We also use other techniques to prove: 5) that there is uncountably many extensions of the logic of generalised Epstein models and, following the well-known phenomenon of Kripke-incompleteness in modal logics, we show that 6) there is uncountably many Epstein-incomplete logics. Finally, we prove that 7) the logic of generalised Epstein models has the interpolation property.