The mathematics of Epstein semantics

Source abstract

Abstract This paper is a mathematical investigation of Epstein semantics. One of the main tools of the present paper is the model-theoretic $\textsf{S}$-set construction introduced in [19]. We use it to prove several results: (1) that each Epstein model has uncountably many equivalent Epstein models, (2) that the logic of Epstein models is the $\textsf{S}$-set invariant fragment of $\textsf{CPL}$, (3) that several sets of Epstein models are undefinable, (4) that logics of undefinable sets of relations can be finitely axiomatized. We also use other techniques to prove (5) that there are uncountably many Epstein-incomplete logics and that (6) the logic of Epstein models has the interpolation property.