Toward a general frame semantics for modal many-valued logics

Frame semantics, given by Kripke or neighborhood frames, do not give completeness theorems for all modal logics extending, respectively, K and E. Such shortcoming can be overcome by means of general frames, i.e., frames equipped with a collection of admissible sets of worlds (which is the range of possible valuations over such frame). We export this approach from the classical paradigm to modal many-valued logics by defining general $${\varvec{A}}$$A-frames over a given residuated lattice $${\varvec{A}}$$A (i.e., the usual frames with a collection of admissible $${\varvec{A}}$$A-valued sets). We describe in detail the relation between general Kripke and neighborhood $${\varvec{A}}$$A-frames and prove that, if the logic of $${\varvec{A}}$$A is finitary, all extensions of the corresponding logic E of $${\varvec{A}}$$A are complete w.r.t. general neighborhood frames. Our work provides a new approach to the current research trend of generalizing relational semantics for non-classical modal logics to circumvent axiomatization problems.