Modal Weak Kleene Logics Through Variables Inclusion

Abstract This paper presents a novel internal modal weak Kleene semantics and its derived logics. Our approach offers an intuitive understanding of modal operators as first-order weak Kleene quantifiers, drawing inspiration from the standard translation of classical modal logic. We explore the properties of this semantics and its associated logics. Our primary contribution lies in characterization theorems for some of these modal weak Kleene logics, leveraging classical modal logic, augmented with a refined variables inclusion requirement. These results not only extend the established characterization of non-modal weak Kleene logics but also provide fresh insights into the interpretations of our modal weak Kleene logics. Specifically, building on these technical findings, we propose philosophical interpretations for our logics and their semantics that coherently extend those of non-modal weak Kleene logics.