Modal logics of uncertainty with two-layer syntax: A general completeness theorem

Modal logics with two syntactical layers (both governed by classical logic) have been proposed as logics of uncertainty following Hamblin's seminal idea of reading the modal operator □φ as 'probably φ', meaning that the probability of φ is bigger than a given threshold. An interesting departure from that (classical) paradigm has been introduced by Hájek with his fuzzy probability logic when, while still keeping classical logic as interpretation of the lower syntactical layer, he proposed to use Łukasiewicz logic in the upper one, so that the truth degree of □φ could be directly identified with the probability of φ. Later, other authors have used the same formalism with different kinds of uncertainty measures and other pairs of logics, allowing for a treatment of uncertainty of vague events (i.e. also changing the logic in the lower layer). The aim of this paper is to provide a general framework for two-layer modal logics that encompasses all the previously studied two-layer modal fuzzy logics, provides a general axiomatization and a semantics of measured Kripke frames, and prove a general completeness theorem. © 2014 Springer-Verlag.