Combining logics using two-layer modal syntax

Two-layer modal syntax is based on two propositional languages (lower and upper one) and a modal language (a collection of modalities together with their arities) and features three kinds of formulae: (i) non-modal formulae build using lower propositional language, (ii) atomic modal formulae obtained by applying the modalities to non-modal ones, and (iii) complex modal formulae built from the atomic ones using the upper propositional language (i.e., the modalities cannot be nested and propositional languages cannot be mixed). Early examples of logics employing this kind of syntax were modal logics of uncertainty inspired by Hamblin’s seminal idea of reading the modal operator Pφ as ‘probably φ’ [6], meaning that the probability of φ is bigger than a given threshold (later elaborated by Fagin, Halpern and many others (see e.g. [2, 5]). These initial examples used the classical logic to govern the behavior of formulae of both modal and non-modal layer. An interesting departure from the classical paradigm has been proposed by Hájek and Harmancová in [4] (later developed in Hájek’s monograph [3]): they kept the classical logic as interpretation of the lower syntactical layer, but proposed to use Lukasiewicz logic in the upper layer, so that the truth degree of Pφ could be directly identified with the probability of φ. Later, other authors changed even the logic governing in the lower layer (e.g., to other fuzzy logic in order to allow for a treatment of uncertainty of vague events). This research gave rise to an interesting way of combining logics which allows to use one logic to reason about formulae (rules) of other logic. The aim of this talk is to propose foundations for further research in this promising area. We define an abstract notion of a two-layer syntax and logic, a general semantics of measured Kripke frames and prove two forms of completeness theorem.