Non-truth-functional fibred semantics

Until recently, truth-functionality has been considered essential to the mechanism for combining logics known as fibring. Following the first efforts towards extending fibred semantics to logics with non-truth-functional operators, this paper aims to clarify the subject at the light of ideas borrowed from the theory of general logics as institutions and the novel notion of non-truth-functional room. Besides introducing the relevant concepts and constructions, the paper presents a detailed worked example combining classical first-order logic with the paraconsistent propositional system C-1, for which a meaningful semantics is obtained. The possibility of extending this technique to build first-order versions of further logics of formal inconsistency is also discussed