The class of projective propositional logics is defined by a certain format of the definition of truth functions for their connectives with respect to a semantic theory. All finite valued logics, but also in finite valued Gödel logic are shown to be projective. Analytic Gentzen type calculi are uniformly derived for all projective logics. Admissibility of cut rules and other structural rules is investigated. The special case of Gödel logics is exemplified in detail and compared with the previous approach of Avron (based on hypersequents).
Baaz, M., & Fermüller, C. G. (1999). Analytic calculi for projective logics. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 1617, pp. 36–51). Springer Verlag. https://doi.org/10.1007/3-540-48754-9_8