This paper is a contribution to the study of two distinct kinds of logics for modelling uncertainty. Both approaches use logics with a two-layered modal syntax, but while one employs classical logic on both levels and infinitely-many multimodal operators, the other involves a suitable system of fuzzy logic in the upper layer and only one monadic modality. We take two prominent examples of the former approach, the probability logics P_{lin} and Pr_{pol} (whose modal operators correspond to all possible linear/polynomial inequalities with integer coefficients), and three logics of the latter approach: Pr^{Ł}, Pr^{ŁΔ} and Pr^{PŁΔ }(given by the Łukasiewicz logic and its expansions by the Baaz–Monteiro projection connective Δ and also by the product conjunction). We describe the relation between the two approaches by giving faithful translations of Pr_{lin} and Pr_{pol} into, respectively, Pr^{ŁΔ} and Pr^{PŁΔ}, and vice versa. We also contribute to the proof theory of two-layered modal logics of uncertainty by introducing a hypersequent calculus HPr^{Ł} for the logic Pr^{Ł}. Using this formalism, we obtain a translation of Pr_{lin} into the logic Pr^{Ł}, seen as a logic on hypersequents of relations, and give an alternative proof of the axiomatization of Pr_{lin}.

P. Baldi, P.Cintula, C.Noguera. Classical and Fuzzy Two-Layered Modal Logics for Uncertainty: Translations and Proof-Theory, *International Journal of Computational Intelligence Systems,*

https://doi.org/10.2991/ijcis.d.200703.001