Solution of some open problems in the arithmetical complexity of first-order fuzzy logics

This short paper addresses the open problems left in the paper F. Montagna and C. Noguera. Arithmetical complexity of first-order predicate fuzzy logics over distinguished semantics. Journal of Logic and Computation, 20(2):399–424, 2010. Besides giving solutions to these two problems, some clarification concerning the role of the full vocabulary (including functional symbols) in the proofs there given is also discussed.