If we view fuzzy logic as a logic, i.e., as a particular case of a multivalued logic, then one of the most natural questions to ask is whether the corresponding propositional logic is decidable, i.e., does there exist an algorithm that, given two propositional formulas F and G, decides whether these two formulas always have the same truth value. It is known that the simplest fuzzy logic, in which & = min and ∨ = max, is decidable. In this paper, we prove a more general result: that all propositional fuzzy logics with algebraic operations are decidable, We also show that this result cannot be generalized further: e.g., no deciding algorithm is possible for logics in which operations are algebraic with constructive (non-algebraic) coefficients.