The paper is a contribution to the development of fuzzy logic, namely to the arithmetic properties of the monadic fuzzy predicate logic. The author first recalls basic results in classical monadic predicate logic, which is complete and decidable, and then shows that in fuzzy logic this problem is much more complicated and interesting. He first focuses on Łukasiewicz logic and shows that the set TAUT (tautologies w.r.t. standard semantics) as well as the set fTAUT (tautologies on all finite models) are \(\Pi_1\) and the set SAT (satisfiable formulas w.r.t. standard semantics) is \(\Pi_1\)-complete and fSAT is \(\Sigma_1\). For monadic Gödel logic, TAUT is \(\Sigma_1\), fTAUT and SAT are \(\Pi_1\) and fSAT is recursive. For monadic product logic, the properties of TAUT and SAT are (up to now) unknown and fTAUT is \(\Pi_1\) and fSAT is recursive. Finally, for monadic BL-fuzzy logic, the properties of TAUT and SAT are unknown, fTAUT is \(\Pi_1\) and fSAT is \(\Sigma_1\). Some further results concerning arithmetical properties are added. The paper also points to open questions in this field.