 For Gödel–Dummett logic the axiomatic system is strongly complete with respect to its semantics. However, the set of tautologies of the semantics for Łukasiewicz logic is not recursively axiomatizable as shown by Bruno Scarpellini (1962). Emil Ragaz (1981) proved that it is actually \(\Sigma_2\)-complete in the sense of arithmetical hierarchy. The situation is even worse in the case of Product logic and Hájek’s basic fuzzy logic BL, where the sets of first-order tautologies of all structures given by continuous t-norms are as complex as true arithmetics (Montagna 2001). Completeness can be achieved either by adding a suitable infinitary inference rule to the Hilbert-style proof system, as done by Louise Hay (1963) for Łukasiewicz logic, or by generalizing the set of truth-values (see next section).