Structural completeness properties are investigated for a range of popular t-norm based fuzzy logics—including Łukasiewicz Logic, Gödel Logic, Product Logic, and Hájek's Basic Logic—and their fragments. General methods are defined and used to establish these properties or exhibit their failure, solving a number of open problems.

Humberstone asks whether every theorem of BCI provably implies $\phi\to\phi$ for some formula $\phi$. Meyer conjectures that the axiom $\mathbf{B}$ does not imply any such "self-implication." We prove a slightly stronger result, thereby confirming Meyer's conjecture.