Fuzzy Equational Logic

Presented is a completeness theorem for fuzzy equational logic with truth values in a complete residuated lattice: Given a fuzzy set Σ of identities and an identity p ≈ q, the degree to which p ≈ q syntactically follows (is provable) from Σ equals the degree to which p ≈ q semantically follows from Σ. Pavelka style generalization of well-known Birkhoff’s theorem is therefore established.