MANY-VALUED LOGICS
The logics complete not for a single variety but for a family of them: the matrices with more than two truth values, where consequence is preservation of designated values. The algebras range over MV, BL, and MTL algebras, De Morgan and Kleene lattices, and bilattices.
Entries are organized by the algebra. Finitely-valued matrices—the strong and weak Kleene systems, Łukasiewicz's finite logics, Post algebras—sit alongside the continuum-valued fuzzy logics complete for t-norm varieties: Łukasiewicz over MV-algebras, Gödel and product logics, the monoidal t-norm logic MTL and Hájek's basic logic BL, with nilpotent-minimum and Rational Pavelka refinements. The four-valued De Morgan matrix of first-degree entailment, and the bilattices, treat gaps and gluts together; the paraconsistent matrices—the Logic of Paradox and da Costa's calculi—admit gluts, while partial logic admits gaps. Where a logic is complete for a subvariety already named under Heyting—Gödel logic, Nelson's strong negation—it is cross-listed rather than duplicated.