We investigate intermediate logics that retain a weak form of contraction. Whereas intermediate logics are generally constructive and well-understood proof-theoretically, the same cannot be said for logics with restricted contraction. This is partly because such systems have a rich semantic motivation, being many-valued or "fuzzy." The result is that the majority of work in such logics focus on algebraic and semantic aspects, downplaying questions of proof. Indeed, the lack of a sufficiently worked-out proof theory is even worse in the case of so-called intermediate logics with fuzzy semantics.
Generalized Basic Logic (GBL) is one such logic, restricting the Basic Logic (BL) of Hajek by omitting pre-linearity from the axioms. We have succeeded in extending an algebraic semantics of Urquhart to BL (Hajek's Basic Logic), have proven soundness for BL under this semantics, and are currently working on the completeness result. Surprisingly, we have found a connection with Kripke semantics in the work of Bova-Montagna which could help simplify the existent approaches to fuzzy logic. We present our outstanding problem in relating Kripke structures in Totally Ordered Commutative Monoids.