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Kripke semantics for BL and GBL

### Andrew Lewis-Smith

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.