Proof Theory of Fuzzy Logics: Urquhart's C and Related Logics

We investigate the proof theory of Urquhart's C and other logics underlying the most prominent fuzzy logics, such as Godel, Product, and lukasiewicz logic. All these logics share the property that their truth values are linearly ordered. We define hypersequent calculi for such logics, and show the following results: (1) Contraction-free counterparts of intuitionistic logic and Godel logic (including C) admit cut-elimination. (2) Validity in these logics is decidable. (3) Hajek's basic fuzzy logic BL properly extends the contraction-free Godel logic; the axiom for commutativity of the minimum is independent from the other axioms of BL. (4) All abovementioned logics are distinct from each other.