Calculi for Symmetric Generalized Galois Logics
Symmetric generalized Galois logics are motivated by proof theoretical, categorical and linguistic regularities, among others. This paper focuses on the proof theory of these logics. First, the connection between “hemi-distributivity” and the non-associative Lambek calculus with multiple right-hand sides is scrutinized. The admissibility of the single algebraic cut for the fusion-fission fragment of this calculus is proven. Next, axiomatizations are given for all the symmetric generalized Galois logics that result from the non-associative bi-Lambek calculus via an addition of a Grishin inequation. Lastly, symmetric generalized Galois logics are formalized as Belnap-style display calculi. The cut rule is proven admissible in these calculi too.