Proof Systems for 3-valued Logics Based on Gödel's Implication

The logic G3<-L was introduced in Robles and Mendez (2014, Logic Journal of the IGPL, 22, 515-538) as a paraconsistent logic which is based on Gödel's 3-valued matrix, except that Kleene-Łukasiewicz's negation is added to the language and is used as the main negation connective. We show that G3<-L is exactly the intersection of G31-L and G31,0.5, the two truth-preserving 3-valued logics which are based on the same truth tables. (In G3L the set D of designated elements is 1, while in G31,0.5 D= 1,0.5 We then construct a Hilbert-type system which has (MP) for to as its sole rule of inference, and is strongly sound and complete for G3<L. Then we show how, by adding one axiom (in the case of G31L or one new rule of inference (in the case of G31,0.5 L, we get strongly sound and complete systems for G31-L and G31,0.5-L. Finally, we provide quasi-canonical Gentzen-type systems which are sound and complete for those logics and show that they are all analytic, by proving the cut-elimination theorem for them.