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

Arnon Avron
2021 Logic Journal of the IGPL
The logic $G3^{&lt;}_{{{}^{\scriptsize{-}}}\!\!\textrm{L}}$ was introduced in Robles and Mendéz (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 $G3^{&lt;}_{{{}^{\scriptsize{-}}}\!\!\textrm{L}}$ is exactly the intersection of $G3^{\{1\}}_{{{}^{\scriptsize{-}}}\!\!\textrm{L}}$ and
more » ... ptsize{-}}}\!\!\textrm{L}}$, the two truth-preserving 3-valued logics which are based on the same truth tables. (In$G3^{\{1\}}_{{{}^{\scriptsize{-}}}\!\!\textrm{L}}$the set${\cal D}$of designated elements is$\{1\}$, while in$G3^{\{1,0.5\}}_{{{}^{\scriptsize{-}}}\!\!\textrm{L}}{\cal 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^{&lt;}_{{{}^{\scriptsize{-}}}\!\!\textrm{L}}$. Then we show how, by adding one axiom (in the case of$G3^{\{1\}}_{{{}^{\scriptsize{-}}}\!\!\textrm{L}}$) or one new rule of inference (in the case of$G3^{\{1,0.5\}}_{{{}^{\scriptsize{-}}}\!\!\textrm{L}}$), we get strongly sound and complete systems for$G3^{\{1\}}_{{{}^{\scriptsize{-}}}\!\!\textrm{L}}$and$G3^{\{1,0.5\}}_{{{}^{\scriptsize{-}}}\!\!\textrm{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.