Degree-preserving Gödel logics with an involution: intermediate logics and (ideal) paraconsistency

Abstract

In this paper, we study intermediate logics between the logic $\mathrm{G}_{\sim}^{\leq}$, the degree-preserving companion of Gödel fuzzy logic with involution $\mathrm{G}_{\sim}$, and classical propositional logic CPL, as well as the intermediate logics of their finite-valued counterparts $\mathrm{G}_{n \sim}^{\leq}$. Although $\mathrm{G}_{\sim}^{\leq}$ and $\mathrm{G}_{n \sim}^{\leq}$are explosive w.r.t. Gödel negation $\neg$, they are paraconsistent w.r.t. the involutive negation $\sim$. We introduce the notion of saturated paraconsistency, a weaker notion than ideal paraconsistency, and we fully characterize the ideal and the saturated paraconsistent logics between $\mathrm{G}_{n \sim}^{\leq}$and CPL. We also identify a large family of saturated paraconsistent logics in the family of intermediate logics for degree-preserving finite-valued $\L$ukasiewicz logics.