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

TL;DR

This paper explores intermediate logics between degree-preserving Godel fuzzy logic with involution and classical logic, characterizing paraconsistent variants and their properties.

Contribution

It introduces saturated paraconsistency, fully characterizes ideal and saturated paraconsistent logics, and links these to finite-valued Lukasiewicz logics.

Findings

Degree-preserving Godel logics are explosive with respect to negation but paraconsistent with involutive negation.

Full characterization of ideal and saturated paraconsistent logics between Godel fuzzy logic and CPL.

Identification of a large family of saturated paraconsistent logics in finite-valued Lukasiewicz logic.

Abstract

In this paper we study intermediate logics between the degree preserving companion of Godel fuzzy logic with an involution and classical propositional logic CPL, as well as the intermediate logics of their finite-valued counterparts. Although these degree-preserving Godel logics are explosive with respect to Godel negation, they are paraconsistent with respect to the involutive negation. 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 the degree-preserving n-valued Godel fuzzy logic with an involution and CPL. We also identify a large family of saturated paraconsistent logics in the family of intermediate logics for degree-preserving finite-valued Lukasiewicz logics.