Paraconsistent da Costa Weakening of Intuitionistic Negation: What does it mean?

In this paper we consider the systems of weakening of intuitionistic negation logic mZ, introduced in [1], [2], which are developed in the spirit of da Costa's approach. We take a particular attention on the philosophical considerations of the paraconsistent mZ logic w.r.t. the constructive semantics of the intuitionistic logic, and we show that mZ is a subintuitionistic logic. Hence, we present the relationship between intuitionistic and paraconsistent subintuitionistic negation used in mZ. Then we present a significant number of examples for this subintuitionistic and paraconsistent mZ logics: Logic Programming with Fiting's fixpoint semantics for paraconsistent weakening of 3-valued Kleene's and 4-valued Belnap's logics. Moreover, we provide a canonical construction of infinitary-valued mZ logics and, in particular, the paraconsistent weakening of standard Zadeh's fuzzy logic and of the Godel-Dummet t-norm intermediate logics.