This paper examines several strategies for extending da Costa’s calculus \(C_1\) in a manner that avoids collapsing into classical propositional logic. Our primary emphasis is on the gently paraconsistent extensions, that is, those that admit the principle of gentle explosion. We commence by reviewing some results related to \(C_1\) and Sette’s calculus \(P_1\) , which is expected to serve as the top extension among all the gently paraconsistent calculi analysed in this study. The subsequent discussion will focus on da Costa’s calculus in relation to De Morgan’s laws and intuitionistic implication. It is well-known that the negation-free fragment of \(C_1\) is positive classical logic. As an alternative approach, we propose a weakening of \(C_1\) such that its negation-free fragment corresponds to positive intuitionistic propositional logic. The weakening presents several noteworthy characteristics, one of which is that extending it with the law of non-contradiction does not result in a collapse into classical propositional logic.
