In this paper, we examine several lesser-known properties of da Costa’s calculi \(C_n\) , where \(1 \leqslant n < \omega \) . We demonstrate that the pair of formulas: \({\sim }(\alpha \wedge {\sim }\alpha )\) and \(\alpha \wedge {\sim }\alpha \) , is not the sole pair from which any conclusion can be derived. We provide a proof, using \(C_1\) as an example, that a form of ’reduction of negation’ holds true. Additionally, we propose several alternative axiomatizations for \(C_1\) , which lead to the definition of its weakenings and the introduction of da Costa-like hierarchies of paraconsistent calculi axiomatized over the positive fragment of intuitionistic propositional logic.
