Sette's calculus P1 and some hierarchies of paraconsistent systems

Abstract

The necessary condition for a calculus to be paraconsistent is that its consequence relation is not explosive. This results in rejection of the principle of ex contradictione sequitur quodlibet. In 1973, Sette presented a calculus, denoted as $P^1$, which is paraconsistent only at the atomic level, i.e. $\alpha $ and ${\sim }\alpha $ yield any $\beta $ if, and only if the formula $\alpha $ is not a propositional variable. The calculus has been viewed as one of the noteworthy paraconsistent calculi since then. The objective of this paper is to propose a new semantics for Sette’s calculus and present some hierarchies of the paraconsistent calculi, which are based on $P^1$. We demonstrate that $P^1$ is sound and complete with respect to the semantics and so are all the calculi under consideration.