A Weakly-Intuitionistic Logic I1

Abstract

In 1995, Sette and Carnielli presented a calculus, $I1$, which is intended to be dual to the paraconsistent calculus $P1$. The duality between $I1$ and $P1$ is reflected in the fact that both calculi are maximal with respect to classical propositional logic and they behave in a special, non-classical way, but only at the level of variables. Although some references are given in the text, the authors do not explicitly define what they mean by ‘duality’ between the calculi. For instance, no definition of the translation function from the language of $I1$ into the language of $P1$ (or from $P1$ to $I1$) was provided (see [4], pp. 88–90) nor was it shown that the calculi were functionally equivalent (see [13], pp. 260–261).
The purpose of this paper is to present a new axiomatization of $I1$ and briefly discuss some results concerning the issue of duality between the calculi.