Axiomatizing a Minimal Discussive Logic

Abstract

In the paper we analyse the problem of axiomatizing the minimal variant of discussive logic denoted as {textsf {D}}_{textsf {0}}. Our aim is to give its axiomatization that would correspond to a known axiomatization of the original discussive logic {textsf {D}}_{textsf {2}}. The considered system is minimal in a class of discussive logics. It is defined similarly, as Jaśkowski’s logic {textsf {D}}_{textsf {2}} but with the help of the deontic normal logic textbf{D}. Although we focus on the smallest discussive logic and its correspondence to {textsf {D}}_{textsf {2}}, we analyse to some extent also its formal aspects, in particular its behaviour with respect to rules that hold for classical logic. In the paper we propose a deductive system for the above recalled discussive logic. While formulating this system, we apply a method of Newton da Costa and Lech Dubikajtis—a modified version of Jerzy Kotas’s method used to axiomatize {textsf {D}}_{textsf {2}}. Basically the difference manifests in the result—in the case of da Costa and Dubikajtis, the resulting axiomatization is pure modus ponens-style. In the case of {textsf {D}}_{textsf {0}}, we have to use some rules, but they are mostly needed to express some aspects of positive logic. {textsf {D}}_{textsf {0}} understood as a set of theses is contained in {textsf {D}}_{textsf {2}}. Additionally, any non-trivial discussive logic expressed by means of Jaśkowski’s model of discussion, applied to any regular modal logic of discussion, contains {textsf {D}}_{textsf {0}}.