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 \( {\textsf {D}}_{\textsf {0}}\).
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Acknowledgements
Krystyna Mruczek-Nasieniewska and Marek Nasieniewski benefited from support provided by Polish National Science Centre (NCN), Grant No. 2016/23/B/HS1/00344. This research has been partly supported by the Interdisciplinary Scientific and Educational School of Moscow University “Brain, Cognitive Systems, Artificial Intelligence”.
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Presented by Heinrich Wansing; Received May 19, 2022.
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Grigoriev, O., Nasieniewski, M., Mruczek-Nasieniewska, K. et al. Axiomatizing a Minimal Discussive Logic. Stud Logica 111, 855–895 (2023). https://doi.org/10.1007/s11225-023-10042-9
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DOI: https://doi.org/10.1007/s11225-023-10042-9
Keywords
- Discussive logics
- The smallest discussive logic
- Discussive operators
- Accessibility relations
- Modal logic
- da Costa and Dubikajtis
- Embedding