In the paper we analyse the problem of axiomatizing the minimal variant of discussive logic denoted as $$ {\textsf {D}}_{\textsf {0}}$$ D 0. Our aim is to give its axiomatization that would correspond to a known axiomatization of the original discussive logic $$ {\textsf {D}}_{\textsf {2}}$$ D 2. The considered system is minimal in a class of discussive logics. It is defined similarly, as Jaśkowski’s logic $$ {\textsf {D}}_{\textsf {2}}$$ D 2 but with the help of the deontic normal logic (...) $$\textbf{D}$$ D. Although we focus on the smallest discussive logic and its correspondence to $$ {\textsf {D}}_{\textsf {2}}$$ D 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 $$ {\textsf {D}}_{\textsf {2}}$$ D 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 $$ {\textsf {D}}_{\textsf {0}}$$ D 0, we have to use some rules, but they are mostly needed to express some aspects of positive logic. $$ {\textsf {D}}_{\textsf {0}}$$ D 0 understood as a set of theses is contained in $$ {\textsf {D}}_{\textsf {2}}$$ D 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 $$ {\textsf {D}}_{\textsf {0}}$$ D 0. (shrink)

In Jaśkowski’s model of discussion, discussive connectives represent certain interactions that can hold between debaters. However, it is not possible within the model for participants to use explicit modal operators. In the paper we present a modal extension of the discussive logic $\textbf{D}_{\textbf{2}}$ that formally corresponds to an extended version of Jaśkowski’s model of discussion that permits such a use. This logic is denoted by $\textbf{m}\textbf{D}_{\textbf{2}}$. We present philosophical motivations for the formulation of this logic. We also give syntactic characterizations (...) of the logic and propose a comparison with certain other modal systems. In particular, we prove that $\textbf{m}\textbf{D}_{\textbf{2}}$ is neither normal nor regular. On the basis of the axiomatization of $\textbf{D}_{\textbf{2}}$, we give an axiomatization of $\textbf{m}\textbf{D}_{\textbf{2}}$. We also give another axiomatization which is not based on the axiomatization of $\textbf{D}_{\textbf{2}}$. Furthermore, we give a natural Kripke-style semantics for $\textbf{m}\textbf{D}_{\textbf{2}}$ and prove the respective adequacy theorems. (shrink)

This book contains a selection of the papers presented at the Logic, Reasoning and Rationality 2010 conference in Ghent. The conference aimed at stimulating the use of formal frameworks to explicate concrete cases of human reasoning, and conversely, to challenge scholars in formal studies by presenting them with interesting new cases of actual reasoning. According to the members of the Wiener Kreis, there was a strong connection between logic, reasoning, and rationality and that human reasoning is rational in so far (...) as it is based on logic. Later, this belief came under attack and logic was deemed inadequate to explicate actual cases of human reasoning. Today, there is a growing interest in reconnecting logic, reasoning and rationality. A central motor for this change was the development of non-classical logics and non-classical formal frameworks. The book contains contributions in various non-classical formal frameworks, case studies that enhance our apprehension of concrete reasoning patterns, and studies of the philosophical implications for our understanding of the notions of rationality. (shrink)

Pietruszczak (Bull Sect Log 38(3/4):163–171, 2009) proved that the normal logics K45 , KB4 (=KB5), KD45 are determined by suitable classes of simplified Kripke frames of the form ⟨W,A⟩ , where A⊆W. In this paper, we extend this result. Firstly, we show that a modal logic is determined by a class composed of simplified frames if and only if it is a normal extension of K45. Furthermore, a modal logic is a normal extension of K45 (resp. KD45; KB4; S5) if (...) and only if it is determined by a set consisting of finite simplified frames (resp. such frames with A≠∅; such frames with A=W or A=∅; such frames with A=W). Secondly, for all normal extensions of K45, KB4, KD45 and S5, in particular for extensions obtained by adding the so-called “verum” axiom, Segerberg’s formulas and/or their T-versions, we prove certain versions of Nagle’s Fact (J Symbol Log 46(2):319–328, 1981) (which concerned normal extensions of K5). Thirdly, we show that these extensions are determined by certain classes of finite simplified frames generated by finite subsets of the set N of natural numbers. In the case of extensions with Segerberg’s formulas and/or their T-versions these classes are generated by certain finite subsets of N. (shrink)