A modal extension of Jaśkowski’s discussive logic $\textbf{D}_\textbf{2}$

Abstract

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.

A Kotas-Style Characterisation of Minimal Discussive Logic

“…Moreover, it can be generalised to any compound formula which results in the below Lemma. Its proof goes similarly as the proof of Reference [1] (Fact 4.6). However, there are essential changes for the case of discussive operators, so we present it for the sake of completeness of considerations.…”

“…For not so straight history of axiomatisation of D 2 see References [16,17].) The same method was used inter alia in Reference [1] to axiomatize a variant of D 2 with modal operators.…”

“…Intuitively, during discussions participants can contradict each other, but a possible external observer as well as particular participants would not conclude that everything follows from such discussion. (Some analysis on this matter can be found in [1]. )…”

“…Moreover, in Jaśkowski's intuitive model, operators take only auxiliary role and modal operators are not present in the object language of the discussive logic. Such a variant seems to be natural and has been considered in [1].…”

On Paracomplete Versions of Jaśkowski's Discussive Logic

“…4 On the other hand, we do not mean here Jaśkowski's discussive logic could not tolerate non-standard approaches. See, for example, [32], where an extended version of Jaśkowski's model of discussion with debaters employing modal operators explicitly is presented.…”