Axiomatizing a Minimal Discussive Logic

Abstract

In the paper we analyse the problem of axiomatizing the minimal variant of discussive logic denoted as $$ {\textsf {D}}_{\textsf {0}}$$ 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 .

On Paracomplete Versions of Jaśkowski's Discussive Logic

“…Last, but not least, the present approach could be generalized by using other (weaker) modal logics as a basis for corresponding systems, similar to how the minimal variant D 0 of D 2 is axiomatized with the help of the deontic normal logic D in [19].…”

“…• ¬(A ∧ ¬B) → w d (A → w d ¬(B → w d ⊥) • (A → w d ¬(B → w d ⊥) → w d ¬(¬(A → w d ⊥) ∧ ¬¬(B → w d ⊥)) • ¬(A ∧ ¬B) → w d ¬(¬(A → w d ⊥) ∧ ¬¬(B → w d ⊥)) • ¬ ¬(¬(A ∧ ¬B) → w d ⊥) ∧ ¬¬(¬(A → w d ⊥) ∧ ¬¬(B → w d ⊥)) ,The case of Ax 12 is obvious from (Imp cl ) and classical logic expressed in the language with ¬ and ∧ 19. Since ∨ l d is definable in the considered language, for the language with ∨ l d , one could just add two axioms: ¬((A∨ l d B)∧((A → w d ⊥)∧¬B)), ¬(¬(A∨ l d B)∧¬((A → w d ⊥)∧¬B)).…”