On modal logics defining Jaśkowski's D2-consequence

Marek Nasieniewski

Andrzej Pietruszczak

pp. 141-161

in: Koji Tanaka, Francesco Berto, Edwin D. Mares, Francesco Paoli (eds), Paraconsistency, Berlin, Springer, 2013

Abstract

Jaśkowski's logic D ₂ (as a set of formulae) was formulated with the help of the modal logic S5 (see Jaśkowski, Stud Soc Sci Torun I(5):57–77, 1948; Stud Soc Sci Torun I(8):171–172, 1949). In Furmanowski (Stud Log 34:39–43, 1975), Perzanowski (Rep Math Log 5:63–72, 1975), Nasieniewski and Pietruszczak (Bull Sect Logic 37(3–4):197–210, 2008) it was shown that to define D ₂ one can use normal and regular logics weaker than S5. In his paper Jaśkowski used a deducibility relation which we will denote by⊢_{D 2} and which fulfilled the following condition: A ₁,…,A _n ⊢;_{D 2} B iff (ulcorner lozenge {A}_{1}^{ullet } ightarrow (ldots ightarrow (lozenge {A}_{n}^{ullet } ightarrow lozenge {B}^{ullet })ldots ,)urcorner in mathbf{S5}), where (−)^∙ is a translation of discussive formulae into the modal language. We indicate the weakest normal and the weakest regular modal logic which define D ₂ -consequence.

Cited authors

Jaśkowski Stanisław