On modal logics defining jaśkowski’s D2-consequence

Abstract

Jaśkowski’s logic D2 (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 D2 one can use normal and regular logics weaker than S5. In his paper Jaśkowski used a deducibility relation which we will denote by-D2 and which fulfilled the following condition: A1,…, An;D2B iff ┌⋄◊A•1→(…→(⋄◊A•n→⋄◊B•)…)┐∈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 D2-consequence.