In the late forties, Stanislaw Jaśkowski published two papers on the discursive (or discussive) sentential calculus, D 2 . He provided a definition of it by an interpretation in the language of S5 of Lewis. The known axiomatization of D 2 with discursive connectives as primitives was introduced by da Costa, Dubikajtis and Kotas in 1977. It turns out, however, that one of the axioms they used is not a thesis of the real Jaśkowski’s calculus. In fact, they built a new system, D* 2 for short, that differs from D 2 in many respects. The aim of this paper is to introduce a direct Kripke-type semantics for the system, axiomatize it in a new way and prove soundness and completeness theorems. Additionally, we present labelled tableaux for D* 2 .