Negations in the Adjunctive Discursive Logic

In the logical literature, Discursive (or Discussive) Logic introduced by Stanis law Jaśkowski is seen as one of the earliest examples of the so-called paraconsistent logic. There is some confusion over what is actually discursive logic nevertheless. One of the possible sources of the confusion is easy to discern; it comes from the fact that Jaśkowski published his two papers in Polish and their English translations appeared many years later. Up till 1999, no one but a Polish reader was able to read Jaśkowski‘s paper on the discursive conjunction and consequently some authors took discursive logic to be a foremost example of a non-adjunctive logic. The situation became even more complicated when da Costa, Dubikajtis and Kotas presented an axiomatization with discursive connectives as primitive symbols. It turned out that a connective of the discursive conjunction they considered did not coincide with any connective presented by Jaśkowski. Moreover, their axiomatization contained some axiom schemata that were not generally valid in Jaśkowski‘s logic. The purpose of this paper is to clarify the confusion surrounding the discursive logic. We will present a new (direct) semantics and axiomatization of Jaśkowski‘s adjunctive discursive logic and show how to define and axiomatize two additional connectives of negation.