In this note we describe a discussive set theory which is related to a certain modal set theory as discussive propositional logic is related to the system S5 of Lewis.

Jaśkowski [3] presented a new propositional calculus labeled “discussive propositional calculus”, to serve as an underlying basis for inconsistent but non-trivial theories. This system was later extended to lower andhigher order predicate calculus . Jaśkowski’s system of discussiveor discursive propositional calculus can actually be extended to predicatecalculus in at least two ways. We have the intention using this calculus ofbuilding later as a basis for a discussive theory of sets. One way is thatstudied by Da Costa and Dubikajtis. Another one (...) is developed in this paperas a solution to a problem formulated by Da Costa. In this work we study afirst order discussive predicate calculus J∗∗.The paper consists of three parts. In the first part we introduce thecalculus J∗∗ and, following Prof. D. Makinson’s suggestion, we show that itis not identical with the predicate calculus [2] of Da Costa and Dubikajtis.An axiomatization of J∗∗ is presented. In the second one, we introduce newdiscussive connectives and study some of the properties. We observe thatthe usual Kripke semantics can be adapted to the calculus J∗∗. (shrink)