Some results on Jaśkowski’s discursive logic
Jaśkowski  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 and higher order predicate calculus (, ). Jaśkowski’s system of discussive or discursive propositional calculus can actually be extended to predicate calculus in at least two ways. We have the intention using this calculus of building later as a basis for a discussive theory of sets. One way is that studied by Da Costa and Dubikajtis. Another one is developed in this paper as a solution to a problem formulated by Da Costa. In this work we study a first order discussive predicate calculus J∗∗ .The paper consists of three parts. In the first part we introduce the calculus J∗∗ and, following Prof. D. Makinson’s suggestion, we show that it is not identical with the predicate calculus  of Da Costa and Dubikajtis. An axiomatization of J∗∗ is presented. In the second one, we introduce new discussive connectives and study some of the properties. We observe that the usual Kripke semantics can be adapted to the calculus J∗∗ .
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