Definability and interpolation in extensions of Johansson's minimal logic

We herein study analogs of Beth's theorem on implicit definability [1] as well as Craig's interpolation property CIP in the family E(J) of extensions of Johansson's minimal logic J [2]. The logic J is paraconsistent unlike the intuitionistic logic Int. At the same time, J and Int have the same positive fragment J(+), and we can use known methods and results on superintuitionistic logics in our study of the family E(J). Note that the paraconsistent systems C-n ,1 < n < w, and CCw introduced by N.C.A. da Costa [3, 4] and R.Sylvan [5] are extensions of J(+) but incomparable with J. It follows from [6] that all extensions of J+ or of J have the Beth property BP. In 1962 K.Schutte [7] proved CIP for the intuitionistic predicate logic. His proof implies CIP also for J and J(+). It is known that in the family E(Int) of superintuitionistic logics there exist exactly sixteen logics with the projective Beth property PBP [8], among them eight logics have CIP [9]. In the present paper we find many examples of logics with CIP or PBP in E(J) which do not contain Int. Simultaneously we get exhaustive lists of positive logics in E(J(+)) with the interpolation property or with PBP. Also we find all positively axiomatizable logics with PBP in E(J) and describe them.