On Interpolation Problem in Paraconsistent Extensions of the Minimal Logic

Interpolation problem over Johansson's minimal logic J is investigated. We present a semantic method for proving interpolation and apply it to extensions of J. A modification of Kripke-style semantics for the logic J is offered and completeness theorem for a number of propositional J-logics is proved. We find a sufficient condition for interpolation in J-logics in terms of Kripke models. We propose a construction of matched product of models, which allows to prove interpolation theorem in a number of known extensions of the minimal logic. We show that the behavior of interpolation in the family of J-logics differs from that in superintuitionistic and positive logics. It is known that intersection of two incomparable superinstuitionistic or positive logics never has Craig's interpolation property CIP, and the sum of such logics with CIP always has CIP. We prove that the sum of J-logics with CIP can be without CIP and even without the restricted interpolation property IPR. On the other hand, there are examples of incomparable J-logics whose intersection has CIP. We use a classification of paraconsistent logics built by S.Odintsov in order to describe location of paraconsistent logics with CIP.