Recognizable logics

We look into the recognition problem for extensions of Johansson's minimal logic J. It is proved that certain of the known logics are recognizable over J. Namely, recognizability over J is revealed for all well-composed logics possessing Craig's interpolation property (CIP), the restricted interpolation property (IPR), or the projective Beth property (PBP). It is proved that the logic JF is not reliably recognizable over J. Furthermore, we establish a link between the algebraic and the modified Kripke semantics, and give a criterion for being reliably recognizable in terms of characteristic formulas.