Strong decidability and strong recognizability

Extensions of Johansson's minimal logic J are considered. It is proved that families of negative and nontrivial logics and a series of other families are strongly decidable over J. This means that, given any finite list $Rul$ of axiom schemes and rules of inference, we can effectively verify whether the logic with axioms and schemes, $J+Rul$, belongs to a given family. Strong recognizability over J is proved for known logics Neg, Gl, and KC as well as for logics LC and NC and all their extensions.