Table admissible inference rules

A recursive basis of inference rules is described which are instantaneously admissible in all table (residually finite) logics extending one of the logics $\mathrm{Int}$ and $Grz$. A rather simple semantic criterion is derived to determine whether a given inference rule is admissible in all table superintuitionistic logics, and the relationship is established between admissibility of a rule in all table (residually finite) superintuitionistic logics and its truth values in $\mathrm{Int}$.