We introduce and study the notions of a PAC-substructure of
a stable structure, and a bounded substructure of an
arbitrary substructure, generalizing [10]. We give
precise definitions and equivalences, saying what it means for
properties such as PAC to be first order, study some examples
(such as differentially closed fields) in detail, relate the
material to generic automorphisms, and generalize a “descent
theorem” for pseudo-algebraically closed fields to the stable
context. We also point out that the elementary invariants of
pseudo-algebraically closed fields from [6] are
also valid for pseudo-differentially closed fields.