This is the second of two related papers. In "Revising Z: Part I - logic and semantics" (this journal) we introduced a simple specification logic ZC comprising a logic and a semantics (in ZF set theory). We then provided an interpretation for (a rational reconstruction of) the specification language Z within ZC. As a result we obtained a sound logic for Z, including the basic schema calculus. In this paper we extend the basic framework with more sophisticated features (including schema operations) and we mount a critique of a number of concepts used in Z. We further demonstrate that the complications and confusions which these concepts introduce can be avoided without compromising expressibility.