A free IPC is a natural logic: Strong completeness for some intuitionistic free logics

IPC, the intuitionistic predicate calculus, has the property

1. Vc(Γ?A ^c /x) ? Γ??xA.Furthermore, for certain important Γ, IPC has the converse property
2. Γ??xA ? Vc(Γ?A ^c /x).
3. may be given up in various ways, corresponding to different philosophic intuitions and yielding different systems of intuitionistic free logic. The present paper proves the strong completeness of several of these with respect to Kripke style semantics. It also shows that giving up (i) need not force us to abandon the analogue of (ii).