 Beginning, then, with the preservation behaviour of rules (as in Garson 1990 and 2001) it is necessary first to distinguish a (sequent-to-sequent) rule’s having a local preservation characteristic with respect to a class V of valuations—that for each v ∈ V, the rule preserves the property of holding on v—from its having a global preservation characteristic with respect to V: if the premiss-sequents hold on every v ∈ V, then the conclusion-sequent holds on every v ∈ V. For brevity, a sequent will be called V-valid when it holds on each v ∈ V. Then we distinguish the local range of a rule ρ, denoted Loc(ρ) from its global range, Glo(ρ), taking the definitions as relative to a background language the class of all valuations for which we denote by U: