 The existence and uniqueness questions for connectives turn out in the case of sequent calculus rules to be intimately connected with the structural rules (Id) and (Cut). Let us call the left and right insertion rules governing an n-ary connective #—or derivatively, the connective thereby governed—Id-inductive if the provability of #(φ1,…,φn) ≻ #(φ1,…,φn) follows from the provability of each φi ≻ φi (i = 1,…,n) by means of those rules (without assistance from other structural rules). Thus means that no provable sequents are lost if the rule (Id) itself is restricted to having applications φ ≻ φ for φ a propositional variable. And let us say that the rules in question are Cut-inductive if an application of the rule (Cut) with the formula #(φ1,…,φn) as cut-formula (i.e., as the φ of our representation of the rule (Cut) in Section  2  or the analogous rule in Set-Set or the multiset based frameworks) can be dispensed with if the premiss sequents for the application of the rule have been derived by (# Left) and (# Right) and (Cut) is allowed with the components φ1,…,φn as cut-formulas (but again, not allowing the use of other structural rules).