 Let us sum up the lessons for the relations between existence and Cut-inductivity and between uniqueness and Id-inductivity that can be extracted from the foregoing discussion. Sequent calculus rules which are Id-inductive uniquely characterize the connective they govern, but not necessarily conversely: the Tonk rules are not Id-inductive, but they do uniquely characterize Tonk, since they are already so strong that any sequent with at least one formula on the left and at least one on the right can be proved with their aid, and hence, in the reduplicated system with Tonk′, we can certainly prove φ Tonk ψ ≻ φ Tonk′ ψ and the converse sequent—without even using the Tonk′ rules! (This is the kind of thing we have in mind in calling this a non-canonical form of uniqueness.) On the ‘existence’ side: it is not clear to the author whether any Cut-inductive rules governing a connective guarantee that they are conservative over the (system consisting of the) rules governing other connective—since Cut Elimination may fail for other reasons; nor do the examples just considered settle the converse, as we see from the following tabular summary of our findings, in which (a “Yes” by) “unique” means the rules uniquely characterize the connective, and “conservative” means that we have a conservative extension of any system of sequent-calculus rules not involving the given connective; Tonk* is not listed separately since it patterns the same way as Tonk: