 Section 3 picks up the theme of sequent-to-sequent rules, concentrating on semantic analysis in the style of J. W. Garson of such rules, and also on the existence and uniqueness of connectives satisfying given rules (a topic introduced by H. Hiż). These last two are seen to be connected with what we call cut-inductivity and id-inductivity of specifically sequent-calculus rules, via, in the former case, a proposal identifying the existence of a connective (with prescribed logical behaviour, given the perspective provided by a logic not explicitly providing such a connective) with conservative extension (of the given logic by rules for the new connective detailing that behaviour). This proposal was made by N. D. Belnap; some problems for conservative extension as a sufficient condition for granting the existence or intelligibility of a new connective with such-and-such logical properties are noted. Section 4 collects several cases of interest in which one might wonder about the existence of connectives with various such properties. For instance (this being Example 3 out of a total of seven) one might ask if it makes sense to have a 1-ary connective with the property that two applications of this connective are equivalent to a single application of (classical) negation. Section 5 assembles supplementary notes and references.