Propositional variables occurring exactly once In candidate modal axioms

One does not often encounter a proposed axiom for extending one modal logic to another with the following feature: in the axiom in question some propositional variable (sentence letter) appears only once. Indeed, for a large range of modal
logics L, which includes all normal modal logics, the sole occurrence of such a sentence letter can be replaced by a propositional truth or falsity constant, to give an arguably simpler axiom yielding the same extension of L, explaining the rarity
of such ‘variable-isolating’ axioms in the literature. But the proof of this simple (and in one form or another, well-known) result – appearing here as Lemma 2.1 – is sensitive to the choice of modal primitives. It breaks down, for example, when, instead of necessity (or possibility), the sole non-Boolean primitive is taken to be noncontingency (or contingency), the main topic of Sections 0 and 4, the latter closing with a selection of the main problems left open. Between these, which we shall have occasion, inter alia, to observe that the (routine) proof of the lemma referred to (which is postponed to a final Appendix, Section 5) is also sensitive to
the choice of Boolean primitives (Section 3).