The Lindenbaum lemma saying that completely meet-irreducible closed sets form a basis of any finitary closure system is an easy-to-prove yet crucial result transcending algebraic logic. While the finitarity restriction is crucial for its usual proof, it is not necessary: there are indeed works proving it (or its variant for a larger class of finitely meet-irreducible closed sets) for non-finitary closure systems arising from particular infinitary logics (i.e., substitution-invariant consequence relations). There is also a general result proving it for a wide class of logics with strong p-disjunction and a countable Hilbert-style axiomatization. Identifying the essential properties of strong p-disjunctions we prove a variant of the Lindenbaum lemma for closure systems which are 1) defined over countable sets, 2) countably axiomatized, and 3) frames (in the order-theoretic sense) but not necessarily substitution-invariant.

A one-premiss rule is said to be archetypal for a consequence relation when not only is the conclusion of any application of the rule a consequence (according to that relation) of the premiss, but whenever one formula has another as a consequence, these formulas are respectively equivalent to a premiss and a conclusion of some application of the rule. We are concerned here with the consequence relation of classical propositional logic and with the task of extending the above notion of archetypality to rules with more than one premiss, and providing an informative characterization of the set of rules falling under the more general notion.