THE JACOBSON RADICAL OF A PROPOSITIONAL THEORY

Abstract Alongside the analogy between maximal ideals and complete theories, Jacobson radical carries over from of commutative rings to theories propositional calculi. This prompts a variant Lindenbaum’s Lemma that relates classical validity intuitionistic provability, syntactical counterpart which is Glivenko’s Theorem. The in fact turns out coincide with deductive closure. As by-product we obtain possible interpretation logic axioms-as-rules conservation criterion for multi-conclusion Scott-style entailment relation single-conclusion one.