The Jacobson Radical of a Propositional Theory

Abstract

Alongside the analogy between maximal ideals and complete theories, the Jacobson radical carries over from ideals of commutative rings to theories of propositional calculi. This prompts a variant of Lindenbaum's Lemma that relates classical validity and intuitionistic provability, and the syntactical counterpart of which is Glivenko's Theorem. The Jacobson radical in fact turns out to coincide with the classical deductive closure. As a by-product we obtain a possible interpretation in logic of the axioms-as-rules conservation criterion for a multi-conclusion Scott-style entailment relation over a single-conclusion one. MSC 2010: 03F03, 03F65.

A General Glivenko–Gödel Theorem for Nuclei

“…By an extension ⊲ ′ of an entailment relation ⊲ we mean in general an entailment relation ⊲ ′ such that ⊲ ⊆ ⊲ ′ . We say that an extension ⊲ ′ of ⊲ is conservative if also ⊲ ⊇ ⊲ ′ and thus ⊲ = ⊲ ′ altogether [61,62,31,32].…”

“…As compared to recent literature on Glivenko's theorem [36,28,52,51,57,46,56,42,31,34], 1 the purpose of the present paper is to generalise Glivenko's theorem from double negation to an arbitrary nucleus, from provability in a calculus to an abstract consequence relation, and from propositional logic to any set of objects whatsoever.…”

“…In logic this gives us a multi-purpose conservation criterion (Theorem 5.2), by which propositional and predicate logic can be handled in parallel. The prime instance of course is Glivenko's theorem (Application 5.3(i)) as a syntactical conservation theorem (see also [31,32]):…”