Abstract
AbstractAlongside 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.
Highlights
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
Glivenko’s theorem from 1929 says that if a propositional formula φ is provable in classical logic, its double negation ¬¬φ is provable in intuitionistic logic
In the present note we follow the analogy between maximal ideals and complete theories to carry over the Jacobson radical from ideals of commutative rings to theories of propositional calculi (Section 5.2), where it turns out to coincide with the stable closure or with the closure with respect to classical logic (Proposition 3, Corollary 1)
Summary
Glivenko’s theorem from 1929 says that if a propositional formula φ is provable in classical logic, its double negation ¬¬φ is provable in intuitionistic logic. In the present note we follow the analogy between maximal (proper) ideals and complete (consistent) theories to carry over the Jacobson radical from ideals of commutative rings to theories of propositional calculi (Section 5.2), where it turns out to coincide with the stable closure or with the closure with respect to classical logic (Proposition 3, Corollary 1) This prompts a variant of Lindenbaum’s Lemma that relates classical validity and intuitionistic provability (Proposition 2), and the syntactical counterpart of which happens to be Glivenko’s Theorem in the form recalled above (Theorem 2). As a by-product we obtain a possible interpretation in logic (Theorem 3) of the axioms-as-rules conservation criterion (Theorem 1) for a multi-conclusion Scott-style entailment relation over a single-conclusion one This criterion has proved to be the common core of many a syntactical counterpart of a semantic conservation theorem corresponding to one of the aforementioned intersection theorems. As for the latter, disjunction elimination plays a central role in the proof of the former, together with some notorious features of (double) negation in intuitionistic logic and of provability in classical propositional logic (Lemmas 1 and 2)
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
