On a Generalization of Glivenko’s Theorem

Abstract

We offer a generalization of the well-known Glivenko’s theorem on double-negation translation. In “Sur quelques points de la Logique de M. Brouwer” V.I. Glivenko got a result, which is now called Glivenko’s theorem and which establishes the equivalence between a statement that a formula belongs to classical propositional logic and a statement that a double-negation of this formula belongs to intuitionistic propositional logic. Glivenko’s theorem is an important achievement in the field of research concerning links between logics conducted using the embedding operation. Here we propose a generalization of Glivenko’s theorem and describe a method which is based on this generalization for constructing analogues of the statements that is some special form of Glivenko’s theorem. In this paper we used author’s original sublogics of classical propositional logic. In particular, logic $Int_{<\omega,\omega>}$ played a principal role (it is, also, a sublogic of intuitionistic porpositional logic). The use of this logic made it possible to give such a generalization of Glivenko’s theorem that covers some extensive (cardinality of the continuum) class of sublogics of intuitionistic propositional logic.