An Approach to Glivenko’s Theorem in Algebraizable Logics

Studia Logica 88 (3):349-383 (2008)
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In a classical paper [15] V. Glivenko showed that a proposition is classically demonstrable if and only if its double negation is intuitionistically demonstrable. This result has an algebraic formulation: the double negation is a homomorphism from each Heyting algebra onto the Boolean algebra of its regular elements. Versions of both the logical and algebraic formulations of Glivenko’s theorem, adapted to other systems of logics and to algebras not necessarily related to logic can be found in the literature (see [2, 9, 8, 14] and [13, 7, 14]). The aim of this paper is to offer a general frame for studying both logical and algebraic generalizations of Glivenko’s theorem. We give abstract formulations for quasivarieties of algebras and for equivalential and algebraizable deductive systems and both formulations are compared when the quasivariety and the deductive system are related. We also analyse Glivenko’s theorem for compatible expansions of both cases.



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Citations of this work

Algebraic semantics for the (↔,¬¬)‐fragment of IPC.Katarzyna Słomczyńska - 2012 - Mathematical Logic Quarterly 58 (1-2):29-37.

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References found in this work

Protoalgebraic Logics.Janusz Czelakowski - 2001 - Kluwer Academic Publishers.
Glivenko Theorems for Substructural Logics over FL.Nikolaos Galatos & Hiroakira Ono - 2006 - Journal of Symbolic Logic 71 (4):1353 - 1384.
Universal Algebra.George Grätzer - 1982 - Studia Logica 41 (4):430-431.

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