Studia Logica

, Volume 88, Issue 3, pp 349–383

An Approach to Glivenko’s Theorem in Algebraizable Logics

Article

DOI: 10.1007/s11225-008-9109-6

Cite this article as:
Torrens, A. Stud Logica (2008) 88: 349. doi:10.1007/s11225-008-9109-6

Abstract

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.

Keywords

Glivenko’s Theorem quasivariety algebraic semantics equivalential deductive system algebraizable logic compatible expansion 

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Departament de Probabilitat, Lògica i Estadística, Facultat de MatemàtiquesUniversitat de BarcelonaBarcelonaSpain

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