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.
Similar content being viewed by others
References
Agliano, P., An algebraic investigation of linear logic, Rapporto Matematico n. 297 (1996), Università di Siena.
Bezhanishvili G. (2001) ‘Glivenko Type Theorems for Intuitionistic Modal Logics’. Studia Logica 67, 98–109
Birkhoff, G., Lattice Theory, 3rd edition. Colloq. Publ., vol. 25, Amer. Math. Soc., Providence, 1967.
Blok, W., and D. Pigozzi, Algebraizable Logics, Memoirs of the American Mathematical Society 396, Providence, 1989.
Blok W.J., Raftery J.G. (1997) ‘Varieties of commutative residuated integral pomonoids and their residuation subreducts’. J. Algebra, 190, 280–328
Burris S., Sankappanavar H.P. (1981) A Course in Universal Algebra, Graduate Texts in Mathematics. Springer-Verlag, New York
Cignoli R., Torrens A. (2002) ‘Free algebras in varieties of BL-algebras with a Boolean retract’. Algebra Universalis, 48, 55–79
Cignoli R., Torrens A. (2003) ‘Hájek basic fuzzy logic and Łukasiewicz infinite–valued logic’. Arch. Math. Logic, 42, 361–370
Cignoli R., Torrens A. (2004) ‘Glivenko like theorems in natural expansions of BCK-logics’. Math. Log. Quart. 50 2, 111–125
Czelakowski J. (2003) Protoalgebraic logics. Kluwer Academic Pub., Dordrecht
Epstein R.L. (1995) Propositional Logics, Second Edition. Oxford University Press, New York
Font J.M., Jansana R. (2003) ‘A General Algebraic Semantics for Sentential Logics’. Kluwer Academic Pub., Dordrecht
Frink O. (1962) ‘Pseudo-complements in semi-lattices’. Duke Math. J. 29, 505–514
Galatos N., Ono H. (2006) ‘Glivenko theorems for substructural logics over FL’. J. Symbolic Logic 71(4): 1353–1384
Glivenko V. (1929) ‘Sur quelques points de la logique de M. Brouwer’. Bull. Acad. des Sci. de Belgique, 15, 183–188
Grätzer G. (1979). Universal Algebra, Second Edition. Springer-Verlag, New York
Grätzer G. (1998). General theory of lattices, Second Edition. Birkhäuser, Basel
Los J., Suszko R. (1958). ‘Remarks on sentential logic’. Indag. Math. 20, 177–183
Troelstra, A., Lectures in Linear Logic, CSLI Lecture Notes, 1991
Wójcicki, R., Theory of Logical Calculi, Basic theory of consequence operations, vol 100 of Synthese Library, Reidel, Dordrecht, 1988.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Torrens, A. An Approach to Glivenko’s Theorem in Algebraizable Logics. Stud Logica 88, 349–383 (2008). https://doi.org/10.1007/s11225-008-9109-6
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11225-008-9109-6