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Original Paper
Glivenko like theorems in natural expansions of BCK-logic
Article first published online: 3 FEB 2004
DOI: 10.1002/malq.200310082
Copyright © 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
1521-3870/asset/cover.gif?v=1&s=3df32dbed8d9153ac6eafe1eaa786854b9b48345 "Mathematical Logic Quarterly")
Additional Information
How to Cite
Cignoli, R. and Torrens Torrell, A. (2004), Glivenko like theorems in natural expansions of BCK-logic. Mathematical Logic Quarterly, 50: 111–125. doi: 10.1002/malq.200310082
Publication History
- Issue published online: 3 FEB 2004
- Article first published online: 3 FEB 2004
- Manuscript Accepted: 10 OCT 2003
- Manuscript Revised: 9 OCT 2003
- Manuscript Received: 18 JUL 2003
- Abstract
- References
- Cited By
Keywords:
- Bounded BCK-algebra;
- involutive BCK-algebra;
- bounded pocrim;
- algebraic semantics;
- natural expansion of a quasivariety;
- natural expansion of a logic;
- regular element;
- Glivenko's theorem;
- bounded BCK-logic
Abstract
The classical Glivenko theorem asserts that a propositional formula admits a classical proof if and only if its double negation admits an intuitionistic proof. By a natural expansion of the BCK-logic with negation we understand an algebraizable logic whose language is an expansion of the language of BCK-logic with negation by a family of connectives implicitly defined by equations and compatible with BCK-congruences. Many of the logics in the current literature are natural expansions of BCK-logic with negation. The validity of the analogous of Glivenko theorem in these logics is equivalent to the validity of a simple one-variable formula in the language of BCK-logic with negation. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)