Abstract
In this paper we consider the implicational fragment of Abelian logic \({{{\sf A}_{\rightarrow}}}\). We show that although the Abelian groups provide an semantics for the set of theorems of \({{{\sf A}_{\rightarrow}}}\) they do not for the associated consequence relation. We then show that the consequence relation is not algebraizable in the sense of Blok and Pigozzi (Mem Am Math Soc 77, 1989). In the second part of the paper, we investigate an extension of \({{{\sf A}_{\rightarrow}}}\) in the same language and having the same set of theorems and show that this new consequence relation is algebraizable with the Abelian groups as its equivalent algebraic semantics. Finally, we show that although \({{{\sf A}_{\rightarrow}}}\) is not algebraizable, it is order-algebraizable in the sense of Raftery (Ann Pure Appl Log 164:251–283, 2013).
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References
Anderson, A., and N. D. Belnap, Entailment: The Logic of Relevance and Necessity, Vol. I, Princeton University Press, Princeton, 1975.
Blok W. J., Jónsson B.: Equivalence of consequence operations. Studia Logica 83, 91–110 (2006)
Blok, W. J., and D. Pigozzi, Algebraizable logics, Memoirs of the American Mathematical Society 77, 1989.
Blyth, T. S., Lattices and Ordered Algebraic Structures, Universitext, Springer-Verlag, New York, 2005.
Butchart S., Kowalski T.: A note on monothetic BCI. Notre Dame Journal of Formal Logic 47(4), 541–544 (2006)
Casari, E., Comparative logics and Abelian l-groups, in R. Ferro et al. (eds.), Logic Colloquium ’88, North Holland, Amsterdam, 1989.
Daoji M.: BCI-algebras and Abelian groups. Mathematica Japonica 32, 693–696 (1987)
Galatos, N., P. Jipsen, T. Kowalski., and H. Ono, Residuated Lattices: An Algebraic Glimpse at Substructural Logics, Studies in Logic and the Foundations of Mathematics, vol. 151, Elsevier, Amsterdam, 2007.
Galli A., Lewin A., Sagastume M.: The logic of equilibrium and Abelian lattice ordered groups. Archive for Mathematical Logic 43, 141–158 (2004)
Humberstone L.: Variations on a theme of Curry. Notre Dame Journal of Formal Logic 47(1), 101–131 (2006)
Humberstone, L., The Connectives, MIT Press, Cambridge, 2011.
Kalman J. A.: Axiomatizations of logics with values in groups. London Math Society 14, 193–199 (1976)
Meyer R. K., Ono H.: The finite model property for BCK and BCW. Studia Logica 53, 107–118 (1994)
Meyer, R. K., and J. Slaney, Abelian logic from A to Z, in R. Routley, G. Priest, and J. Norman (eds.), Paraconsistent Logic: Essays on the Inconsistent, Analytica, Philosophia Verlag, Munich, 1989.
Meyer, R. K., and J. Slaney, A, still adorable, in M. Coniglio, W. Carnielli, and I. D’Ottaviano (eds.), Paraconsistency: The Logical Way to the Inconsistent, Marcel Dekker, New York, 2002.
Paoli F.: Logic and groups. Logic and Logical Philosophy 9, 109–128 (2001)
Paoli F., Spinks M., Veroff R.: Abelian logic and the logics of pointed lattice-ordered varieties. Logica Universalis 2, 209–233 (2008)
Raftery J.: Order algebraizable logics. Annals of Pure and Applied Logic 164, 251–283 (2013)
Restall, G., An Introduction to Substructural Logics, Routledge, London, 2000.
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Butchart, S., Rogerson, S. On the Algebraizability of the Implicational Fragment of Abelian Logic. Stud Logica 102, 981–1001 (2014). https://doi.org/10.1007/s11225-013-9515-2
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DOI: https://doi.org/10.1007/s11225-013-9515-2