Abstract.
We consider the class of pointed varieties of algebras having a lattice term reduct and we show that each such variety gives rise in a natural way, and according to a regular pattern, to at least three interesting logics. Although the mentioned class includes several logically and algebraically significant examples (e.g. Boolean algebras, MV algebras, Boolean algebras with operators, residuated lattices and their subvarieties, algebras from quantum logic or from depth relevant logic), we consider here in greater detail Abelian ℓ-groups, where such logics respectively correspond to: i) Meyer and Slaney’s Abelian logic [31]; ii) Galli et al.’s logic of equilibrium [21]; iii) a new logic of “preservation of truth degrees”.
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This paper was written while the second author was a Visiting Professor in the Department of Education at the University of Cagliari. The facilities and assistance provided by the University and by the Department are gratefully acknowledged.
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Paoli, F., Spinks, M. & Veroff, R. Abelian Logic and the Logics of Pointed Lattice-Ordered Varieties. Log. univers. 2, 209–233 (2008). https://doi.org/10.1007/s11787-008-0034-2
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DOI: https://doi.org/10.1007/s11787-008-0034-2