Abelian Logic and the Logics of Pointed Lattice-Ordered Varieties

Logica Universalis 2 (2):209-233 (2008)
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”
Keywords Abelian logic  lattice-ordered variety  assertional logic  logics preserving degrees of truth
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DOI 10.1007/s11787-008-0034-2
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Lloyd Humberstone (2013). Replacement in Logic. Journal of Philosophical Logic 42 (1):49-89.
Lloyd Humberstone (2014). Prior’s OIC Nonconservativity Example Revisited. Journal of Applied Non-Classical Logics 24 (3):209-235.
Lloyd Humberstone (2013). Aggregation and Idempotence. Review of Symbolic Logic 6 (4):680-708.

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