Logica Universalis

, Volume 2, Issue 2, pp 209–233

Abelian Logic and the Logics of Pointed Lattice-Ordered Varieties

  • Francesco Paoli
  • Matthew Spinks
  • Robert Veroff
Article

DOI: 10.1007/s11787-008-0034-2

Cite this article as:
Paoli, F., Spinks, M. & Veroff, R. Log. univers. (2008) 2: 209. doi:10.1007/s11787-008-0034-2

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 -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 

Mathematics Subject Classification (2000).

Primary 03G25 Secondary 03G10, 03C05 

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Copyright information

© Birkhaeuser 2008

Authors and Affiliations

  • Francesco Paoli
    • 1
  • Matthew Spinks
    • 2
  • Robert Veroff
    • 3
  1. 1.Department of EducationUniversity of CagliariCagliariItaly
  2. 2.Mathematical InstituteUniversity of BernBernSwitzerland
  3. 3.Department of Computer ScienceUniversity of New MexicoAlbuquerqueUSA

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