Implicit operations in MV-algebras and the connectives of Łukasiewicz logic

Authors:
Xavier Caicedo
Affiliations:
Department of Mathematics, Universidad de los Andes, Bogotá
Venue:
Algebraic and proof-theoretic aspects of non-classical logics
Year:
2007

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Abstract

It is shown that a conservative expansion of infinite valued Łukasiewicz logic by new connectives univocally determined by their axioms does not necessarily have a complete semantics in the real interval [0,1]. However, such extensions are always complete with respect to valuations in a family of MV-chains. Rational Łukasiewicz logic being the largest one that has a complete semantics in [0,1]. In addition, this logic does not admit expansions by axiomatic implicit connectives that are not already explicit. Similar results are obtained for n-valued Łukasiewicz logic and for the logic of abelian lattice ordered groups. These and related results are obtained by the study of compatible operations implicitly defined by identities in the varieties of MV-algebras and abelian l-groups; the pertaining algebraic results having independent interest.