Product Łukasiewicz logic. (English) Zbl 1059.03011

The authors develop a so-called product Łukasiewicz logic \(P{\L}\) which has the class of product MV-algebras as its semantic counterpart. These algebras have been investigated, e.g., by A. Di Nola and A. Dvurečenskij [Mult.-Valued Log. 6, No. 1–2, 193–215 (2001; Zbl 1016.06006)]. Also, for the sake of obtaining the standard completeness, an extension \(P{\L}'\) of the above logic is introduced. It is proved that the algebras of \(P{\L}'\) coincide with Montagna’s \(PMV^+\) algebras. Several completeness theorems are proved. The extensions of product Łukasiewicz logics by a unary connective \(\triangle\) are investigated as well. The overall exposition of product Łukasiewicz logics in this paper is smooth and rather self-contained. The authors found various connections between their product logics and other fuzzy logical systems.


03B50 Many-valued logic
03B52 Fuzzy logic; logic of vagueness
06D35 MV-algebras


Zbl 1016.06006
Full Text: DOI