## 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.

### MSC:

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

Zbl 1016.06006
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