Type fuzzy quantifiers determined by fuzzy measures on residuated lattices. Part I. Basic definitions and examples

We study fuzzy quantifiers of type 〈 1 , 1 〉 defined using fuzzy measures and integrals. Residuated lattices are considered as structures of truth values. We present basic notions on fuzzy measures over algebras of fuzzy subsets of a given fuzzy set, and the definition and necessary properties of so-called ⊙-fuzzy integrals. Residuated lattice operations are established to derive operations between fuzzy sets describing degrees in which formulas expressing relations between fuzzy sets are true. Finally, a general definition of type 〈 1 , 1 〉 fuzzy quantifiers determined by fuzzy measures and integrals is introduced and several examples of important natural language quantifiers are modeled using this approach.