Type 〈1,1〉〈1,1〉 fuzzy quantifiers determined by fuzzy measures on residuated lattices. Part I. Basic definitions and examples

We study fuzzy quantifiers of type 〈1,1〉〈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〉〈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.