Available online at www.sciencedirect.com ScienceDirect Fuzzy Sets and Systems 242 (2014) 31–55 www.elsevier.com/locate/fss Type 〈1, 1〉 fuzzy quantifiers determined by fuzzy measures on residuated lattices. Part I. Basic definitions and examples ✩ Antonín Dvoˇ rák ∗ , Michal Holˇ capek National Supercomputing Center IT4Innovations, Division of University of Ostrava, Institute for Research and Applications of Fuzzy Modeling, 30. dubna 22, 701 03 Ostrava 1, Czech Republic Available online 17 October 2013 Abstract 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 defi- nition 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.  2013 Elsevier B.V. All rights reserved. Keywords: Fuzzy measure; Fuzzy integral; Fuzzy logic; Fuzzy quantifier 1. Introduction We propose a general model of fuzzy quantifiers (with two arguments) determined by fuzzy measures and integrals. Besides its potential for use in reasoning (a logical theory for these quantifiers in line with previous studies [27,28] is in preparation), it can also be used in various applications. Using various fuzzy measure spaces, we cover a wide class of fuzzy quantifiers discussed in the literature (see the examples in Section 5). However, in this series of papers we are mainly interested in establishing a theoretical framework for a sufficiently general class of fuzzy quantifiers that can be tailored to individual applications with important theoretical properties guaranteed. The study of generalized quantifiers evolved from pioneering work by Mostowski [26], Lindström [23], Barwise and Cooper [1], and van Benthem [39] into a large research field. Peters and Westerståhl provide an overview of the field and have described many results [34]. In the classical setting, a quantifier Q of type 〈1, 1〉 (such as “every”, “many”, etc.) is usually modeled, given a universe M , as a mapping Q M from the Cartesian product of power sets P (M) × P (M) to the set {false, true} or, equivalently, as a binary relation on subsets of M . As previously discussed [11], when we think about the definition and properties of generalized quantifiers such as many, few, and others, we feel that their truth values should not change abruptly if we gradually change the cardinality ✩ This work was supported by the European Regional Development Fund in the IT4Innovations Centre of Excellence project (CZ.1.05/1.1.00/02.0070). * Corresponding author. Tel.: +420 597091406; fax: +420 596120478. E-mail addresses: antonin.dvorak@osu.cz (A. Dvoˇ rák), michal.holcapek@osu.cz (M. Holˇ capek). 0165-0114/$ – see front matter  2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.fss.2013.10.003