Characterizing Quantifier Fuzzification Mechanisms: A behavioral guide for applications

Rúa Jenaro de la Fuente, s/n –Campus Vida – Universidade de Santiago de Compostela -15782 Santiago de Compostela – citius.usc.es © 2017, Elsevier. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/ Characterizing Quantifier Fuzzification Mechanisms: A behavioral guide for applications Díaz-Hermida, F., Pereira-Fariña, M., Vidal, J., Ramos-Soto, A. Version: accepted manuscript Diaz-Hermida, F., Pereira-Fariña, M., Vidal, J., Ramos-Soto, A. (2018) Characterizing Quantifier Fuzzification Mechanisms: A behavioral guide for applications. Fuzzy Sets And Systems, 345: 1-23. Doi: 10.1016/j.fss.2017.07.017 How to cite: Copyright information: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 Characterizing Quantifier Fuzzification Mechanisms: a behavioral guide for applications F. Diaz-Hermidaa,∗, M. Pereira-Fari˜nab,c,JuanC.Vidal a, A. Ramos-Sotoa aCentro de Investigaci´on en Tecnolox´ıas da Informaci´on (CITIUS), University of Santiago de Compostela, Campus Vida, E-15782, Santiago de Compostela, Spain bCentre for Argument Technology, University of Dundee, Dundee DD1 4HN, UK cDepartment of Philosophy and Anthropology, Universidade de Santiago de Compostela, Pz. de Mazarelos, Santiago de Compostela 15782, Spain Abstract Important advances have been made in the fuzzy quantification field. Nevertheless, some problems remain when we face the decision of selecting the most convenient model for a specific application. In the literature, several desirable adequacy properties have been proposed, but theoretical limits impede quantification models from simultaneously fulfilling every adequacy property that has been defined. Besides, the complexity of model definitions and adequacy properties makes very difficult for real users to understand the particularities of the different models that have been presented. In this work we will present several criteria conceived to help in the process of selecting the most adequate Quantifier Fuzzification Mechanisms for specific practical applications. In addition, some of the best known well-behaved models will be compared against this list of criteria. Based on this analysis, some guidance to choose fuzzy quantification models for practical applications will be provided. Keywords: fuzzy quantification, determiner fuzzification schemes, theory of generalized quantifiers, quantifier fuzzification mechanism, applications of fuzzy quantification 1. Introduction The evaluation of fuzzy quantified expressions is a topic that has been widely dealt with in literature [2, 7, 8, 10, 11, 12, 13, 14, 16, 17, 19, 20, 23, 29, 33, 35, 37, 39, 40]. The range of applications of fuzzy quantification includes fuzzy control [36], temporal reasoning in robotics [26], fuzzy databases [5], information retrieval [4, 13, 24, 25], data fusion [21, 37], syllogistic reasoning [27, 28] and more recently data-to-text applications [15, 30, 31, 32]. Moreover, the definition of adequate models to evaluate quantified expressions is fundamental to perform ‘computing with words’, topic that was suggested by Zadeh [41] to express the ability of programming systems in a linguistic way. ∗Corresponding author. Tel.: +0034 881816410 Email addresses: [email protected] (F. Diaz-Hermida), [email protected] (M. Pereira-Fari˜na), [email protected] (Juan C. Vidal), [email protected] (A. Ramos-Soto) Preprint submitted to Fuzzy Sets and Systems June 6, 2017 *Manuscript 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 In general, most approaches to fuzzy quantification use the concept of fuzzy linguistic quantifier to represent absolute or proportional fuzzy quantities. Zadeh [40] defined quantifiers of the first type as quantifiers used for representing absolute quantities (by using fuzzy numbers on N), and quantifiers of the second type as quantifiers used for representing relative quantities (defined by using fuzzy numbers on [0,1]). For analyzing the behavior of fuzzy quantification models different properties of convenient or necessary fulfillment have been defined [7, 9, 12, 20]. However, most of the approaches fail to exhibit a plausible behavior as it has been proved through the different reviews that have been published [2, 7, 8, 9, 20] and only a few [7, 11, 19, 20] seem to exhibit an adequate behavior in the general case. In this work, we will follow Gl¨ockner’s approximation to fuzzy quantification [20]. In his approach, the author generalizes the concept of generalized classic quantifier [3] (second order predicates or set relationships) to the fuzzy case; that is, a fuzzy quantifier is a fuzzy relationship between fuzzy sets. And then he rewrites the fuzzy quantification problem as the problem of looking for a mechanism to transform semi-fuzzy quantifiers (quantifiers in a middle point between generalized classic quantifiers and fuzzy quantifiers, used to specify the meaning of quantified expressions) into fuzzy quantifiers. The author calls these transformation mechanisms Quantifier Fuzzification Mechanisms (QFMs). Being based in the linguistic Theory of Generalized Quantifiers (TGQ) [3], this approach is able to handle most of the quantification phenomena of natural language. In addition, including quantification into a common theoretical framework following TGQ, it also allows the translation of most of the analysis that has been made from a linguistic perspective to the fuzzy case, and facilitates the definition and the test of adequacy properties. Gl¨ockner has also defined a rigorous axiomatic framework to ensure the good behavior of QFMs. Models fulfilling this framework are called Determiner Fuzzification Schemes (DFSs) and they comply with a broad set of properties that guarantee a good behavior from a linguistic and fuzzy point of view. See the recent [33] or [20] for a comparison between Zadeh’s and Gl¨ockner’s approaches. The DFS framework has supposed a notable advance and several well behaved QFMs have been identified [20], [9]. However, important problems still remain when we must face the decision of selecting an specific QFM for a practical application. First, it has been proved that no model can fulfill every desirable adequacy property that have been proposed [20], and as a consequence, a ‘perfect model’ cannot exist. Besides, the complexity of the definition of the models and adequacy properties makes really difficult for a user to decide which one is the most convenient for a certain application. In addition, as we will show along the exposition, there are some criteria that have not been previously taken into account for analyzing the plausible models and, even for the cases in which some criteria had been previously considered, a complete comparison among the behavior of at least the best-behaved models has not been done. In this work we will focus in, to the best of our knowledge, the best behaved QFMs: models FMD,FI,FA[9] and models M,MCX and Fowa [20] with the objective of establishing a set of criteria that facilitates the understanding of the behavioral differences among them and helps in the process of selecting the more convenient model for applications. All the selected models, being QFMs, present a more general definition than models following Zadeh’s framework [40]. Furthermore, some of them generalize other known approaches, as the ones based on the Sugeno or Choquet integrals. Thus, selected models comprise a really good representation of the ‘state of the art’ of fuzzy quantification. We refer the reader to the exhaustive and recent revision in [8] for a thoroughly comparative analysis of fuzzy quantification proposals. Previous state of the art 2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 revisions about the fuzzy quantification field can be found in [2, 7, 9, 20]. Before continuing, we would remark that only the models FA,M,MCX and Fowa fulfill the strict DFS framework, being alpha-cut based models like FMD and FI, previously considered as non plausible from the point of view of the DFS framework [20, section 7.2]. In order to understand the differences between these models and DFSs, we will first compare the selected models against the main properties considered into the QFM framework. Once the main differences derived from the properties described in [20] have been presented, we will introduce the new set of criteria that will allow us to improve the comparison between the different models and to prove that, for some problems, alpha-cut based models FMD and FIcan be superior to known DFSs. Moreover, as we will argue when we analyze the different models against the set of criteria introduced in this paper, ‘a clear winner’ cannot be identified, being the general situation that some models are more appropriate for some applications than others. The paper is organized as follows. Section 2 will summarize Gl¨ockner’s approach to fuzzy quantification, based on QFMs. In section 3, we will present the definition of the models FMD, FI,FA,M,MCX and Fowa. Section 4 will present the main properties considered in the QFM framework [20] and a brief comparison of the models FMD,FI,FA,M,MCX and Fowa, with the objective of clearly identifying the behavioral differences of these models with respect to these properties. Section 5 will be devoted to establish the set of criteria that will allow us to improve the comparison of the considered models, and to analyze the different models against this new set of criteria. Section 6 summarizes the results and establishes some criteria to guide in the model selection for applications. The paper is closed with some conclusions. 2. The fuzzy quantification framework To overcome Zadeh’s framework to fuzzy quantification Gl¨ockner [20] rewrote the problem of fuzzy quantification as the problem of looking for adequate ways to convert specification means (semi-fuzzy quantifiers) into operational means (fuzzy quantifiers). Fuzzy quantifiers are just a fuzzy generalization of crisp or classic quantifiers. Before giving the definition of fuzzy quantifiers, we will show the definition of classic quantifiers according to TGQ. Definition 1. A two valued (generalized) quantifier on a base set E ∅is a mapping Q : P(E)n−→ 2,wheren∈Nis the arity (number of arguments) of Q, 2={0,1}denotes the set of crisp truth values, and P(E)is the powerset of E. Examples of some definitions of classic quantifiers are: all (Y1,Y2)=Y1⊆Y2 at least80% (Y1,Y2)=|Y1∩Y2| |Y1|≥0.80 Y1∅ 1Y1=∅ In a fuzzy quantifier, arguments and results can be fuzzy. A fuzzy quantifier assigns a gradual result to each choice of X1,...,Xn∈ P(E), where by  P(E)we denote the fuzzy powerset of E. In the case Eis finite we will denote |E|=m. Definition 2. [20, definition 2.6] An n-ary fuzzy quantifier  Q on a base set E ∅is a mapping  Q: P(E)n−→ I=[0,1]. 3 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 For example, the fuzzy quantifier  all : P(E)2−→ Icould be defined as:  all (X1,X2)=inf max 1−μX1(e),μ X2(e):e∈E where by μX(e)we denote the membership function of X∈ P(E)and by in f we are denoting the infimum. Although a certain consensus may be achieved to accept the previous expression as a suitable definition for  all this has not to be the unique possible one. The problem of establishing consistent fuzzy definitions for quantifiers (e.g., ‘at least eighty percent’) is faced in [20] by introducing the concept of semi-fuzzy quantifiers. A semi-fuzzy quantifier represents a medium point between classic quantifiers and fuzzy quantifiers. Semi-fuzzy quantifiers are similar but far more general than Zadeh’s linguistic quantifiers [40]. A semi-fuzzy quantifier only accepts crisp arguments, as classic quantifiers, but let the result range over the truth grade scale I, as for fuzzy quantifiers. Definition 3. [20, definition 2.8] An n-ary semi-fuzzy quantifier Q on a base set E ∅is a mapping Q :P(E)n−→ I. Qassigns a gradual result to each pair of crisp sets (Y1,...,Yn). Examples of semi-fuzzy quantifiers are: about 5(Y1,Y2)=T2,4,6,8(|Y1∩Y2|)(1) at least about80% (Y1,Y2)=S0.5,0.8|Y1∩Y2| |Y1|X1∅ 1X1=∅ where T2,4,6,8(x)and S0.5,0.8(x)represent the common trapezoidal and Sfuzzy numbers1. Semi-fuzzy quantifiers are much more intuitive and easier to define than fuzzy quantifiers, but they do not solve the problem of evaluating fuzzy quantified sentences. In fact, additional mechanisms are needed to transform semi-fuzzy quantifiers into fuzzy quantifiers, i.e., mappings with domain in the universe of semi-fuzzy quantifiers and range in the universe of fuzzy quantifiers: Definition 4. [20, definition 2.10]A quantifier fuzzification mechanism (QFM) Fassigns to each semi-fuzzy quantifier Q :P(E)n→Ia corresponding fuzzy quantifier F(Q): P(E)n→Iof the same arity n ∈Nand on the same base set E. 3. Some paradigmatic QFMs 3.1. Standard DFSs based on trivalued cuts In this section we will present the three main Gl¨ockner’s approaches [20]. All the models that have been proposed by Gl¨ockner are standard DFSs, and as a consequence they show an excellent 1Functions Ta,b,c,dand Sα,γ are defined as Ta,b,c,d(x)=⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 0x≤a x−a b−aa<x≤b 1b<x≤c 1−x−c d−cc<x≤d 0d<x ,Sα,γ (x)=⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 0x<α 2(x−α) (γ−α)2α<x≤α+γ 2 1−2(x−γ) (γ−α)2α+γ 2<x≤γ 1γ<x 4 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 theoretical behavior. These models are called standard because they induce the standard tnorm min and the standard tconorm max.2 Before presenting models M[20, definition 7.22], MCX [20, definition 7.56] and Fowa [20, definition 8.13] we need to introduce some additional definitions. Definition 5. [20, definition 7.15]Let E be a referential set, X ∈ P(E)a fuzzy set, and γ∈I. Xmin γ,Xmax γ∈P(E)are defined as: Xmin γ=X>1 2:γ=0 X≥1 2+1 2γ:γ>0 Xmax γ=X≥1 2:γ=0 X>1 2−1 2γ:γ>0 where X≥α={e∈E:μX(e)≥α}is the alpha-cut of level αof X and X>α ={e∈E:μX(e)>α} is the strict alpha-cut of level α. In previous expression, Xmin γrepresents the elements that without doubt, belong to the fuzzy set Xfor the ‘cautiousness’ level γwhilst Xmax γincludes also the elements whose belongingness to the cautiousness level γis undefined. Elements that are not in Xmax γdo not belong to the cautiousness level γ. The cautiousness cut can be interpreted as a ‘trivalued set’ in which elements in Xmin γhave membership function of 1, whilst for elements in Xmax γ/Xmin γbelongingness is undefined (membership degree of 1 2). Membership function for elements that are not in Xmax γis 0. For the definition of M,MCX and Fowa we also need the fuzzy median operator: Definition 6. Fuzzy median med1 2:I×I−→ Iis defined as: med1 2(u1,u2)=⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ min (u1,u2):min (u1,u2)>1 2 max (u1,u2):max (u1,u2)<1 2 1 2:otherwise The following definitions extends the fuzzy median to fuzzy sets: Definition 7. Operator m1 2:P(I)→Iis defined as m1 2X=med1 2inf X,sup X for all X ∈P(I), where by inf X,sup X we are denoting the infimum and supremum of X, respectively. The set that contains all the possible images of a quantifier over the range defined by a three valued cut of level γis defined as [19, page 100]: Definition 8. Let Q :P(E)→Ibe a semi-fuzzy quantifier, X1,...,Xn∈ P(E)fuzzy sets and γ∈[0,1] a cautiousness level. S Q,X1,...,Xn(γ):[0,1] →Iis defined as: SQ,X1,...,Xn(γ)( X1,...,Xn)=Q(Y1,...,Yn):(Xi)min γ⊆Yi⊆(Xi)max γ 2In [20, section 3.4] it is explained how semi-fuzzy quantifiers can be used ‘to embed’ the classical logical functions. By means of the application of a QFM F, we can study if Ftransforms the classical logical functions into appropriate fuzzy logical functions. 5 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 Supremum and infimum of SQ,X1,...,Xn(γ)are represented by means of the following notation: Definition 9. Let Q : P(E)→Ibe a semi-fuzzy quantifier, X1,...,Xn∈ P(E)fuzzy sets and γ∈[0,1] a cautiousness level. Q,X1,...,Xn(γ):[0,1] →Iis defined as: Q,X1,...,Xn(γ)=sup SQ,X1,...,Xn(γ) Definition 10. Let Q : P(E)→Ibe a semi-fuzzy quantifier, X1,...,Xn∈ P(E)fuzzy sets and γ∈[0,1] a cautiousness level. ⊥Q,X1,...,Xn(γ):[0,1] →Iis defined as: ⊥Q,X1,...,Xn(γ)=inf SQ,X1,...,Xn(γ) Using previous definitions we present the three paradigmatic DFSs : Definition 11. [20, definition 7.22] Standard DFS M:(Q:P(E)→I)→( Q: P(E)→I)is defined as M(Q)( X1,...,Xn)=1 0 med1 2Q,X1,...,Xn(γ),⊥Q,X1,...,Xn(γ)dγ Definition 12. [20, definition 7.56, theorem 7.87] Standard DFS MCX :(Q:P(E)→I)→  Q: P(E)→Iis defined as MCX (Q)( X1,...,Xn)=sup QL V,W(X1,...,Xn):V1⊆W1,...,Vn⊆Wn where QL V,W(X1,...,Xn)=min ΞV,W(X1,...,Xn),inf {Q(Y1,...,Yn):Vi⊆Yi⊆Wi} ΞV,W(X1,...,Xn)= n min i=1min inf μXi(e):e∈Vi,inf 1−μXi(e):eWi Definition 13. [20, definition 8.13] Standard DFS Fowa :(Q:P(E)→I)→ Q: P(E)→Iis defined as Fowa (Q)( X1,...,Xn)=1 21 0 Q,X1,...,Xn(γ)dγ+1 21 0 ⊥Q,X1,...,Xn(γ)dγ 3.2. Alpha-cut based QFMs FIand FMD Now, we will present the two QFMs based on alpha-cuts FIand FMD. Definition 14. [11, section 2.1], [9, chapter 3]Let Q :P(E)n→Ibe a semi-fuzzy quantifier over a base set E. The QFM FMD is defined as: FMD (Q)(X1,...,Xn)=1 0 Q(X1)≥α,...,(Xn)≥αdα for every X1,...,Xn∈ P(E). When fuzzy sets X1,...,Xn∈ P(E)are normalized and we limit ourselves to the unary and binary quantifiers considered in the Zadeh’s framework, FMD coincides with the quantification model GD defined in [6, page 281], [34, section 3.3.2. and section 3.4.1.], [7, page 37]. In this way, FMD generalizes the GD model to the Gl¨ockner’s framework. Let us now present the definition of the FImodel. 6 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 Definition 15. [10], [11, section 2.2],[9, chapter 3] Let Q :P(E)n→Ibe a semi-fuzzy quantifier over a base set E. The QFM FIis defined as: FI(Q)( X1,...,Xn)=1 0 ...1 0 Q(X1)≥α1,...,(Xn)≥αndα1...dαn for every X1,...,Xn∈ P(E). 3.3. Non standard DFS FA The definition of the QFM FAis based on a probabilistic interpretation of fuzzy sets in which we interpret membership degrees as probabilities [9],[12]. However, the FAmodel can also be defined by means of fuzzy operators without any reference to probability theory. The QFM FAfulfills the axioms of the DFS framework but it is not a standard DFS, as the logic operators induced by the FAmodel are the product tnorm and the probabilistic sum tconorm. Definition 16. Let X ∈ P(E)be a fuzzy set, E finite. The probability of the crisp set Y ∈P (E) of being a representative of the fuzzy set X ∈ P(E)is defined as mX(Y)= e∈Y μX(e) e∈E\Y (1−μX(e)) As we have stated above, it is possible to make a similar definition without making any reference to probability theory. If we consider the product tnorm (∧(x1,x2)=x1·x2)andthe Lukasiewicz implication then mX(Y)is the equipotence between Yand X[1]: Eq(Y,X)=∧e∈E(μX(e)→μY(e)) ∧(μY(e)→μX(e)) Using the previous definition we define the FADFS as: Definition 17. [14, pag. 1359]Let Q :P(E)n→Ibe a semi-fuzzy quantifier, E finite. The DFS FAis defined as FA(Q)( X1,...,Xn)= Y1∈P(E) ...  Yn∈P(E) mX1(Y1)...mXn(Yn)Q(Y1,...,Yn)(2) for all X1,...,Xn∈ P(E). The next expression is an alternative definition of the model FA: FA(X1,...,Xn)= Y1∈P(E) ...  Yn∈P(E) Eq(Y1,X1)∧...∧Eq(Yn,Xn)∧Q(Y1,...,Yn) where ∨is the Lukasiewicz tconorm (∨(x1,x2)=min (x1+x2,1)), ∧is the product tnorm (∧(x1,x2)=x1·x2)andEq(Y,X)is the equipotence between the crisp set Yand the fuzzy set X. In this way, the definition of FAcan be done by means of common fuzzy operators. 7 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 4. The DFS axiomatic framework In this section we will present the DFS axiomatic framework [20]. In the previous reference, the author has dedicated the whole third and fourth chapters to describe the framework and the properties that are a consequence of it. For the sake of brevity, we will only give a general overview of the framework and some intuitions about the set of properties derived from it. We refer the reader to the previous publication for full detail. Definition 18. AQFMFis called a determiner fuzzification scheme (DFS) if the following conditions are satisfied for all semi-fuzzy quantifiers Q :P(E)n→I: Correct generalization U(F(Q)) =Qif n≤1(Z-1) Projection quantifiers F(Q)=πeif Q=πefor some e∈E(Z-2) Dualisation FQ =F(Q) n>0(Z-3) Internal joins F(Q∪)=F(Q) ∪n>0(Z-4) Preservation of monotonicity If Qis nonincreasing in the n-th arg, then (Z-5) F(Q)is nonincreasing in n-th arg, n>0 Functional application FQ◦n × i=1 fi=F(Q)◦n × i=1 F(fi)(Z-6) where f1,..., fn:E→E,E∅ 4.1. Main properties derived from the DFS framework We will only make a brief exposition of the main properties derived from the DFS framework. Full detail can be found in the aforementioned reference [20, chapters three and four.]. •Correct generalization (P1): perhaps, the most important property derived from the DFS framework is the correct generalization property. Correct generalization requires that the behavior of a fuzzy quantifier F(Q)over crisp arguments is the expected; that is, results obtained with a fuzzy quantifier F(Q)and with the corresponding semi-fuzzy quantifier Qmust coincide over crisp arguments. It is included in the DFS axiomatic framework for semi-fuzzy quantifiers of arities 0 and 1 (Z-1). •Quantitativity (P2): quantitative quantifiers do not depend on any particular characteristic of the elements of the base set. In the finite case, quantitative quantifiers can always be defined as a function of the cardinality of the boolean combinations of the argument sets. A QFM Fpreserves quantitativity if quantitative semi-fuzzy quantifiers are translated into quantitative fuzzy quantifiers by F. Most typical examples of quantifiers we find in the literature are quantitative (e.g., ‘around five’,‘at least 80%’, etc.). Non-quantitative quantifiers involve the reference to specific elements of the referential (e.g., ‘John’ in a set of people). •Projection quantifier (P3): Axiom Z-2 guarantees that the projection crisp quantifier πe(Y)(that returns 1 if e∈Yand 0 in other case) is generalized to the fuzzy projection quantifier πe(X)(that returns μX(e)). •Induced propositional logic (P4): we will say that a QFM comply with the induced propositional logic if crisp logical functions (¬(x),∧(x1,x2),∨(x1,x2),→(x1,x2)), that can be embedded into the definition of semi-fuzzy quantifiers, are generalized to acceptable fuzzy logical functions; that is, a negation operator, a tnorm,atconorm and a fuzzy 8 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 translated to the fuzzy case [9, chapter 3]. This property is not fulfilled by any DFS. However, loosing the internal negation property and as a consequence, the maintenance of the relationships of the ‘Aristotelian square’ seems more relevant than the fulfillment of the conservativity property. Model FI.Model FIlooses the internal meets property. Moreover, the internal negation property is only fulfilled in the finite case (see [9, chapter 3]). Loosing the internal meets property, FImodel does not guarantee absolute unary/binary transformations. For example, FI(about 10)( hard workers,well paid)and FI(about 10)(ha rd workers ∩well paid) are not equivalent, and then “about 10 hard workers are well paid” (evaluated by means of the binary absolute quantifier “about 10”)and“about 10 employees are hard workers and are well paid” (evaluated by means of the application of the unary version of the absolute quantifier ‘about 10’ and the induced tnorm of the model used to compute the intersection of ‘hard workers’ and ‘well paid’) will not produce the same results. 5.2. Aggregative behavior for low degrees of membership Aggregative behavior for low degrees of membership is one of the main critiques that has been made to the Zadeh’s count model [38],[20, section A.3], [2]. The intuition around aggregative behavior is that in evaluating quantified expressions, a large amount of elements fulfilling a property with ‘low degree’ of membership should not be confused with a small amount of elements fulfilling a property with ‘high degree’ of membership. In the case of the Zadeh’s model is easy to understand the meaning of aggregative behavior as: count (∃)( {0.01/e1,...,0.01/e100})=count (∃)( {1/e1,0/e2,...,0/e100})=1 in words, ‘exist one tall person’ can be fulfilled if there exists exactly ‘one tall person’,orifthere exist 100 people being ‘0.01 tall’. Although intuitions against aggregative behavior seem clear, giving up models presenting aggregative behavior will force us to discard non-standard DFS Fassociated to archimedean tconorms4.BeingF(∃)equal to [20, Theorem 4.61]: F(∃)( X)=sup  ∨m i=1(ai):A={a1,...,am}∈P(E)finite,aiajif i j for all X∈ P(E),thenF(∃)(X)will always present aggregative behavior for every non-standard DFS associated to an archimedean tconorm ∨.Archimedean tconorms are a very relevant class of tconorm operators, including most of the common examples of tconorm operators. To the best of our knowledge, a clear definition of aggregative behavior has not been presented in the literature, that has limited itself to present examples with existential quantifiers and/or with proportional quantifiers representing small proportions (e.g., ‘about 10%’). In this discussion, we will limit us to consider aggregative behavior for existential quantifiers, as it is enough to characterize the models we are considering. 4For an archimedean tconorm, limn−→ ∞ ∨(c/e1,...,c/en)=1. Every continuous tconorm such that ∨(x,x)>x,x∈ (0,1)is archimedean. 15 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 5.2.1. Analysis of the models Model FA.FAmodel presents aggregative behavior as a consequence of inducing the nonstandard probabilistic sum tconorm  ∨(a,b)=a+b−ab.FortheFAmodel: F(∃)( X)= ∨e∈EμX(e) Moreover, FAmodel tends to the Zadeh’s Sigma-count model when the size of the referential Etends to infinite [12]; that is: lim |E|→∞ FA(Q)(X)=μQe∈EμX(e) |E| In this way, FAshares the critiques of aggregative behavior that has been made to the Zadeh’s model for large referential sets. Models M,MCX,Fowa,FMD and FI.None of the rest of the models show aggregative behavior. For all of them, F(∃)(X)=sup {μX(e):e∈E},Efinite. We will give some intuitions about the reasons for which these models do not present aggregative behavior. Model MCX has been proved to be extremely stable. In [20, section 7.12] it is proved that a change in the arguments that does not exceed a given Δwill not change the result of the quantifier by more than Δ. Then, F(Q)( ∅)and F(Q)( {c/e1,c/e2,...,c/eN}), with c‘small’, will produce approximately the same results. With respect to models M,Fowa,FMD and FIwe should take into account that all of their definitions are made by using an integration process over the alpha-cuts or the three-valued cuts of the argument sets. In the case of alpha cuts, only alpha cuts in the integration interval (0,c] could be altered by modifications in degrees of membership of elements with membership degree μX(e)≤cthat are maintained in (0,c]. In the case of three-valued cuts, only the integration interval [1 −2c,1] could be altered by modifications in degrees of membership in the same interval. As the results of the integral do not change out the integration range, effects of modifications lower or equal than cwill be limited to c(in the case of alpha-cuts) or 2cin the case of three-valued cuts5. 5.3. Propagation of fuzziness Propagation of fuzziness is related with the transmission of imprecision from the inputs (arguments and quantifiers) to the outputs (results of evaluating quantified expressions). We will reproduce the main definitions in [20, section 5.2 and 6.3]. Let be ca partial order in I×Idefined as xcy⇔y≤x≤1 2or 1 2≤x≤y for x,y∈I. ccan be extended to fuzzy sets, semi-fuzzy quantifiers and fuzzy quantifiers in the following way: 5It can be proved that differences in the integration process for Mand FCh are also limited to c, but we are only interested in giving an intuitive explanation of the reasons for which these models do not present aggregative behaviour. 16 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 XcX⇔μX(e)cμX(e),for all e∈E QcQ⇔Q(Y1,...,Yn)cQ(Y1,...,Yn),for all Y1,...,Yn∈P(E)  Qc Q⇔ Q(X1,...,Xn)c Q(X1,...,Xn),for all X1,...,Xn∈ P(E) Definition 19. [20, section 6.3]. Let a QFM Fbe given. a. We say that Fpropagates fuzziness in arguments if the following property is satisfied for all Q:P(E)n→Iand X1,...,Xn,X 1,...,X n∈ P(E).IfXicX ifor all i=1,...,nthen F(Q)(X1,...,Xn)cF(Q)X 1,...,X n. b. We say that Fpropagates fuzziness in quantifiers if F(Q)cF(Q)whenever QcQ. Propagation of fuzziness in arguments and in quantifiers is considered as optional but really convenient in [20, section 6.3]. Intuitively, from an user point of view, fuzzier inputs or fuzzier quantifiers should not produce more specific outputs. Although both propagation of fuzziness properties seem natural, we should note that most basic tnorms and tconorms do not fulfill propagation of fuzziness properties (e.g. product tnorm and Lukasiewicz tnorm and their corresponding tconorms) do not fulfill propagation of fuzziness. This fact is relevant, as every DFS embeds basic logic operators [20, section 3.4]. Moreover, fulfillment of propagation of fuzziness properties have strong negative consequences for the ranking of objects (see section 5.6). 5.3.1. Analysis of the models Models M,MCX.Models Mand MCX are the paradigmatic examples of standard DFSs fulfilling propagation of fuzziness properties (see [20, chapter 7]. Using Mand MCX assure that when presented with fuzzier inputs or quantifiers, we will always obtain fuzzier outputs. Models Fowa,FA,FMD and FI.Model Fowa is the paradigmatic example of an standard DFSs that does not fulfill both propagation of fuzziness properties. FAmodel does not fulfill propagation of fuzziness in arguments, as it is not fulfilled by the induced product tnorm and the induced probabilistic sum tconorm of the model (see [9, chapter 3]) and it is easy to find counterexamples for propagation of fuzziness in quantifiers. FMD and FIdo not fulfill the property of propagation of fuzziness in arguments (see [9, chapter 3]) and it is also trivial to find counterexamples for the property of propagation of fuzziness in quantifiers. 5.4. Identity quantifier: as many as possible First, we will define the identity semi-fuzzy quantifier. We will limit us to the proportional case: Definition 20. The unary semi-fuzzy quantifier identity:P(E)→Iis defined as identity (Y)=|Y| |E|,Y∈P(E) For the identity semi-fuzzy quantifier, adding one element increments the result in 1 m. Thus, the increase in the output obtained with the addition of elements to the argument set is linear, making possible to interpret identity (Y)as ‘as many as possible’ or ‘the more the better’.In 17 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 other way, the identity semi-fuzzy quantifier measures the relative weight of the input set Ywith respect to the referential set E.Thatis,identity (Y)=|Y|/|E|. A plausible fuzzy counterpart of the identity quantifier should also produce a linear increase in the output for a linear increase in the input. Definition 21. [9, chapter 3] We will say that a QFM Ffulfills the average property for the identity quantifier if: F(identity)( X)=1 m m  j=1 μXej As a result of the fulfillment of the average property for the identity quantifier, the improvement obtained in F(identity)( X)is linear with respect to the increases of the membership degrees of the argument fuzzy set. This property allows us to enquire if this intuition is translated to the fuzzy case, assuring that in the fuzzy case we will obtain a measure of the relative weight of X∈ P(E)with respect to E. 5.4.1. Analysis of the models Models Fowa,FMD,FIand FA.Model Fowa [20, chapter 8], and models FMD,FI[9] generalize the OWA approach, and then they trivially fulfill the property of averaging for the identity quantifier. Model FAalso fulfills this property [9],[12]. Models Mand MCX.Models Mand MCX do not fulfill this property, as a direct consequence of fulfilling the propagation of fuzziness in the arguments. For Mand MCX models, if M(X)=a or MCX (X)=a,a≥0.5, then M(X),MCX (X)∈[0.5,a]forXcX(Xfuzzier than X). More clearly: M(identity)(( {1/e1,1/e2,0/e3,0/e4})) =MCX (identity)(( {1/e1,1/e2,0/e3,0/e4})) =MCX (identity)(( {0.5/e1,0.5/e2,0.5/e3,0.5/e4})) =MCX (identity)(( {1/e1,1/e2,0.5/e3,0.5/e4})) =MCX (identity)(( {0.5/e1,0.5/e2,0/e3,0/e4})) =0.5 that is, as M(identity)(( {1/e1,1/e2,0/e3,0/e4})) =0.5 then for every possible Xsuch that XcXthe result will be at least as fuzzier as 0.5, but as 0.5 is the fuzzier possible output, M(identity)(X)=0.5. In figure 1 we show a graphic representation of this behavior. Although we would expect a high degree of fulfillment for the identity quantifier in case 1 and a low degree in case 2, results of applying Mor MCX to the identity quantifier for both inputs is 0.5 and for every intermediate case between case 1) and case 2) is also 0.5. 5.5. Evaluating quantifiers over ‘quantified partitions’ In this section we will analyze the behavior of the models when we simultaneously evaluate a set of fuzzy quantifiers associated to a ‘quantified partition’ of the quantification universe. Let us consider the set of quantification labels presented in figure 2. For reasons we will see later on, we will restrict us to a set of labels such that μQi(x)+ μQi+1(x)=1forsomei. In any case, this is a very common way of dividing the reference 18 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 Case 1) Case 2) Figure 1: Indiscernible situations for the identity quantifier Figure 2: Partition of the quantified universe. universe in practical applications. We will refer to quantified partitions fulfilling this property as ‘Ruspini quantified partitions’. When we consider the simultaneous evaluation of a set of quantifiers defined by means of a quantified partition, the behavior of standard DFSs (M,MCX Fowa)andQFMs(FA,FIand FMD) present strong differences. For some situations, models M,MCX and Fowa tend to produce the 0.5 output for every quantifier in the partition. Contrasting with this behavior, models FA, FIand FMD guarantee that the sum of the evaluation results equals 1, producing ‘a distribution’ of the truth between the set of quantified labels. 5.5.1. Analysis of the models Models M,MCX and Fowa.As a consequence of being based in trivalued cuts models M,MCX and Fowa present a tendency to produce 0.5 evaluation results for some situations. Let us consider a fuzzy set Xsuch that X={0.5/e1,0.5/e2,...,0.5/em} then, if for a semi-fuzzy quantifier Qiis fulfilled that there exist r,jsuch that qi(r)=1and qi(j)=0 (note that every quantifier in figure 2 fulfill this property) then M(X)=MCX (X)=Fowa (X)=0.5 as it can be easily checked. This behavior is independent of the granularity of the partition. That is, finer partitions will continue to produce a 0.5 output in each situation in which we could find some i,jsuch that qi(i)=1andqi(j)=0. 19 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 Model FA.Before presenting the behavior of the FAmodel, we need to introduce some definitions to precise the meaning of a ‘Ruspini quantified partition’. Definition 22. We will say that a set of semi-fuzzy quantifiers Q1,...,Qr:Pn(E)→Iforms a Ruspini partition of the quantification universe if for all Y1,...,Yn∈P(E)it holds that Q1(Y1,...,Yn)+...+Qr(Y1,...,Yn)=1 Example 23. The next set of quantifiers form a Ruspini partition of the quantification universe: Qi(Y1,Y2)=labeli|Y1∩Y2| |Y1|Y1∅ 1 5Y1=∅ where labelirepresents the ‘i-th’ fuzzy number in the partition. This set of fuzzy numbers forms a Ruspini partition of the quantification universe as iQi(Y1,Y2)=1forallY1,Y2∈P(E). Definition 24. [9, chapter 3]We will say that a QFM Ffulfills the property of probabilistic interpretation of quantifiers if for all the Ruspini partitions of the quantification universe Q1,...,Qr:P(E)n→Iit holds that F(Q1)( X1,...,Xn)+...+F(Qr)(X1,...,Xn)=1 This property is very interesting because it allows us to interpret the result of evaluating fuzzy quantified expressions as probabilities over the labels related to the quantifiers6.FA,FMD and FImodels fulfill this property [9, chapter three], [12]. Thus, we can interpret that these models tend to distribute the truth between the set of labels of the partition, assuring that the sum of the evaluation results associated to each label adds to 1. In addition, in [12] it has been proved that for unary quantifiers the FAmodel tends to the Zadeh’s Sigma-count model when the size of the referential Etends to infinite; that is: lim |E|→∞ FA(Q)(X)=μQe∈EμX(e) |E| In this way, for a big |E|we have FA(Q)( X)≈μQe∈EμX(e) |E|.As e∈EμX(e) |E|is a punctual value, when we apply this result to a Ruspini quantified partition like the one presented in figure 2, the weights of the evaluation of the quantified expressions tend to concentrate themselves in one quantified label qi(being FA(Qi)( X)≈1) or two contiguous ones qi,qi+1(being FA(Qi)+ FA(Qi+1)≈1). Models FMD and FI.Models FMD and FIalso fulfill the property of probabilistic interpretation of quantifiers. Hence, the result of evaluating a set of quantifiers Q1,...,Qrforming a Ruspini quantified partition can be interpreted as a probability defined over the quantified labels of the quantifiers. Thus, we can interpret that these models tend to distribute the weight of evaluating quantified sentences over the set of labels used to define the fuzzy quantifiers. 6In [22] a probabilistic interpretation of quantifiers is also used under the label semantics interpretation of fuzzy sets. 20 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 Moreover, the following result proves that in the unary case, for a ‘perfectly’ distributed fuzzy set, models FMD and FItend to assign to each quantifier a probability weight proportional to its area. Let us define an equispaced fuzzy set over [0,1] as: μX(a1)=0<μ X(a2)=h<μ X(a2)=2h,...,μ X(am)=1 Then, if we restrict ourselves to piecewise continuous functions, it is fulfilled that: lim m→∞ 1 0 Q(X≥α)dα=lim m→∞ μQ1 mh+μQ2 mh+...+μQm−1 mh=1 0 μQ(x)dx as we are simply computing the area of the quantifier. As a consequence, when we evaluate a set of unary quantifiers Q1,...,Qr:P(E)→Iover a fuzzy set following an identity function (μX(ei)=i m) we obtain: FMD (Qi)( X)=FI(Qi)( X)≈area μQi This property is related to the probabilistic alpha-cut interpretation of models FMD and FI. In this interpretation, if membership values of Xare perfectly distributed, then ‘weights’ of the alpha cuts are perfectly distributed over the quantification universe. In this way, quantifiers of greater areas tend to ‘collect’ more weight than quantifiers with smaller areas. This also means that, for ‘finer’ quantifier partitions, weights tend to be more distributed between the quantifiers of the partition. 5.6. Fine distinction between objects In applications of fuzzy quantifiers for ranking generation, we generally have a set of objects o1,...,oNfor which the fuzzy fulfillment of a set of criteria p1,...,pmis known Xoi= {μXi(p1)/p1,...,μ Xi(pm)/pm},whereμXipj/pjrepresents the fulfillment of the criteria pjby the object oi. Additionally, we generally have a set of weights W={μW(p1)/p1,...,μ W(pm)/pm} indicating the relative importance of the criteria p1,...,pm. Fuzzy quantification can be used to generate a ranking by means of the assignment of a weight to each object, computed using an unary proportional quantified expression, roi= Q(Xoi) when a vector of weights is not involved, or computed using a binary proportional quantified expression roi= Q(W,Xoi)when there exists a vector of weights Wto indicate the relative importance of each criteria. Hence, when we compute roifor each i=1,...,N, we can rank each object with respect to ‘how  Q’ criteria it fulfills (e.g., for  Q=many,‘how many’). Fuzzy quantifiers seem specially convenient for ranking applications. As roiindicates ‘how good’ is the object iin fulfilling ‘  Qcriteria’. We can easily adjust the quantifiers to prioritize objects fulfilling ‘most of the criteria’,‘some of them’,‘at least 10’,etc. Ranking applications usually demand a great discriminative power between objects. In general, we should expect that even small variations in the inputs would produce some effect in the outputs. In order to analyze the discriminative power of QFMs, we will need some definitions: Definition 25. Let h (x):[0,1] →Ian strictly increasing continuous mapping; i.e., h (x)>h(y) for every x >y. We define the unary and binary semi-fuzzy quantifiers Qh:P(E)→Iand 21 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 Qh:P(E)2→Ias Qh(Y)=h(|Y|),Y∈P(E) Qh(Y1,Y2)=h|Y1∩Y2| |Y1|Y1∅ 1Y1=∅ For assuring the discriminative power of QFMs, we will require that in the case of unary quantifiers, any increase in the fulfillment of a criteria will increase F(Qh). In the binary case, we will require that any increase in the fulfillment of a criteria associated with a strictly positive weight will also increase F(Qh).Thatis,ashis strictly increasing, we expect that an increase in the values of the inputs is translated into an increase in the result of the evaluation. Definition 26. Let us consider X1,X2∈ P(E),i=1,...,m, j ∈{1,...,m}such that μX1(ei)= μX2(ei),ij, μX1(ei)<μ X2(ei),i=j. We say that a QFM Ffulfills the property of discriminative ranking generation for unary quantifiers if: F(Qh)( X2)>F(Qh)( X1) for h (x)strictly increasing. Definition 27. Let us consider W,X1,X2∈ P(E),i=1,...,m, j ∈{1,...,m}such that μX1(ei)= μX2(ei),ij, μX1(ei)<μ X2(ei),μ W(i)>0,i=j. We say that a QFM Ffulfills the property of discriminative ranking generation for binary quantifiers if it fulfills: F(Qh)( W,X2)>F(Qh)( W,X1) for h (x)strictly increasing. 5.6.1. Analysis of the models Models Mand MCX.Fulfillment of propagation of fuzziness properties makes Mand MCX very inconvenient for ranking applications. Examples presented in section 5.4 have shown that these models are piecewise constant, and that they are not able to differentiate between really large regions of the input space. As a consequence, these models are incapable of making fine distinction between objects. Model Fowa.Model Fowa have been presented in [20, chapter 8] as the paradigmatic example of a standard DFS non-fulfilling the properties of propagation of fuzziness. Thus, the author consider the Fowa model convenient for applications needing an ‘enhanced discriminatory force’. As Fowa model generalizes OWA, it adequately deals with the fine distinction between objects in the unary case. But in the binary case, Fowa is piecewise constant, as it proves the following example: Fowa (Qid)( {1/e1,1/e2,0.5/e3,0.5/e4},{1/e1,1/e2,0/e3,0/e4}) =0.75 =Fowa (Qid)( {1/e1,1/e2,0.5/e3,0.5/e4},{1/e11/e2,0.5/e3,0.5/e4}) In previous example, for 0.5 weights of e3,e4, we can modify object fulfillment in the [0,0.5] range without obtaining any difference in the output. 22 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 Model FMD.A similar problem happens with the FMD model. As the FMD fulfills the strong conservativity property (see [9, chapter 3]) we have FMD (Qid)( W,Xoi)=FMD (Qid)W,W ∩Xoi and then, FMD (Qid)( {1/e1,1/e2,0.5/e3,0.5/e4},{1/e1,1/e2,1/e3,1/e4}) =FMD (Qid)( {1/e1,1/e2,0.5/e3,0.5/e4},{1/e1,1/e2,0.5/e3,0.5/e4}) =1 Coinciding with the FIQFM in the unary case, FMD model fulfills the property for unary quantifiers (see below). Model FI.Model FIfulfills the property of discriminative ranking generation. The proof is shown in the Apendix. Model FA.The FAmodel also fulfills the property of discriminative ranking generation. The proof is shown in the Apendix. 5.7. Some recommendations for selecting QFMs for applications In table 2 we synthesize the behavior of the QFMs FMD,FI,FA,M,MCX and Fowa with respect to the set of additional criteria we have presented. We summarize some recommendations for the selection of convenient models for applications: 1. In applications that require a fine distinction between objects (e.g., ranking applications) only models FIand FAshould be used for non unary quantifiers. In the unary case FMD and Fowa coincide with the FImodel for increasing quantifiers, and are also acceptable. 2. In applications in which aggregative behavior is not acceptable, FAshould be avoided. 3. For maximal coherence with linguistic criteria, models MCX and FAare the preferred ones. Models Mand Fowa show a good behavior as well. Model FI, being inferior to DFSs with respect to linguistic coherence, conserves linguistic transformations of the ‘Aristotelian square’ in the finite case. 4. If propagation of fuzziness is required, the only viable options are Mand MCX. 5. In order to preserve the intuitions underneath the identity quantifier, guaranteeing that a linear increase in the inputs produces a linear increase in the outputs, models FA,FMD,FI or Fowa should be selected. 6. When taken into account the behavior of QFMs over quantified partitions, if we expect more undefined results for fuzzier fuzzy sets, standard DFSs should be used. In the case of prefering that QFMs could be interpreted as probabilities over quantified labels, distributing the ‘degree of fulfilment’ between the different labels, the convenient models are FMD, FIand FA. Summing up, the model FAis a really convenient model for all the applications in which aggregative behavior is not an impediment. MCX is the perfect model for applications in which preservation of fuzziness properties are required, but presents the handicap that is very inadequate for ranking applications and it does not maintain the linguistic intuitions under the ‘identity 23 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 FMD FIFAM MCX Fowa Linguistic Compatibility partial partial DFS+FAI DFS DFS+FAI DFS Aggregative behavior No No Yes No No No Identity Quantifier Yes Yes Yes No No Yes Propagation of Fuzziness No No No Yes Yes No Quantified Partitions Pr Pr Pr Ind Ind Ind Fine differentiation No Yes Yes No No No Table 2: Summary of the behaviour of the QFMs. FAI: Fuzzy Argument Insertion, Pr: probability interpretation of quantified labels, Ind: tendency to 0.5 in the evaluation results over quantified labels. quantifier’. Additionally, it has been proved that the MCX model presents a very stable behavior [20, section 7.12], which assures a certain insensitivity against modifications in the memberships degrees. If we need a model guaranteeing a fine distinction between objects but avoiding aggregative behavior, the best option is the FImodel. FIalso guarantees linguistic intuitions associated to the identity quantifier, allows to interpret quantified partitions as probabilities, and for fuzzy sets whose membership degrees are maximally distributed over the referential set, evaluation results provided by FItend to the area of the quantifier. Moreover, FIpreserves internal an external negation properties (this last property in the finite case), assuring the conservation of the linguistic relations of the ‘Aristotelian square’. Although FIis not a DFS, it is a remarkable model that presents a great equilibrium between the fulfillment of the different criteria. FMD, sharing some of the behavior of the FImodel, is not adequate for achieving a fine differentiation of objects in the binary case. We consider linguistic behavior of FImodel superior to the linguistic behavior of FMD, as this last model does not preserve linguistic transformations of the Aristotelian square. Mmodel shares most of the behavior of the MCX model, presenting the same problems but loosing some properties, as Fuzzy Argument Insertion. Fowa model has been presented as the paradigmatic example of a standard DFS convenient for ranking applications, but we have seen that this model is not adequate for achieving a fine differentiation between objects with binary quantifiers. However, if we were interested in preserving the properties of standard DFSs guaranteeing some discriminative power, then the Fowa model is the convenient option. Finally, the way in which QFMs behave over quantified partitions can guide us in our decision between standard DFSs and the remaining models. Standard DFS will tend to produce more undefined results (in the sense of closeness to 1 2) for fuzzier fuzzy sets (in the sense of closeness to 1 2of their membership degrees). FA,FMD and FIgenerate results that can be interpreted as probabilities, dividing the ‘evaluation weight’ between the different quantifiers in the partition. FMD and FIalso preserve the intuition of ‘weight of the quantifier’ (in the sense of the coverage of the quantification universe by the labels) for a perfect distribution of membership degrees. That is, FMD and FItend to produce a result proportional to the area of the quantifier for fuzzy sets whose membership degrees tend to be equally distributed over [0,1]. 24