Characterizing Quantifier Fuzzification Mechanisms: a behavioral guide for practical applications
F. Diaz-Hermida, M. Pereira-Fariña, Juan C. Vidal, A. Ramos-Soto
aa r X i v : . [ c s . A I] M a y Characterizing Quantifier Fuzzification Mechanisms: a behavioralguide for practical applications
F. Diaz-Hermida ∗ , M. Pereira-Fari˜na, Juan C. Vidal, A. Ramos-Soto Centro de Investigaci´on en Tecnolox´ıas da Informaci´on (CITIUS), University of Santiago de Compostela, Campus Vida,E-15782, Santiago de Compostela, Spain
Abstract
Important advances have been made in the fuzzy quantification field. Nevertheless, some prob-lems remain when we face the decision of selecting the most convenient model for a specificapplication. In the literature, several desirable adequacy properties have been proposed, buttheoretical limits impede quantification models from simultaneously fulfilling every adequacyproperty that has been defined. Besides, the complexity of model definitions and adequacy prop-erties makes very di ffi cult for real users to understand the particularities of the di ff erent modelsthat have been presented. In this work we will present several criteria conceived to help in theprocess of selecting the most adequate Quantifier Fuzzification Mechanisms for specific practicalapplications. In addition, some of the best known well-behaved models will be compared againstthis list of criteria. Based on this analysis, some guidance to choose fuzzy quantification modelsfor 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 inliterature [2, 7, 8, 27, 10, 11, 14, 12, 15, 17, 18, 21, 33, 24, 29, 31, 35]. The range of applicationsof fuzzy quantification includes fuzzy control [30], temporal reasoning in robotics [23], fuzzydatabases [5], information retrieval [4, 22, 13], data fusion [31, 19] and more recently data-to-text applications [26, 25].Moreover, the definition of adequate models to evaluate quantified expressions is fundamen-tal to perform ‘computing with words’, topic that was suggested by Zadeh [36] to express theability of programming systems in a linguistic way.In general, most approaches to fuzzy quantification use the concept of fuzzy linguistic quan-tifier to represent absolute or proportional fuzzy quantities. Zadeh [35] defined quantifiers of the ∗ Corresponding author. Tel.: + 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 September 4, 2018 rst 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 (definedby using fuzzy numbers on [0 , ff erent properties of convenientor necessary fulfillment have been defined [7, 9, 12, 18]. However, most of the approaches failto exhibit a plausible behavior as has been proved through the di ff erent reviews that have beenpublished [2, 7, 8, 18, 9] and only a few [7, 11, 17, 18] seem to exhibit an adequate behavior inthe general case.In this work, we will follow Gl¨ockner’s approximation to fuzzy quantification [18]. In hisapproach, the author generalizes the concept of generalized classic quantifier [3] (second orderpredicates or set relationships) to the fuzzy case; that is, a fuzzy quantifier is a fuzzy relation-ship between fuzzy sets. And then he rewrites the fuzzy quantification problem as the problemof looking for a mechanism to transform semi-fuzzy quantifiers (quantifiers in a middle pointbetween generalized classic quantifiers and fuzzy quantifiers, used to specify the meaning ofquantified expressions) into fuzzy quantifiers. The author calls these transformation mechanisms Quantifier Fuzzification Mechanism ( QFMs ). Being based in the linguistic
Theory of Gener-alized Quantifiers (TGQ) [3], this approach is capable of handling most of the quantificationphenomena of natural language. In addition, including quantification into a common theoreticalframework following TGQ, it also allows the translation of most of the analysis that has beenmade from a linguistic perspective to the fuzzy case, and facilitates the definition and the test ofadequacy properties.Gl¨ockner has also defined a rigorous axiomatic framework to ensure the good behavior ofQFMs. Models fulfilling this framework are called
Determiner fuzzification schemes (DFSs) andthey comply with a broad set of properties that guarantee a good behavior from a linguistic andfuzzy point of view. See the recent [27] or [18] for a comparison between Zadeh’s and Gl¨ockner’sapproaches.The DFS framework has supposed a notable advance and several well behaved QFMs havebeen identified [18], [9]. However, important problems still remain when we must face thedecision of selecting an specific QFM for a practical application. First, it has been proved thatno model can fulfill every desirable adequacy property that has been proposed [18], and as aconsequence, a ‘perfect model’ cannot exist. Besides, the complexity of the definition of themodels and adequacy properties makes really di ffi cult for a user to decide which one is the mostconvenient for a certain application. In addition, as we will show along the exposition, thereare some criteria that have not been previously taken into account for analyzing the plausiblemodels and, even for the cases in which some of these criteria had been previously considered, acomplete comparison among the behavior of at least the best-behaved models has not been done.In this work we will focus on, to the best of our knowledge, the best-behaved QFMs: models F MD , F I , F A [9] and models M , M CX and F owa [18] with the objective of establishing a set ofcriteria that facilitates the understanding of the behavioral di ff erences among them and helpingwith 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 [35].Furthermore, some of them generalize other known approaches, as the ones based on the Sugenoor Choquet integrals. Thus, selected models comprise a really good representation of the ‘stateof 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 artrevisions about the fuzzy quantification field can be found in [2, 7, 18, 9].Before continuing, we would remark that only the models F A , M , M CX and F owa fulfill the2trict DFS framework, being alpha-cut based models like F MD and F I , previously consideredas non plausible from the point of view of the DFS framework [18, section 7.2]. In order tounderstand the di ff erences between these models and DFSs, we will first compare the selectedmodels against the main properties considered into the QFM framework. Once the main di ff er-ences derived from the properties described in [18] have been presented, we will introduce thenew set of criteria that will allow us to improve the comparison between the di ff erent models andto prove that, for some problems, alpha-cut based models F MD and F I can be superior to knownDFSs.Moreover, as we will argue when we analyze the di ff erent models against the set of criteriaintroduced in this paper, a ‘clear winner’ cannot be identified, being the general situation thatsome models are more appropriate for some applications than others.The paper is organized as follows. Section 2 will summarize Gl¨ockner’s approach to fuzzyquantification, based on quantifier fuzzification mechanisms. In section 3, we will present thedefinition of the models F MD , F I , F A , M , M CX and F owa . Section 4 will present the mainproperties considered in the QFM framework [18] and a brief comparison of the models F MD , F I , F A , M , M CX and F owa , with the objective of clearly identifying the behavioral di ff erencesof these models with respect to these properties. Section 5 will be devoted to establish the set ofcriteria that will allow us to improve the comparison of the considered models, and to analyze thedi ff erent models against this new set of criteria. Section 6 summarizes the results and establishessome criteria to guide in the model selection for applications. The paper is closed with someconclusions.
2. The fuzzy quantification framework
To overcome Zadeh’s framework to fuzzy quantification Gl¨ockner, [18] rewrote the problemof fuzzy quantification as the problem of looking for adequate ways to convert specificationmeans (semi-fuzzy quantifiers) into operational means (fuzzy quantifiers).Fuzzy quantifiers are just a fuzzy generalization of crisp or classic quantifiers. Before givingthe definition of fuzzy quantifiers, we will show the definition of classic quantifiers according toTGQ.
Definition 1.
A two valued (generalized) quantifier on a base set E , ∅ is a mapping Q : P ( E ) n −→ , where n ∈ N is the arity (number of arguments) of Q, = { , } denotes the set ofcrisp truth values, and P ( E ) is the powerset of E. Examples of some definitions of classic quantifiers are: all ( Y , Y ) = Y ⊆ Y at least
80% ( Y , Y ) = ( | Y ∩ Y || Y | ≥ . Y , ∅ Y = ∅ In a fuzzy quantifier, arguments and results can be fuzzy. A fuzzy quantifier assigns a gradualresult to each choice of X , . . . , X n ∈ e P ( E ), where by e P ( E ) we denote the fuzzy powerset of E . Definition 2. [18, definition 2.6] An n-ary fuzzy quantifier e Q on a base set E , ∅ is a mapping e Q : e P ( E ) n −→ I = [0 , . f all : e P ( E ) −→ I could be defined as: f all ( X , X ) = inf (cid:8) max (cid:0) − µ X ( e ) , µ X ( e ) (cid:1) : e ∈ E (cid:9) where by µ X ( e ) we denote the membership function of X ∈ e P ( E ).Although a certain consensus may be achieved to accept previous expression as a suitabledefinition for f all this is not the unique possible one. The problem of establishing consistentfuzzy definitions for quantifiers (e.g., ‘at least eighty percent’ ) is faced in [18] by introducing theconcept of semi-fuzzy quantifiers. A semi-fuzzy quantifier represents a medium point betweenclassic quantifiers and fuzzy quantifiers. Semi-fuzzy quantifiers are similar but far more generalthan Zadeh’s linguistic quantifiers [35]. 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. [18, definition 2.8] An n-ary semi-fuzzy quantifier Q on a base set E , ∅ is amapping Q : P ( E ) n −→ I .Q assigns a gradual result to each pair of crisp sets ( Y , . . . , Y n ). Examples of semi-fuzzyquantifiers are: about 5 ( Y , Y ) = T , , , ( | Y ∩ Y | ) (1) at least about
80% ( Y , Y ) = ( S . , . (cid:16) | Y ∩ Y || Y | (cid:17) X , ∅ X = ∅ where T , , , ( x ) and S . , . ( x ) represent the common trapezoidal and S fuzzy numbers .Semi-fuzzy quantifiers are much more intuitive and easier to define than fuzzy quantifiers, butthey do not solve the problem of evaluating fuzzy quantified sentences. In fact, additional mech-anisms are needed to transform semi-fuzzy quantifiers into fuzzy quantifiers, i.e., mappings withdomain in the universe of semi-fuzzy quantifiers and range in the universe of fuzzy quantifiers: Definition 4. [18, definition 2.10]A quantifier fuzzification mechanism (QFM) F assigns to eachsemi-fuzzy quantifier Q : P ( E ) n → I a corresponding fuzzy quantifier F ( Q ) : e P ( E ) n → I of thesame arity n ∈ N and on the same base set E.
3. Some paradigmatic QFMs
In this section we will present the three main Gl¨ockner’s approaches [18]. All the modelsthat have been proposed by Gl¨ockner are standard
Determiner Fuzzification Schemes , and as Functions T a , b , c , d and S α,γ are defined as T a , b , c , d ( x ) = x ≤ a x − ab − a a < x ≤ b b < x ≤ c − x − cd − c c < x ≤ d d < x , S α,γ ( x ) = x < α (cid:16) ( x − α )( γ − α ) (cid:17) α < x ≤ α + γ − (cid:16) ( x − γ )( γ − α ) (cid:17) α + γ < x ≤ γ γ < x tnorm min and the standard tconorm max . Before presenting models M [18, definition 7.22], M CX [18, definition 7.56] and F owa [18,definition 8.13] we need to introduce some additional definitions. Definition 5. [18, definition 7.15]Let E be a referential set, X ∈ e P ( E ) a fuzzy set, and γ ∈ I . X min γ , X max γ ∈ P ( E ) are defined as:X min γ = ( X > : γ = X ≥ + γ : γ > X max γ = ( X ≥ : γ = X > − γ : γ > where X ≥ α = { e ∈ E : µ X ( e ) ≥ α } is the alpha-cut of level α of X and X >α = { e ∈ E : µ X ( e ) > α } is the strict alpha-cut of level α . In previous expression, X min γ represents the elements that without doubt, belong to the fuzzyset X for the ‘cautiousness’ level γ whilst X max γ includes also the elements whose belongingnessto the cautiousness level γ is undefined. Elements that are not in X max γ do not belong to the cau-tiousness level γ . The cautiousness cut can be interpreted as a ‘trivalued set’ in which elementsin X min γ have membership function of 1, whilst for elements in X max γ / X min γ belongingness is un-defined (membership degree of ). Membership function for elements that are not in X max γ is0. For the definition of M , M CX and F owa we also need the fuzzy median operator: Definition 6.
Fuzzy median med : I × I −→ I is defined as:med ( u , u ) = min ( u , u ) : min ( u , u ) > max ( u , u ) : max ( u , u ) < : otherwise The following definitions extends the fuzzy median to fuzzy sets:
Definition 7.
Operator m : P ( I ) → I is defined asm X = med (cid:0) inf X , sup X (cid:1) for all X ∈ P ( I ) . The set that contains all the possible images of a quantifier over the range defined by a threevalued cut of level γ is defined as [17, page 100]: Definition 8.
Let Q : P ( E ) → I be a semi-fuzzy quantifier, X , . . . , X n ∈ e P ( E ) fuzzy sets and γ ∈ [0 , a cautiousness level. S Q , X ,..., X n ( γ ) : [0 , → I is defined as:S Q , X ,..., X n ( γ ) ( X , . . . , X n ) = n Q ( Y , . . . , Y n ) : ( X i ) min γ ⊆ Y i ⊆ ( X i ) max γ o In [18, 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 F transforms the classical logical functions into approppriatefuzzy logical functions. S Q , X ,..., X n ( γ ) are represented by means of the following notation: Definition 9.
Let Q : e P ( E ) → I be a semi-fuzzy quantifier, X , . . . , X n ∈ e P ( E ) fuzzy sets and γ ∈ [0 , a cautiousness level. ⊤ Q , X ,..., X n ( γ ) : [0 , → I is defined as: ⊤ Q , X ,..., X n ( γ ) = sup S Q , X ,..., X n ( γ ) Definition 10.
Let Q : e P ( E ) → I be a semi-fuzzy quantifier, X , . . . , X n ∈ e P ( E ) fuzzy sets and γ ∈ [0 , a cautiousness level. ⊥ Q , X ,..., X n ( γ ) : [0 , → I is defined as: ⊥ Q , X ,..., X n ( γ ) = inf S Q , X ,..., X n ( γ )Using previous definitions we present the three paradigmatic DFSs : Definition 11. [18, definition 7.22] Standard DFS M : ( Q : P ( E ) → I ) → ( e Q : e P ( E ) → I ) isdefined as M ( Q ) ( X , . . . , X n ) = Z med (cid:0) ⊤ Q , X ,..., X n ( γ ) , ⊥ Q , X ,..., X n ( γ ) (cid:1) d γ Definition 12. [18, definition 7.56, theorem 7.87] Standard DFS M CX : ( Q : P ( E ) → I ) → (cid:16) e Q : e P ( E ) → I (cid:17) is defined as M CX ( Q ) ( X , . . . , X n ) = sup n Q LV , W ( X , . . . , X n ) : V ⊆ W , . . . , V n ⊆ W n o where Q LV , W ( X , . . . , X n ) = min (cid:0) Ξ V , W ( X , . . . , X n ) , inf { Q ( Y , . . . , Y n ) : V i ⊆ Y i ⊆ W i } (cid:1) Ξ V , W ( X , . . . , X n ) = n min i = min (cid:0) inf (cid:8) µ X i ( e ) : e ∈ V i (cid:9) , inf (cid:8) − µ X i ( e ) : e < W i (cid:9)(cid:1) Definition 13. [18, definition 8.13] Standard DFS F owa : ( Q : P ( E ) → I ) → (cid:16) e Q : e P ( E ) → I (cid:17) isdefined as F owa ( Q ) ( X , . . . , X n ) = Z ⊤ Q , X ,..., X n ( γ ) d γ + Z ⊥ Q , X ,..., X n ( γ ) d γ F I and F MD Now, we will present the two QFMs based on alpha-cuts F I and F MD . Definition 14. [11, section 2.1], [9, chapter 3]Let Q : P ( E ) n → I be a semi-fuzzy quantifierover a base set E. The QFM F MD is defined as: F MD ( Q ) ( X , . . . , X n ) = Z Q (cid:0) ( X ) ≥ α , . . . , ( X n ) ≥ α (cid:1) d α for every X , . . . , X n ∈ e P ( E ) . When fuzzy sets X , . . . , X n ∈ e P ( E ) are normalized and we limit ourselves to the unary andbinary quantifiers considered in the Zadeh’s framework, F MD coincides with the quantificationmodel GD defined in [6, page 281], [28, section 3.3.2. and section 3.4.1.], [7, page 37]. In thisway, F MD generalizes the GD model to the Gl¨ockner’s framework.Let us now present the definition of the F I model.6 efinition 15. [10], [11, section 2.2],[9, chapter 3] Let Q : P ( E ) n → I be a semi-fuzzy quanti-fier over a base set E. The QFM F I is defined as: F I ( Q ) ( X , . . . , X n ) = Z . . . Z Q (cid:16) ( X ) ≥ α , . . . , ( X n ) ≥ α n (cid:17) d α . . . d α n for every X , . . . , X n ∈ e P ( E ) .3.3. Non standard DFS F A The definition of the QFM F A is based on a probabilistic interpretation of fuzzy sets in whichwe interpret membership degrees as probabilities [9],[12]. However, the F A model can also bedefined by means of fuzzy operators without any reference to probability theory.The QFM F A fulfills the axioms of the DFS framework but it is not a standard DFS, asthe logic operators induced by the F A model are the product tnorm and the probabilistic sum tconorm . Definition 16.
Let X ∈ e 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 ∈ e P ( E ) is defined asm X ( Y ) = Y e ∈ Y µ X ( e ) Y e ∈ E \ Y (1 − µ X ( e ))As we have stated above, it is possible to make a similar definition without making anyreference to probability theory. If we consider the product tnorm ( ∧ ( x , x ) = x · x ) and theLukasiewicz implication then m X ( Y ) is the equipotence between Y and X [1]: Eq ( Y , X ) = ∧ e ∈ E ( µ X ( e ) → µ Y ( e )) ∧ ( µ Y ( e ) → µ X ( e ))Using the previous definition we define the F A DFS as:
Definition 17. [14, pag. 1359]Let Q : P ( E ) n → I be a semi-fuzzy quantifier, E finite. The DFS F A is defined as F A ( Q ) ( X , . . . , X n ) = X Y ∈P ( E ) . . . X Y n ∈P ( E ) m X ( Y ) . . . m X n ( Y n ) Q ( Y , . . . , Y n ) (2) for all X , . . . , X n ∈ e P ( E ) . The next expression is an alternative definition of the model F A : F A ( X , . . . , X n ) = _ Y ∈P ( E ) . . . _ Y n ∈P ( E ) Eq ( Y , X ) ∧ . . . ∧ Eq ( Y n , X n ) ∧ Q ( Y , . . . , Y n )where ∨ the Lukasiewicz tconorm ( ∨ ( x , x ) = min ( x + x , ∧ is the product tnorm ( ∧ ( x , x ) = x · x ) and Eq ( Y , X ) is the equipotence between the crisp set Y and the fuzzy set X . In thisway, definition of F A can be done by means of common fuzzy operators.7 . The DFS axiomatic framework In this section we will present the DFS axiomatic framework [18]. In the previous reference,the author has dedicated the whole third and fourth chapters to describe the framework and theproperties that are consequence of it. For the sake of brevity, we will only give a general overviewof the framework and some intuitions about the set of properties derived from it. We refer thereader to the previous publication for full detail.
Definition 18.
A QFM F is called a determiner fuzzification scheme (DFS) if the followingconditions are satisfied for all semi-fuzzy quantifiers Q : P ( E ) n → I : Correct generalization U ( F ( Q )) = Q if n ≤ F ( Q ) = e π e if Q = π e for some e ∈ E (Z-2)Dualisation F (cid:16) Q e (cid:3) (cid:17) = F ( Q ) e (cid:3) n > F ( Q ∪ ) = F ( Q ) e ∪ n > Q is nonincreasing in the n -th arg, then (Z-5) F ( Q ) is nonincreasing in n -th arg, n > F (cid:18) Q ◦ n × i = b f i (cid:19) = F ( Q ) ◦ n × i = b F ( f i ) (Z-6)where f , . . . , f n : E ′ → E , E ′ , ∅ 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 [18, chapters three and four.]. • Correct generalization (P1): perhaps, the most important property derived from the DFSframework is the correct generalization property. Correct generalization requires that thebehavior of a fuzzy quantifier F ( Q ) over crisp arguments is the expected; that is, resultsobtained with a fuzzy quantifier F ( Q ) and with the corresponding semi-fuzzy quantifier Q must coincide over crisp arguments. It is included in the DFS axiomatic for semi-fuzzyquantifiers of arities 0 and 1 (Z-1). • Quantitativity (P2): quantitative quantifiers do not depend on any particular characteristicof the elements of the base set. In the finite case, quantitative quantifiers can always bedefined as a function of the cardinality of the boolean combinations of the argument sets.A QFM F preserves quantitativity if quantitative semi-fuzzy quantifiers are translated intoquantitative fuzzy quantifiers by F . • Projection quantifier (P3):
Axiom Z-2 guarantees that the projection crisp quantifier π e ( Y ) (that returns 1 if e ∈ Y and 0 in other case) is generalized to the fuzzy projectionquantifier e π e ( X ) (that returns µ X ( e )). • Induced propositional logic (P4): we will say that a QFM comply with the inducedpropositional logic if crisp logical functions ( ¬ ( x ), ∧ ( x , x ), ∨ ( x , x ), → ( x , x )), thatcan be embedded into the definition of semi-fuzzy quantifiers, are generalized to accept-able fuzzy logical functions; that is, a negation operator, a tnorm , a tconorm and a fuzzyimplication function. 8 External negation (P5): in the common case, external negation of a semi-fuzzy quantifieris computed by the application of the standard negation e ¬ ( x ) = − x . A QFM fulfillingthe external negation property guarantees that F ( e ¬ Q ) is equivalent to e ¬F ( Q ). Thanks tothe external negation property, equivalence of expressions “it is false that at least 80% ofthe hard workers are well paid” and “less than 80% of the hard workers are well paid” isassured. • Internal negation (P6): the internal negation (antonym) of a semi-fuzzy quantifier isdefined as Q ¬ ( Y , . . . , Y n ) = Q ( Y , . . . , ¬ Y n ). For example, ‘no’ is the antonym of ‘all’ because all ( Y , Y ) ¬ = all ( Y , ¬ Y ) = no ( Y , Y ). Fulfillment of the internal negationproperty assures that internal negation transformation are translated to the fuzzy case. • Dualisation (P7): the dualisation property coincides with the Z-3 axiom of the DFS frame-work, being a consequence of the simultaneous fulfillment of the external negation andinternal negation properties. In conjunction with previous properties, equivalences in the‘Aristotelian square’ are maintained in the fuzzy case. As an example, equivalence of F ( all ) ( hard workers , well paid ) and F ( no ) ( hard workers , e ¬ well paid ) is assured, orin words, “all hard workers are well paid” is equivalent to “no hard worker is not wellpaid”. • Union / intersection of arguments (P8) : this property guarantees the compliance withsome transformations that allow to construct new quantifiers by means of unions (andintersections) of arguments. As a particular case, the equivalence between absolute unaryand binary quantifiers is a consequence of this axiom. As an example, the equivalencebetween “about 5 hard workers are well paid” and “about 5 people are hard workers andwell paid” is assured. For QFMs fulfilling the DFS framework, the fulfillment of this prop-erty, in combination with internal and external negation properties, allow the preservationof the boolean argument structure that can be expressed in natural language when none ofthe boolean variables X i occurs more than once [18, section 3.6]. • Coherence with standard quantifiers (P9): by standard quantifiers we refer to the classi-cal quantifiers ∃ , ∀ and their binary versions some and all . We will say that a QFM main-tains coherence with standard quantifiers if the fuzzy versions of the classical quantifiersare the expected. For example, a QFM fulfilling this property complies (where e ∨ , e ∧ , e → arethe logical operators induced by the QFM): F ( ∃ ) ( X ) = sup ( m e ∨ i = µ X ( a i ) : A = { a , . . . , a m } ∈ P ( E ) , a i , a j if i , j ) F ( all ) ( X , X ) = inf ( m e ∧ i = µ X ( a i ) e → µ X ( a i ) : A = { a , . . . , a m } ∈ P ( E ) , a i , a j if i , j ) • Monotonicity in arguments (P10): this property assures the translation of monotonicityin arguments relations from the semi-fuzzy to the fuzzy case. As an example, the binarysemi-fuzzy quantifier ‘most’ is increasing in its second argument (e.g. “most students arepoor” ). This property assures that the fuzzy version of ‘most’ is also increasing in itssecond argument. • Monotonicity in quantifiers (P11): this property assures the preservation of monotonicityrelations in quantifiers. For example, ‘between 4 and 6’ is more specific than ‘between 2 nd 8’ . Fulfilment of this property assures that in the fuzzy case, the specificity relationsbetween quantifiers are preserved. • Crisp argument insertion (P12):
For a semi-fuzzy quantifier Q : P ( E ) n → I , crispargument insertion allow to construct a new quantifier Q : P ( E ) n − → I by means of therestriction of Q by a crisp set A ; that is, the crisp argument insertion Q ⊳ A is definedas Q ⊳ A ( Y , . . . , Y n − ) = Q ( Y , . . . , Y n − , A ). A QF M preserving the property of crispargument insertion assures that F ( Q ⊳ A ) = F ( Q ) ⊳ A ; that is, it is equivalent to firstrestrict the semi-fuzzy quantifier Q by A and then applying the fuzzification scheme F orto first applying the fuzzification mechanism and then restricting the corresponding fuzzyquantifier by A . Crisp argument insertion allow to model the ‘adjectival restriction’ ofnatural language in the crisp case. In [18, chapter six] some additional adequacy properties for characterizing DFSs were de-scribed. These additional properties were not included in the DFS framework in some cases,for being incompatible with it, and in other cases, in order to not excessively restrict the set oftheoretical models fulfilling the framework, which was important for the author for studying thefull set of classes of standard models and their theoretical limits. We will present now the morerelevant: • Continuity in arguments (P13): this property assures the continuity of the models withrespect to the argument sets. It is fundamental to guarantee that small modifications in ar-guments do not provoke high variations in the results of evaluating quantified expressions. • Continuity in quantifiers (P14): this property assures the continuity of the models withrespect to variations in the quantifiers. • Propagation of fuzziness (P15): this property assures that fuzzier inputs (understood asfuzzier input sets) and fuzzier quantifiers produce fuzzier outputs. We will discuss thisproperty in more detail when we introduce the set of criteria we will use to improve thecharacterization of the behavior of the QFMs (see section 5.3). • Fuzzy argument insertion (P16): this property is the fuzzy counterpart of the crisp argu-ment insertion. It is a very restrictive property, that will impose great limitations into theset of models fulfilling the DFS axiomatic framework .
In this section we will make a brief summary of the theoretical analysis of the QFMs F MD , F I , the non-standard DFS F A and the standard DFSs M , M CX and F owa with respect to theset of previous properties. Table 1 summarizes the fulfillment of the properties for the di ff erentmodels. A detailed analysis of these models can be found in [18] and [9].This analysis will allow us to understand the main di ff erences between the models we areconsidering. As we will argue in the following section, although the set of properties previouslypresented allow for a deep analysis of the models, we consider that they are not enough to under-stand the behavioral di ff erences between them and to decide which ones can be more appropriatefor specific applications. The introduction of these new criteria and the analysis of the behaviorof the modes with respect to it will be the objective of the last two sections of this paper.10 M CX F owa F MD F I F A Properties derived from the DFS frameworkP1. Correct Generalization Y Y Y Y Y YP2. Quantitativity Y Y Y Y Y YP3. Projection quantifiers Y Y Y Y Y YP4. Induced propositional logic Y Y Y Y Y YP6. External negation Y Y Y Y Y YP7. Internal negation Y Y Y N finite YP8. Dualisation Y Y Y N finite YP9. Union / intersection of argument Y Y Y Y N YP10. Coherence with standard quantifiers Y Y Y unary unary YP11. Monotonicity in arguments Y Y Y Y Y YP12. Monotonicity in quantifiers Y Y Y Y Y YP13. Crisp Argument Insertion Y Y Y Y Y YAdditional propertiesP14. Continuity in arguments Y Y Y finite finite finiteP15. Continuity in quantifiers Y Y Y Y Y YP16. Propagation of fuzziness Y Y N N N NP17. Fuzzy argument insertion N Y N N Y Y Table 1: Comparison of the behavior of the models against the set of properties in the QFM framework
Before summarizing the behavior of these models against these properties, we would like toemphasize that our point of view is that although models F MD and F I are not DFSs, they arereally competitive with respect to models fulfilling the DFS framework. The main di ff erencesbetween the F MD and F I models when we compare them with DFSs is that they fail to fulfillsome of the linguistic properties derived from the DFS framework. In addition, these modelsalso fail to fulfill some of the QFM properties in the infinite case, which do not a ff ect to most ofthe practical applications of fuzzy quantification . Finally F MD and F I only fulfill the coherencewith the standard quantifiers property in the unary case, although in the specific case of the F I model fulfillment of the property depends on the mechanism we will use to compute the inducedoperators of the model [9, chapter 3].The competitiveness of F MD and F I models with respect to DFSs will become more clearwhen we present the analysis of the models against the new set of criteria, which from our pointof view will prove that in some cases models F MD and F I present some advantages againstmodels fulfilling the DFS framework. M model M model is one of the first models formulated by Gl¨ockner [16] and it is also one of the threemodels for which the author has provided computational algorithms in [18, chapter 11]. Being anstandard DFS M model fulfills the properties derived from the DFS framework. Additionally, the We have the hypothesis that for ‘practical quantifiers’ (i.e., defined by means of continuous fuzzy numbers) mod-els F MD and F I fulfill the continuous in arguments property. We also have the hypothesis that model F I fulfill theinternal negation property for infinite domains in the same cases. The fulfillment of these properties will guarantee theconvenience of these models for infinite domains in the practical cases. model is continuous in arguments and in quantifiers and fulfills the properties of propagationof fuzziness in arguments and in quantifiers . M CX model M CX model is also a standard DFS and another model for which the author has provideda computational implementation in [18, chapter 11]. M CX is continuous in arguments and inquantifiers and fulfills both fuzziness propagation properties . M CX is considered by Gl¨ockneras a model of unique properties: it fulfills the property of fuzzy argument insertion [18, defini-tion 7.82], it is specially robust against modification of membership degrees and generalizes theSugeno integral (see [18, section 7.13] for more details). F owa model F owa model is the paradigmatic example of an standard DFS that does not propagate fuzzinessin arguments or in quantifiers . F owa model is also continuous in arguments and in quantifiers .As it fails to fulfill propagation of fuzziness properties, it is considered as the ideal model for ap-plications in which an improved discriminative power is necessary [18, section 8.1]. F owa modelgeneralizes Choquet integral. It is the third model for which a computational implementation hasbeen provided in [18, definition 7.82]. F MD model F MD is the generalization to QFMs of the GD model proposed by Delgado et al. in [6],[28], [7]. F MD model is not a DFS, failing to fulfill the internal negation property , and as aconsequence, the dualisation axiom of DFSs (Z3) . F MD model is continuous in the arguments inthe finite case and also continuous in the quantifiers . F MD fulfills the properties of probabilisticinterpretation of quantifiers and of averaging for the identity quantifier [9, chapter 3], that will bereintroduced as one of the criteria for comparing the behavior of selected QFMs in the followingsection. F MD does not fulfill any of the propagation of fuzziness properties . F I model F I model is the second alpha-cut based model analyzed in [9]. F I model does not fulfill the internal joins property (axiom Z4) , and then fails to be a DFS. F I is continuous in the argumentsin finite domains and also continuous in the quantifiers . F I model fulfills the dualisation propertyin the finite case . F I model also fulfills the properties of probabilistic interpretation of quanti-fiers and averaging for the identity quantifier [9, chapter 3]. F I does not fulfill propagation offuzziness properties . F A model F A model is, to our knowledge, the unique known non-standard DFS. The fuzzy operatorsinduced by the model are the product tnorm and the probabilistic sum tconorm , making thismodel essentially di ff erent of the standard DFSs presented in [18]. By definition F A is a finitemodel. Moreover, F A is continuous in arguments and in quantifiers , it does not fulfill fuzzinesspropagation properties, but it fulfills probabilistic interpretation of quantifiers and averaging forthe identity quantifier properties . . Some additional criteria to characterize the behavioral di ff erences of the QFMs We have seen that models F MD , F I , F A , M , M CX and F owa fulfill most of the adequacyproperties that has been presented in [18]. If we only took into account properties included inthe QFM framework when selecting a model for an application, we would just choose one of thebest-behaved models (e.g., F A or M CX ) and we will use them in every possible application offuzzy quantification.However, as we will see through this section, properties included in the QFM frameworkare not enough for fully understanding the behavioral di ff erences between the selected models.We will present an analysis that proves that models here discussed have some strong di ff erencesin their behavior. In addition, an aspect we consider specially relevant is that, from an userviewpoint, the complexity of the definition of the models and adequacy properties makes verydi ffi cult for a non-specialist in fuzzy quantification to determine which model should be chosenfor a specific application.Thus, it is essential to establish a set of criteria that help us understand the behavioral dif-ferences between the di ff erent models and facilitate the selection of the more convenient onesfor applications. In general, the set of criteria that we will take into account would not allow usto select ‘a perfect model’, or even ‘a preferred one’ for every possible application. But we areconvinced they are important to (1) clarify the di ff erences between the behavior of the QFMs (2)to select or discard QFMs for specific applications with respect to the behavior we consider moreimportant and (3) to understand the problems that the selection of a specific model could havefor a particular application.The following is a summary of the criteria we will consider: • Linguistic compatibility . By linguistic compatibility we mean the fulfillment of the mostrelevant linguistic properties derived from the DFS framework. In the summary of the be-havior of the main QFMs we have seen that between the selected models, only DFSs fulfillthe main set of properties that have been established to guarantee an adequate behaviorwith respect to the main linguistic expectations. • Aggregative behavior for low degrees of membership: aggregative behavior makes ref-erence to the tendency of a model to confuse one ‘high degree’ membership element with alarge quantity of ‘low degree’ membership elements. It has been one of the main critiquesmade to the P count model [35], [34]. • Propagation of fuzziness:
Propagation of fuzziness is the main property used in [18,section 5.2 and 6.3] to group the di ff erent classes of standard DFSs [18, chapters 7 and 8].Basically, models fulfilling the propagation of fuzziness properties ‘transfer’ fuzziness ininputs to the outputs; that is, they guarantee that fuzzier inputs and / or fuzzier quantifiersproduce fuzzier outputs. • Identity quantifier: the ‘identity semi-fuzzy quantifier’ is defined by means of the identityfunction f ( x ) = xN , x ∈ , . . . , N in the absolute case or by means of f ( x ) = x , x ∈ [0 , Evaluating quantifiers over ‘quantified partitions’.
With this criterion we refer to thebehavior of the models when we apply, simultaneously, a set of quantifiers dividing thequantification universe (e.g., ‘nearly none’ , ‘a few’ , ‘several’ , ‘many’ , ‘nearly all’ ) toa fuzzy set. That is, how the degrees of fulfillment of the evaluation of the quantifiedexpressions are distributed between the labels. • Fine distinction between objects . In applications of fuzzy quantification for rankinggeneration is generally needed that fuzzy quantifiers are able to clearly distinguish betweenobjects fulfilling a set of criteria with di ff erent degrees. Criteria to distinguish the QFMswith respect to their discriminative power are necessary for these applications. With linguistic compatibility we make reference to the main linguistic properties presentedin [18, chapter 4 and 6]. The DFS framework guarantees that the main linguistic transforma-tions, including argument permutations, negation of quantifiers, antonym of quantifiers, dual ofquantifiers, argument insertion, internal meets, etc. are transferred from the semi-fuzzy to thefuzzy case. M CX and F A . Being DFSs, both models fulfill all the semantic linguistic propertiesderived from the DFS framework. Moreover, these models fulfill the fuzzy argument insertionproperty [18, section 6.8], as it can be seen in [18, section 7.13] and in [9, chapter 3].
Models M and F owa . M and F owa models fulfill semantic linguistic properties derived from theDFS framework, but not fuzzy argument insertion. Model F MD . The main di ff erence of F MD model with respect to DFSs is the non-fulfillment ofthe internal negation property . This fact impedes the F MD model to correctly translate antonymrelationships to the fuzzy case and, as a consequence, duality transformations (see [9, chapter 3]).As an example, failing to fulfill the internal negation property the model cannot guarantee theequivalence of F MD ( all ) ( hard workers , well paid ) and F MD ( no ) ( hard workers , e ¬ well paid ).In words, results of evaluating “all hard workers are well paid” and “no hard worker is not wellpaid” are di ff erent. F MD fulfills the strong conservativity property [18, section 6.7] that guarantees that conser-vative semi-fuzzy quantifiers (i.e., quantifiers fulfilling Q ( Y , Y ) = Q ( Y , Y ∩ Y ) are correctlytranslated 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 relation-ships of the ‘Aristotelian square’ seems more relevant than the fulfillment of the conservativityproperty. Model F I . Model F I looses the internal meets property. Moreover, the internal negation prop-erty is only fulfilled in the finite case (see [9, chapter 3]).Loosing the internal meets property F I model does not guarantee absolute unary / binarytransformations. For example, F I (about ) ( hard workers , well paid ) and F I (about ) ( hardworkers e ∩ well paid ) are not equivalent, and then “about 10 hard workers are well paid” (eval-uated by means of the binary absolute quantifier “about 10” ) and “about 10 employees are hardworkers and are well paid” (evaluated by means of the unary version of the absolute quanti-fier ‘about 10’ and the induced tnorm of the model used to compute the intersection of ‘hardworkers’ and ‘well paid’ ) will not produce the same results.14 .2. Aggregative behavior for low degrees of membership Aggregative behavior for low degrees of membership is one of the main critiques that hasbeen made to the Zadeh’s P count model [32],[18, section A.3], [2]. The intuition around ag-gregative behavior is that in evaluating quantified expressions, a large amount of elements ful-filling a property with ‘low degree’ of membership should not be confused with a small amountof elements fulfilling a property with ‘high degree’ of membership. In the case of the Zadeh’smodel is easy to understand the meaning of aggregative behavior as: X count ( ∃ ) ( { . / e , . . . , . / e } ) = X count ( ∃ ) ( { / e , / e , . . . , / e } ) = ‘exist one tall person’ can be fulfilled if there exists exactly ‘one tall person’ , or if thereexist 100 people being ‘0.01 tall’ .Although intuitions against aggregative behavior seem clear, giving up models presentingaggregative behavior will force us to discard non-standard DFS F associated to archimedeantconorms . Being F ( ∃ ) equal to ([18, Theorem 4.61]): F ( ∃ ) ( X ) = sup ne ∨ mi = ( a i ) : A = { a , . . . , a m } ∈ P ( E ) f inite , a i , a j i f i , j o for all X ∈ e P ( E ), then F ( ∃ ) ( X ) will always present aggregative behavior for every non-standardDFS associated to an archimedean tconorm ∨ . Archimedean tconorms are a very relevant classof 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 pre-sented in the literature, that has limited itself to present examples with existential quantifiersand / or with proportional quantifiers representing small proportions (e.g., ‘about 10%’ ). In thisdiscussion, we will limit us to consider aggregative behavior for existential quantifiers, as it isenough to characterize the models we are considering. F A . F A model presents aggregative behavior as a consequence of inducing the non-standard probabilistic sum tconorm e ∨ ( a , b ) = a + b − ab . For the F A model: F ( ∃ ) ( X ) = e ∨ e ∈ E µ X ( e )Moreover, F A model tends to the Zadeh’s Sigma-count model when the size of the referential E tends to infinite [12]; that is:lim | E |→∞ F A ( Q ) ( X ) = µ Q P e ∈ E µ X ( e ) | E | ! In this way, F A shares the critiques of aggregative behavior that has been made to the Zadeh’smodel for large referential sets. For an archimedean tconorm, lim n −→∞ ∨ ( c / e , . . . , c / e n ) =
1. Every continuous tconorm such that ∨ ( x , x ) > x , x ∈ (0 ,
1) is archimedean. odels M , M CX , F owa , F MD and F I . None of the rest of the models show aggregative behavior.For all of them, F ( ∃ ) ( X ) = sup { µ X ( e ) : e ∈ E } , E finite. We will give some intuitions about thereasons for which these models do not present aggregative behavior.Model M CX has been proved to be extremely stable. In [18, section 7.12] it is proved that achange in the arguments that does not exceed a given ∆ will not change the result of the quantifierby more than ∆ . Then, F ( Q ) ( ∅ ) and F ( Q ) ( { c / e , c / e , . . . , c / e N } ), with c ‘small’, will produceapproximately the same results.With respect to models M , F owa , F MD and F I we should take into account that all of theirdefinitions are made by using an integration process over the alpha-cuts or the three-valued cutsof the argument sets.In the case of alpha cuts, only alpha cuts in the integration interval (0 , c ] could be alteredby modifications in degrees of membership of elements with membership degree µ X ( e ) ≤ c thatare maintained in (0 , c ]. In the case of three-valued cuts, only the integration interval [1 − c , ff ects of modifications lower or equalthan c will be limited to c (in the case of alpha-cuts) or 2 c in the case of three-valued cuts . Propagation of fuzziness is related with the transmission of imprecision from the inputs (ar-guments and quantifiers) to the outputs (results of evaluating quantified expressions). We willreproduce the main definitions in [18, section 5.2 and 6.3].Let be (cid:22) c a partial order in I × I defined as x (cid:22) c y ⇔ y ≤ x ≤
12 or 12 ≤ x ≤ y for x , y ∈ I . (cid:22) c can be extended to fuzzy sets, semi-fuzzy quantifiers and fuzzy quantifiers in the followingway: X (cid:22) c X ′ ⇔ µ X ( e ) (cid:22) c µ X ′ ( e ) , for all e ∈ EQ (cid:22) c Q ′ ⇔ Q ( Y , . . . , Y n ) (cid:22) c Q ′ ( Y , . . . , Y n ) , for all Y , . . . , Y n ∈ P ( E ) e Q (cid:22) c f Q ′ ⇔ e Q ( X , . . . , X n ) (cid:22) c f Q ′ ( X , . . . , X n ) , for all X , . . . , X n ∈ e P ( E ) Definition 19. [18, section 6.3]. Let a QF M F be given. a. We say that F propagates fuzziness in arguments if the following property Q (cid:22) c Q ′ issatisfied for all Q : P ( E ) n → I and X , . . . , X n , X ′ , . . . , X ′ n ∈ e P ( E ). If X i (cid:22) c X ′ i for all i = , . . . , n then F ( Q ) ( X , . . . , X n ) (cid:22) c F ( Q ) (cid:16) X ′ , . . . , X ′ n (cid:17) .b. We say that F propagates fuzziness in quantifiers if F ( Q ) (cid:22) c F ( Q ′ ) whenever Q (cid:22) c Q ′ .Propagation of fuzziness in arguments and in quantifiers is considered as optional but reallyconvenient in [18, section 6.3]. Intuitively, from an user point of view, fuzzier inputs or fuzzierquantifiers should not produce more specific outputs. It can be proved that di ff erences in the integration process for M and F Ch are also limited to c , but we are onlyinterested in giving an intuitive explanation of the reasons for which these models do not present aggregative behaviour. tnorms and tconorms do not fulfill propagation of fuzziness properties (e.g. product tnorm and probabilistic sum tconorm ). This fact is relevant, as every DFS embeds basic logic opera-tors [18, section 3.4]. Moreover, fulfillment of propagation of fuzziness properties have strongnegative consequences for the ranking of objects (see section 5.6). M , M CX . Models M and M CX are the paradigmatic examples of standard DFSs fulfill-ing propagation of fuzziness properties (see [18, chapter 7]. Using M and M CX assure that whenpresented with fuzzier inputs or quantifiers, we will always obtain fuzzier outputs. Models F owa , F A , F MD and F I . Model F owa is the paradigmatic example of an standard DFSsthat does not fulfill both propagation of fuzziness properties. F A model does not fulfill propagation of fuzziness in arguments, as it is not fulfilled by theinduced product tnorm and the induced probabilistic sum tconorm of the model (see [9, chapter3]) and it is easy to find counterexamples for propagation of fuzziness in quantifiers. F MD and F I do not fulfill the property of propagation of fuzziness in arguments (see [9, chapter 3]) and itis also trivial to find counterexamples for the property of propagation of fuzziness in quantifiers. First, we will define the identity semi-fuzzy quantifier. We will limit us to the proportionalcase:
Definition 20.
The unary semi-fuzzy quantifier identity : P ( E ) → I is defined as identity ( Y ) = | Y || E | , Y ∈ P ( E )For the identity semi-fuzzy quantifier, adding one element increments the result in 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’ . Inother way, the identity semi-fuzzy quantifier measures the relative weight of the input set Y withrespect to the referential set E . That is, identity ( Y ) = | Y | / | E | .A plausible fuzzy counterpart of the identity quantifier should also produce a linear increasein the output for a linear increase in the input. Definition 21. [9, chapter 3] We will say that a QFM F fulfills the average property for theidentity quantifier if: F ( identity ) ( X ) = m m X j = µ X (cid:16) e j (cid:17) As a result of the fulfillment of the average property for the identity quantifier, the improve-ment obtained in F ( identity ) ( X ) is linear with respect to the increase of the membership gradesof the argument fuzzy set. This property allows us to enquire if this intuition is translated to thefuzzy case, assuring that in the fuzzy case we will obtain a measure of the relative weight of X ∈ e P ( E ) with respect to E . 17 ase 1) Case 2) Figure 1: Indiscernible situations for the identity quantifier F owa , F MD , F I and F A . Model F owa [18, chapter 8], and models F MD , F I [9] gener-alize the OWA approach, and then they trivially fulfill the property of averaging for the identityquantifier. Model F A also fulfills this property [9],[12]. Models M and M CX . Models M and M CX do not fulfill this property, as a direct consequence offulfilling the propagation of fuzziness in the arguments. For M and M CX models, if M ( X ) = a or M CX ( X ) = a , a ≥ .
5, then M ( X ′ ) , M CX ( X ′ ) ∈ [0 . , a ] for X ′ (cid:22) c X ( X ′ fuzzier than X ).More clearly: M ( identity ) (( { / e , / e , / e , / e } )) = M CX ( identity ) (( { / e , / e , / e , / e } )) = M CX ( identity ) (( { . / e , . / e , . / e , . / e } )) = M CX ( identity ) (( { / e , / e , . / e , . / e } )) = M CX ( identity ) (( { . / e , . / e , / e , / e } )) = . M ( identity ) (( { / e , / e , / e , / e } )) = . X ′ such that X ′ (cid:22) c X the result will be at least as fuzzier as 0 .
5, but as 0 . M ( identity ) ( X ′ ) = . M or M CX to the identity quantifier for both inputs is 0 . In this section we will analyze the behavior of the models when we simultaneously evaluatea set of fuzzy quantifiers associated to a ‘quantified partition’ of the quantification universe. Letus 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 µ Q i ( x ) + µ Q i + ( x ) = i . In any case, this is a very common way of dividing the referenceuniverse 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 aquantified partition, behavior of standard DFSs ( M , M CX F owa ) and QFMs ( F A , F I and F MD )present strong di ff erences. For some situations, models M , M CX and F owa tend to produce the0 . F A , F I igure 2: Partition of the quantified universe. and F MD guarantee that the sum of the evaluation results equals 1, producing ‘a distribution’ ofthe truth between the set of quantified labels. M , M CX and F owa . As a consequence of being based in trivalued cuts models M , M CX and F owa present a tendency to produce 0 . X such that X = { . / e , . / e , . . . , . / e m } then, if for a semi-fuzzy quantifier Q i is fulfilled that there exist r , j such that q i ( r ) = q i ( j ) = M ( X ) = M CX ( X ) = F owa ( X ) = . . i , j such that q i ( i ) = q i ( j ) = Model F A . Before presenting the behavior of the F A model, we need to introduce some defini-tions to precise the meaning of a ‘Ruspini quantified partition’. Definition 22.
We will say that a set of semi-fuzzy quantifiers Q , . . . , Q r : P n ( E ) → I forms aRuspini partition of the quantification universe if for all Y , . . . , Y n ∈ P ( E ) it holds thatQ ( Y , . . . , Y n ) + . . . + Q r ( Y , . . . , Y n ) = Example 23.
The next set of quantifiers form a Ruspini partition of the quantification universe: i ( Y , Y ) = ( label i (cid:16) | Y ∩ Y || Y | (cid:17) Y , ∅ Y = ∅ where the ‘i-th’ fuzzy number in the partition is represented by label i represents. This set offuzzy numbers form a Ruspini partition of the quantification universe as P i Q i ( Y , Y ) = Y , Y ∈ P ( E ). Definition 24. [9, chapter 3]We will say that a QFM F fulfills the property of probabilis-tic interpretation of quantifiers if for all the Ruspini partitions of the quantification universeQ , . . . , Q r : P ( E ) n → I it holds that F ( Q ) ( X , . . . , X n ) + . . . + F ( Q r ) ( X , . . . , X n ) = . F A , F MD and F I models fulfill this property [9, chapter three], [12]. Thus, we can interpret that these modelstend to distribute the truth between the set of labels of the partition, assuring that the sum of theevaluation results associated to each label adds to 1.In addition, in [12] it has been proved that for unary quantifiers the F A model tends to theZadeh’s Sigma-count model when the size of the referential E tends to infinite; that is:lim | E |→∞ F A ( Q ) ( X ) = µ Q P e ∈ E µ X ( e ) | E | ! In this way, for a big | E | we have F A ( Q ) ( X ) ≈ µ Q (cid:16) P e ∈ E µ X ( e ) | E | (cid:17) . As P e ∈ E µ X ( e ) | E | is a punctualvalue, when we apply this result to a Ruspini quantified partition like the one presented in figure2, the weights of the evaluation of the quantified expressions tend to concentrate themselves inone quantified label q i (being F A ( Q i ) ( X ) ≈
1) or two contiguous ones q i , q i + (being F A ( Q i ) + F A ( Q i + ) ≈ Models F MD and F I . Models F MD and F I also fulfill the property of probabilistic interpre-tation of quantifiers . Hence, the result of evaluating a set of quantifiers Q , . . . , Q r forming aRuspini quantified partition can be interpreted as a probability defined over the quantified la-bels of the quantifiers. Thus, we can interpret that these models tend to distribute the weight ofevaluating quantified sentences over the set of labels used to define the fuzzy quantifiers.Moreover, the following result proves that in the unary case, for a ‘perfectly’ distributed fuzzyset, models F MD and F I tend to assign to each quantifier a probability weight proportional to itsarea. Let us define an equispaced fuzzy set over [0 ,
1] as: µ X ( a ) = < µ X ( a ) = h < µ X ( a ) = h , . . . , µ X ( a m ) = m →∞ Z Q ( X ≥ α ) d α = lim m →∞ µ Q m ! h + µ Q m ! h + . . . + µ Q m − m ! h = Z µ Q ( x ) dx In [20] a probabilistic interpretation of quantifiers is also used under the label semantics interpretation of fuzzy sets.
20 as we are simply computing the area of the quantifier.As a consequence, when we evaluate a set of unary quantifiers Q , . . . , Q r : P ( E ) → I overa fuzzy set following an identity function ( µ X ( e i ) = im ) we obtain: F MD ( Q i ) ( X ) = F I ( Q i ) ( X ) ≈ area (cid:0) µ Q i (cid:1) This property is related to the probabilistic alpha-cut interpretation of models F MD and F I .In this interpretation, if membership values of X are perfectly distributed, then ‘weights’ of thealpha cuts are perfectly distributed over the quantification universe. In this way, quantifiers ofgreater areas tend to ‘collect’ more weight than quantifiers with smaller areas. This also meansthat, for ‘finer’ quantifier partitions, weights tend to be more distributed between the quantifiersof the partition. In applications of fuzzy quantifiers for ranking generation, we generally have a set of ob-jects o , . . . , o N for which the fuzzy fulfillment of a set of criteria p , . . . , p m is known X o i = { µ X i ( p ) / p , . . . , µ X i ( p m ) / p m } , where µ X i (cid:16) p j (cid:17) / p j represents the fulfillment of the criteria p j bythe object o i . Additionally, we generally have a set of weights W = { µ W ( p ) / p , . . . , µ W ( p m ) / p m } indicating the relative importance of the criteria p , . . . , p m .Fuzzy quantification can be used to generate a ranking by means of the assignment of aweight to each object, computed using an unary proportional quantified expression, r o i = e Q ( X o i )when a vector of weights is not involved, or computed using a binary proportional quantifiedexpression r o i = e Q ( W , X o i ) when there exists a vector of weights W to indicate the relativeimportance of each criteria. Hence, when we compute r o i for each i = , . . . , N , we can rankeach object with respect to ‘how e Q ’ criteria it fulfills (e.g., for e Q = many , ‘how many’ ).Fuzzy quantifiers seem specially convenient for ranking applications. As r o i indicates ‘howgood’ is the object i in fulfilling ‘ e Q criteria’. We can easily adjust the quantifiers to prioritizeobjects fulfilling ‘most of the criteria’ , ‘some of them’ , ‘a least 10’ , etc.Ranking applications usually demand a great discriminative power between objects. In gen-eral, we should expect that even small variations in the inputs would produce some e ff ect in theoutputs. In order to analyze the discriminative power of QFMs, we will need some definitions: Definition 25.
Let h ( x ) : [0 , → I an strictly increasing continuous mapping; i.e., h ( x ) > h ( y ) for every x > y. We define the unary and binary semi-fuzzy quantifiers Q h : P ( E ) → I andQ h : P ( E ) → I as Q h ( Y ) = h ( | Y | ) , Y ∈ P ( E ) Q h ( Y , Y ) = ( h (cid:16) | Y ∩ Y || Y | (cid:17) X , ∅ X = ∅ For assuring the discriminative power of QFMs, we will require that in the case of unaryquantifiers, any increase in the fulfillment of a criteria will increase F ( Q h ). In the binary case,we will require that any increase in the fulfillment of a criteria associated with a strictly positiveweight will also increase F ( Q h ). That is, as h is strictly increasing, we expect that an increasein the values of the inputs is translated into an increase in the result of the evaluation.21 efinition 26. Let us consider X , X such that µ X ( e i ) = µ X ( e i ) , i , j, µ X ( e i ) < µ X ( e i ) , i = j. We say that a QFM F fulfills the property of discriminative ranking generation for unaryquantifiers if: F ( Q h ) ( X ) > F ( Q h ) ( X ) for h ( x ) strictly increasing. Definition 27.
Let us consider W , X , X such that µ X ( e i ) = µ X ( e i ) , i , j, µ X ( e i ) < µ X ( e i ) , µ W ( i ) > , i = j. We say that a QFM F fulfills the property of discriminative ranking generation for bi-nary quantifiers if it fulfills: F ( Q h ) ( W , X ) > F ( Q h ) ( W , X ) for h ( x ) strictly increasing.5.6.1. Analysis of the modelsModels M and M CX . Fulfillment of propagation of fuzziness properties makes M and M CX very inconvenient for ranking applications. Examples presented in section 5.4 have shown thatthese models are piecewise constant, and that they are not able of di ff erentiating between reallylarge regions of the input space. As a consequence, these models are incapable of making finedistinction between objects. Model F owa . Model F owa have been presented in [18, chapter 8] as the paradigmatic example of astandard DFS non-fulfilling the properties of propagation of fuzziness. Thus, the author considerthe F owa model convenient for applications needing an ‘enhanced discriminatory force’.As F owa model generalizes OWA, it adequately deals with the fine distinction between objectsin the unary case. But in the binary case, F owa is piecewise constant, as it proves the followingexample: F owa ( Q id ) ( { / e , / e , . / e , . / e } , { / e , / e , / e , / e } ) = . = F owa ( Q id ) ( { / e , / e , . / e , . / e } , { / e / e , . / e , . / e } )In previous example, for 0 . e , e , we can modify object fulfillment in the [0 , . ff erence in the output. Model F MD . A similar problem happens with the F MD model. As the F MD fulfills the strongconservativity property (see [9, chapter 3]) we have F MD ( Q id ) ( W , X o i ) = F MD ( Q id ) (cid:16) W , W e ∩ X o i (cid:17) and then, F MD ( Q id ) ( { / e , / e , . / e , . / e } , { / e , / e , / e , / e } ) = F MD ( Q id ) ( { / e , / e , . / e , . / e } , { / e , / e , . / e , . / e } ) = odel F I . Model F I fulfills the property of discriminative ranking generation. The proof isshown in the Apendix. Model F A . The F A model also fulfills the property of discriminative ranking generation. Theproof is shown in the Apendix.
6. Some recommendations for selecting QFMs for applications
In table 2 we synthesize the behavior of the QFMs F MD , F I , F A , M , M CX and F owa withrespect to the set of additional criteria we have presented. This allow to establish some recom-mendations for the selection of convenient models for applications:1. In applications that require a fine distinction between objects (e.g., ranking applications)only models F I and F A should be used for non unary quantifiers. In the unary case F MD and F owa coincide with the F I model for increasing quantifiers, and are also acceptable.2. In applications in which aggregative behavior is not acceptable, F A should be avoided.3. For maximal coherence with linguistic criteria, models M CX and F A are the preferredones. Models M and F owa show a good behavior as well. Model F I , being inferiorto 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 M and M CX .5. In order to preserve the intuitions underneath of the identity quantifier, guaranteeing thata linear increase in the inputs produces a linear increase in the outputs, models F A , F MD , F I or F owa should be selected.6. When taken into account the behavior of QFMs over quantified partitions, if we expectmore undefined results for fuzzier fuzzy sets, standard DFSs should be used. In the case ofprefering that QFMs could be interpreted as probabilities over quantified labels, distribut-ing the ‘degree of fulfilment’ between the di ff erent labels, the convenient models are F MD , F I and F A .Summing up, the model F A is a really convenient model for all the applications in whichaggregative behavior is not an impediment. M CX is the perfect model for applications in whichpreservation of fuzziness properties is required, but presents the handicap that is very inadequatefor ranking applications and it does not maintain the linguistic intuitions under the ‘identityquantifier’. Additionally, it has been proved that the M CX models presents a very stable behavior[18, section 7.12], which assures a certain insensitivity against modifications in the membershipsdegrees.If we need a model guaranteeing a fine distinction between objects but avoiding aggregativebehavior, the best option is the F I model. F I also guarantees linguistic intuitions associatedto the identity quantifier, allows to interpret quantified partitions as probabilities, and for fuzzysets whose membership degrees are maximally distributed over the referential set, evaluationresults provided by F I tend to the area of the quantifier. Moreover, F I preserves internal anexternal negation properties (this last property in the finite case), assuring the conservation ofthe linguistic relations of the ‘Aristotelian square’. Although F I is not a DFS, it is a remarkablemodel that presents a great equilibrium between the fulfillment of the di ff erent criteria.23 MD F I F A M M CX F owa Linguistic Compatibility partial partial DFS + FAI DFS DFS + FAI DFSAggregative behavior No No Yes No No NoIdentity Quantifier Yes Yes Yes No No YesPropagation of Fuzziness No No No Yes Yes NoQuantified Partitions Pr Pr Pr Ind Ind IndFine di ff erentiation No Yes Yes No No No Table 2: Summary of the behaviour of the QFMs. FAI: Fuzzy Argument Insertion, Pr: probability interpretation, Ind:tendency to 0.5 evaluation results F MD , sharing some of the behavior of the F I model, is not adequate for fine di ff erentiationof objects in the binary case. We consider linguistic behavior of F I model superior to the lin-guistic behavior of F MD , as this last model does not preserve linguistic transformations of theAristotelian square. M model shares most of the behavior of the M CX model, presenting the same problems butloosing some properties, as Fuzzy Argument Insertion. F owa model has been presented as the paradigmatic example of standard DFS convenient forranking applications, but we have seen that this model is not adequate for achieving a fine di ff er-entiation between objects with binary quantifiers. However, if we were interested in preservingall the properties of standard DFS guaranteeing some discriminative power, then the F owa modelis the convenient option.Finally, the way in which QFMs behave over quantified partitions can guide us in our decisionbetween standard DFSs and the remaining models. Standard DFS will tend to produce moreundefined results (in the sense of closeness to ) for fuzzier fuzzy sets (in the sense of closenessto of their membership degrees). F A , F MD and F I generate results that can be interpreted asprobabilities, dividing the ‘evaluation weight’ between the di ff erent quantifiers in the partition. F MD and F I also preserve the intuition of ‘weight of the quantifier’ (in the sense of the coverageof the quantification universe by the labels) for a perfect distribution of membership degrees.That is, F MD and F I tend to produce a result proportional to the area of the quantifier for fuzzysets whose membership degrees tend to be equally distributed over [0 ,
7. Conclusions
In this work we have advanced in the definition of some criteria to provide a better under-standing of the behavior of the most significative QFMs. First, we have compared the selectedQFMs against the main set of properties presented in [18], with the objective of clarifying thedi ff erences that these models present with respect to the properties proposed in the QFM frame-work.After that, we argued that previous considered properties, while being really convenient toseparate ‘good quantification models’ from ‘bad ones’, are not su ffi cient to clearly distinguish be-tween the set of analyzed QFMs, and specially, to help potential users in the process of selectingthe most convenient model for a specific application.In order to advance in this problem, we have introduced a new set of criteria, specially de-signed to di ff erentiate the behavior of the analyzed models. An in-depth comparative analysis of24he main models has been performed with respect to this new set of criteria. Based on this anal-ysis we have established some recommendations to guide in the selection of the more adequatemodel for specific practical applications.As future work, we consider relevant the possibility of defining new oriented criteria, focusedon specific applications. References [1] W. Bandler and L. Kohout. Fuzzy power sets and fuzzy implication operators.
Fuzzy Sets and Systems , 4:13–30,1980.[2] S. Barro, A. Bugar´ın, P. Cari˜nena, and F. D´ıaz-Hermida. A framework for fuzzy quantification models analysis.
IEEE Transactions on Fuzzy Systems , 11:89–99, 2003.[3] J. Barwise and R. Cooper. Generalized quantifiers and natural language.
Linguistics and Philosophy , 4:159–219,1981.[4] G. Bordogna and G. Pasi. Modeling vagueness in information retrieval. In M. Agosti, F. Crestani, and G. Pasi,editors,
Lectures on Information Retrieval (LNCS 1980) , pages 207–241. Springer-Verlag Berlin Heidelberg, 2000.[5] P. Bosc and O. Pivert. Sqlf: A relational database language for fuzzy querying.
IEEE Transactions on FuzzySystems , 3(1):1–17, 1995.[6] M. Delgado, D. S´anchez, and M. A. Vila. A survey of methods for evaluating quantified sentences. In
Proc. FirstEuropean Society for fuzzy logic and technologies conference (EUSFLAT’99) , pages 279–282, 1999.[7] M. Delgado, D. S´anchez, and M. A. Vila. Fuzzy cardinality based evaluation of quantified sentences.
InternationalJournal of Approximate Reasoning , 23(1):23–66, 2000.[8] M. Delgado, D. S´anchez, and MA Vila. Fuzzy quantification: a state of the art.
Fuzzy Sets and Systems , 242:1–302,2014.[9] F. D´ıaz-Hermida.
Modelos de ?cuantificac?´on borrosa basados en una interpretaci´on probabil´ıstica y su aplicaci´onen recuperaci´on de informaci´on . PhD thesis, Universidad de Santiago de Compostela, 2006.[10] F. D´ıaz-Hermida, A. Bugar´ın, P. Cari˜nena, and S. Barro. Evaluaci´on probabil´ıstica de proposiciones cuantificadasborrosas. In
Actas del X Congreso Espa˜nol Sobre Tecnolog´ıas y L´ogica Fuzzy (ESTYLF 2000) , pages 477–482,2000.[11] F. D´ıaz-Hermida, A. Bugar´ın, P. Cari˜nena, and S. Barro. Voting model based evaluation of fuzzy quantified sen-tences: a general framework.
Fuzzy Sets and Systems , 146:97–120, 2004.[12] F. Diaz-Hermida, A. Bugar´ın, and David E. Losada. The probatilistic quantifier fuzzification mechanism fa: A the-oretical analysis. Technical report, Centro Singular de Investigaci´on en Tecnolox´ıas da Informaci´on, Universidadede Santiago de Compostela, 2014. arXiv preprint arXiv: 1410.7233.[13] F. D´ıaz-Hermida, David. E. Losada, A. Bugar´ın, and S. Barro. A probabilistic quantifier fuzzification mechanism:The model and its evaluation for information retrieval.
IEEE Transactions on Fuzzy Systems , 13(1):688–700, 2005.[14] F. D´ıaz-Hermida, D.E. Losada, A. Bugar´ın, and S. Barro. A novel probabilistic quantifier fuzzification mechanismfor information retrieval. In
Proc. IPMU 2004, the 10th International Conference on Information Prosessing andManagement of Uncertainty in Knowledge-Based Systems , pages 1357,1364, Perugia, Italy, July 2004.[15] D. Dubois and H. Prade. Fuzzy cardinality and the modeling of imprecise quantification.
Fuzzy Sets and Systems ,16:199–230, 1985.[16] I. Gl¨ockner. DFS- an axiomatic approach to fuzzy quantification. TR97-06, Techn. Fakult¨at, Univ. Bielefeld, 1997.[17] I. Gl¨ockner. Evaluation of quantified propositions in generalized models of fuzzy quantification.
InternationalJournal of Approximate Reasoning , 37(2):93–126, 2004.[18] I. Gl¨ockner.
Fuzzy Quantifiers: A Computational Theory . Springer, 2006.[19] I. Gl¨ockner, A. Kn¨oll, and A. Wolfram. Data fusion based on fuzzy quantifiers. In
In: Proceedings of EuroFusion98,International Data Fusion Conference , pages 39–46, 1998.[20] J. Lawry. An alternative approach to computing with words.
International Journal of Uncertainty, Fuzziness andKnowledge Based Systems , 9:3–16, 2001.[21] Y. Liu and E.E. Kerre. An overview of fuzzy quantifiers. (i) interpretations. (ii) reasoning and applications.
FuzzySets and Systems , 95:1–121, 135–146, 1998.[22] D.E. Losada, F. D´ıaz-Hermida, A. A. Bugar´ın, and S. Barro. Experiments on using fuzzy quantified sentences inadhoc retrieval. In
Proc. SAC-04, the 19th ACM Symposium on Applied Computing - Special Track on InformationAccess and Retrieval , pages 1059,1066, Nicosia, Cyprus, March 2004.[23] M. Mucientes, R. Iglesias, C.V. Regueiro, A. Bugar´ın, and S. Barro. A fuzzy temporal rule-based velocity controllerfor mobile robotics.
Fuzzy Sets and Systems , 134(3; Special Issue: Fuzzy Set Techniques for Intelligent RoboticSystems):83–99, 2003.
24] A. L. Ralescu. Cardinality, quantifiers, and the aggregation of fuzzy criteria.
Fuzzy Sets and Systems , 69:355–365,1995.[25] A. Ramos-Soto, A. Bugar´ın, and S. Barro. On the role of linguistic descriptions of data in the building of naturallanguage generation systems.
Fuzzy Sets and Systems , 285:31–51, 2016.[26] A. Ramos-Soto, A. Bugar´ın, S. Barro, and J. Taboada. Linguistic descriptions for automatic generation of textualshort-term weather forecast on real prediction data.
IEEE Transactions on Fuzzy Systems , 23(1):44–57, 2015.[27] MD Ruiz, D. S´anchez, and M. Delgado. On the relation between fuzzy and generalized quantifiers.
Fuzzy Sets andSystems , page In press, 2016.[28] D. S´anchez.
Adquisici´on de relaciones entre atributos en bases de datos relacionales . PhD thesis, Universidad deGranada. E.T.S. de Ingenier´ıa Inform´atica, 1999.[29] R. R. Yager. Quantified propositions in a linguistic logic.
J. Man-Mach. Stud , 19:195–227, 1983.[30] Ronald R. Yager. Approximate reasoning as a basis for rule-based expert systems.
IEEE Transactions on Systems,Man and Cybernetics , 14(4):636–642, 1984.[31] R.R. Yager. On ordered weighted averaging aggregation operators in multicriteria decisionmaking.
IEEE Transac-tions on Systems, Man and Cybernetics , 18(1):183–191, 1988.[32] R.R. Yager. Counting the number of classes in a fuzzy set.
IEEE Transactions on Systems, Man and Cybernetics ,23(1):257–264, 1993.[33] M. Ying. Linguistic quantifiers modeled by sugeno integrals.
Artificial Intelligence , 179:581–600, 2006.[34] L. A. Zadeh. A computational approach to fuzzy quantifiers in natural languages. In R.R. Yager, editor,
Fuzzy setsand applications : Selected papers by L.A. Zadeh , pages 569–613. New York : John Wiley and sons, cop.1987,1987.[35] L.A. Zadeh. A computational approach to fuzzy quantifiers in natural languages.
Comp. and Machs. with Appls. ,8:149–184, 1983.[36] L.A. Zadeh. Fuzzy logic = computing with words. IEEE Transactions on Fuzzy Systems , 4(2):103–111, 1996.
AppendixDiscriminative ranking generation, model F I Proof.
Intuitively, when we increase the fulfillment of a property µ X oi (cid:16) p j (cid:17) associated to a weightgreater than 0 from a to b , we are adding an element to the alpha-cuts in the range ( a , b ]. Asthe weight of p j is greater than 0, the relative cardinality with respect to the alpha-cuts of W containing p j will increase.In detail, let be µ W (cid:16) p j (cid:17) = c > µ X oi (cid:16) p j (cid:17) = a the fulfillment of the criteria p j for the object i . Let us consider a second fuzzy set X o ′ i such that µ X oi ( p z ) = µ X oi ′ ( p z ) for every z , j , and µ X oi ′ (cid:16) p j (cid:17) = b > a .Then, F I ( Q h ) ( W , X o i ) = Z Z Q h (cid:16) W ≥ α , X o i ≥ α (cid:17) d α d α = Z Z a Q h (cid:16) W ≥ α , X o i ≥ α (cid:17) d α d α + Z Z ba Q h (cid:16) W ≥ α , X o i ≥ α (cid:17) d α d α + Z Z b Q h (cid:16) W ≥ α , X o i ≥ α (cid:17) d α d α Expressions R R a Q h (cid:16) W ≥ α , X o i ≥ α (cid:17) d α d α and R R b Q h (cid:16) W ≥ α , X o i ≥ α (cid:17) d α d α are equal for o i and o ′ i . With respect to R R ba Q h (cid:16) W ≥ α , X o i ≥ α (cid:17) d α d α , for alpha-cuts in (0 , c ] × ( a , b ]: Q h (cid:16) W ≥ α , X o i ≥ α (cid:17) > Q h (cid:18) W ≥ α , X o ′ i ≥ α (cid:19) p j ∈ W ≥ α , and p j ∈ X o ′ i ≥ α but p j < X o i ≥ α . And then F I ( Q h ) ( W , X o i ) < F I ( Q h ) (cid:16) W , X o ′ i (cid:17) . Discriminative ranking generation, model F A Proof.
Again, let be µ W (cid:16) p j (cid:17) = c > µ X oi (cid:16) p j (cid:17) = a the fulfillment of the criteria p j for theobject i . Let us consider a second fuzzy set X o ′ i such that µ X oi ( p z ) = µ X oi ′ ( p z ) for every z , j ,and µ X oi ′ (cid:16) p j (cid:17) = b > a . We are trying to prove that: F A ( Q h ) ( W , X o i ) = X Y ∈P ( E ) X Y ∈P ( E ) m W ( Y ) m X oi ( Y ) Q h ( Y , Y ) < X Y ∈P ( E ) X Y ∈P ( E ) m W ( Y ) m X oi ′ ( Y ) Q h ( Y , Y ) = F A ( Q h ) (cid:16) W , X o ′ i (cid:17) Making some computations with F A ( Q h ) ( W , X o i ) we obtain : F A ( Q h ) ( W , X o i ) = X Y ∈P ( E ) X Y ∈P ( E ) m W ( Y ) m X oi ( Y ) Q h ( Y , Y ) = X Y ∈P ( E \{ p i } ) X Y ∈P ( E \{ p i } ) (1 − c ) m W \{ p i } ( Y ) (1 − a ) m X oi \{ p i } ( Y ) Q h ( Y , Y ) (3) + X Y ∈P ( E \{ p i } ) X Y ∈P ( E ) | p i ∈ Y a (1 − c ) m W \{ p i } ( Y ) m X oi \{ p i } ( Y ) Q h ( Y , Y ) (4) + X Y ∈P ( E ) | p i ∈ Y X Y ∈P ( E \{ p i } ) c (1 − a ) m W \{ p i } ( Y ) m X oi \{ p i } ( Y ) Q h ( Y , Y ) (5) + X Y ∈P ( E ) | p i ∈ Y X Y ∈P ( E ) | p i ∈ Y ca × m W \{ p i } ( Y ) m X oi \{ p i } ( Y ) Q h ( Y , Y ) (6)but if p i < Y , then the relative cardinality | Y ∩ C || Y | = | Y ∩ ( C ∪{ p i } ) || Y | for C ∈ P ( E \ { p i } ). Then, the sumof expressions 3 and 4: X Y ∈P ( E \{ p i } ) X Y ∈P ( E ) \{ p i } (1 − c ) m W \{ p i } ( Y ) (1 − a ) m X oi \{ p i } ( Y ) Q h ( Y , Y ) + X Y ∈P ( E \{ p i } ) X Y ∈P ( E ) | p i ∈ Y a (1 − c ) m W \{ p i } ( Y ) m X oi \{ p i } ( Y ) Q h ( Y , Y ) = X Y ∈P ( E \{ p i } ) X C ∈P ( E \{ p i } ) (1 − c ) m W \{ p i } ( Y ) m X oi \{ p i } ( C ) Q h ( Y , C )is not a ff ected by the modification of µ X oi (cid:16) p j (cid:17) ; that is, it will coincide with the equivalent expres-sion for F A ( Q h ) ( W , X o i ′ ). By E \ { e i } we denote E ∩ { e i } ; that is, the set E without the element e i . For fuzzy sets, X \ { p i } is the projection of X eliminating the p i element. Then, in m X \{ p i } ( Y ) the element p i is not taken into account in the computation of theprobability mass of Y .
27e will focus now in 5 and 6. For F A ( Q h ) ( W , X o i ′ ), equivalent expression of 5 and 6 are obtainedby substituting (1 − a ) and a by (1 − b ) and b , respectively; that is, we reduced (1 − a ) by an( b − a ) factor and we increase a by an ( b − a ) factor. As h ( x ) is increasing, (1 − a ) h ( x ) + ah ( y ) < (1 − b ) h ( x ) + bh ( y ) for b > a , y > x . Thus, it is trivial to see that 5 and 6 are lesser than theequivalent expressions for F A ( Q h ) ( W , X o i ′′