The evaluation of fuzzy quantified expressions is a topic that has been widely dealt with in literature [2, 7, 8, 13, 14, 15, 16, 20, 24, 25, 28, 30, 36, 32, 31, 33, 34, 40, 56, 45, 46, 48, 49, 50, 51, 52, 53, 54, 55, 58] since the use of quantified expressions in fields such as fuzzy control , temporal reasoning in robotics, [11, 10, 44, 43], complex fuzzy queries in databases [8, 9], information retrieval [6, 5, 41, 23, 22, 35], data fusion [51, 37], etc. can take advantage of using vague and interpretable quantification models. Moreover, the definition of adequate models to evaluate quantified expressions is fundamental to perform “computing with words”, topic that was suggested by Zadeh  to express the ability of programming systems in a linguistic way. In this paper we analyze the theoretical behavior and some practical consequences of the model defined on [23, 22]111Most of the theoretical results presented in this paper have been previously published in the dissertation , in spanish.. Furthermore, we show that the underlying probabilistic interpretation of this model hints the utility of the model for a number of applications.
In general, approaches to fuzzy quantification in the literature use the concept of fuzzy linguistic quantifier  to represent absolute or proportional fuzzy quantities. Zadeh  defines quantifiers of the first kind as quantifiers used for representing absolute quantities (defined by using fuzzy numbers on ) , and quantifiers of the second kind as quantifiers used for representing relative quantities (defined by using fuzzy numbers on ). In the literature, quantifiers of the first kind are associated to sentences involving only one single fuzzy property (as in “about three men are tall” where “tall” is a fuzzy property); and quantifiers of the second kind are associated to sentences involving two fuzzy properties (as in “about 70% of blond men are tall” where “blond” and “tall” are fuzzy properties). The linguistic quantifier associated to the former sentence denotes the semantics of “about 3” and is defined by using a fuzzy number with domain on . The linguistic quantifier associated to the second sentence represents the semantics of “about 70%” and is defined by using a fuzzy number with domain on .
Moreover, most of the existing approaches for dealing with fuzzy quantification are based on the evaluation of the compatibility between the linguistic quantifier and a scalar, possibilistic or probabilistic cardinality measure for the involved fuzzy sets. Scalar approaches , usually consist of a simple evaluation of the quantifier on the cardinality value. For possibilistic approaches, an overlapping measure SUP-min is generally used [14, 16, 45] whilst for probabilistic approaches, [15, 16, 21] a weighted mean of all the compatibility values is computed. OWA approaches [51, 54] can also be related to the probabilistic interpretation. A different approach is used in [28, 30, 36, 32, 31, 33, 34], where families of models that are based on a three valued interpretation of fuzzy sets are defined.
For analyzing the behavior of fuzzy quantification models different properties of convenient or necessary fulfillment have been defined [16, 28, 30, 36, 32, 31, 33, 34, 46]. Most of the approaches in literature fail to exhibit a plausible behavior [2, 16, 29, 32, 33, 17, 46], and only a few [16, 21, 28, 30, 36, 32, 31, 33] seem to exhibit an adequate behavior in the general case.
In this work we will follow the Glöckner approximation to fuzzy quantification [28, 30, 36, 32, 31, 33, 34]. In his approach, the author generalizes the concept of generalized classic quantifier [3, 27, 38] (second order predicates or set relationships) to the fuzzy case; that is, a fuzzy quantifier is a fuzzy relationship between fuzzy sets. And then rewrites the fuzzy quantification problem as the problem of looking for mechanism to transform semi-fuzzy quantifiers (quantifiers between generalized classic quantifiers and fuzzy quantifiers that are adequate to specify the meaning of quantified expressions) to fuzzy quantifiers.
Moreover, Glöckner has also defined a rigorous axiomatic framework to assure the good behavior of QFMs. Models fulfilling this framework are called Determiner fuzzification schemes (DFSs) and they fulfill an important set of appropriate behavior properties.
The main goal of this work is to analyze the behavior of the model [23, 22, 17]. This model has a very solid theoretical behavior, superior to most of the models defined in the literature. Moreover, we show that the underlying probabilistic interpretation based on likelihood functions [42, 47, 4, 26] has very interesting consequences, that assure its utility for a number of applications. For example, in [23, 22, 17] the application of the model in a information retrieval task was shown, with competitive results. In  the model has been used in a summarization application for the evaluation of quantified temporal expressions. From a theoretical point of view the model is a DFS, although is only defined in finite domains. The fulfillment of the DFS axioms guarantees a very good theoretical behavior. As an important point, the fuzzy operators induced by the model are the product t-norm and the probabilistic sum t-conorm. This fact makes the model essentially different of the models defined in  because all those “standard models” induce the min tnorm and the max tconorm. To our knowledge, the model is the unique known non standard DFSs.
The paper is organized as follows. In the first section, we resume the Glöckner’s approach to fuzzy quantification, based on quantifier fuzzification mechanisms222A complete explanation of the QFM framework can be consulted in the excellent work .. In the second section we explain some of the properties that let us to analyze the behavior of the quantification model. Most of them are a compilation of the properties defined on [33, 34, chapters 3 and 4], but we have added to these properties two very interesting properties fulfilled by the model and by the probabilistic models defined in . In section three the QFM is defined. We also explores the behavior of the model when the cardinality of the base set tends to infinite, with a surprising relation with original Zadeh’s model . Next section is devoted to some interesting consequences of the probabilistic interpretation of the QFM, with relation with a number of application fields. Proofs of the properties and efficient algorithm solutions are collected in two apendixes. A bibliographic analysis of quantification models has not been included as it can be found in [2, 16, 29, 32, 33, 17, 46].
2 Quantifier fuzzification mechanisms
To overcome the Zadeh’s framework to fuzzy quantification Glöckner  rewrites the problem of fuzzy quantification as the problem of looking for adequate means to convert the specification means (semi-fuzzy quantifiers) into the operational means (fuzzy quantifiers) . In this section we explain in some detail the framework proposed by Glöckner to achieve that result.
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 and some examples:
Definition 1 (Classic quantifier.)
[34, pag. 57] A two valued (generalized) quantifier on a base set is a mapping , where is the arity (number of arguments) of , denotes the set of crisp truth values, and is the powerset of .
In this work we assume the base set is finite as the model is only defined on finite base sets.
Examples of some definitions of classic quantifiers are:
Let us consider the evaluation of the sentence “at least eighty percent of the members are lawyers” where the properties “members” and “lawyers” are respectively defined as , and “at least eighty percent” is defined in expression 1. Then .
In a fuzzy quantifier arguments and result can be fuzzy. The definition of a fuzzy quantifier is:
Definition 3 (Fuzzy Quantifier)
[34, pag. 66] An n-ary fuzzy quantifier on a base set is a mapping . Here denotes the fuzzy powerset of .
A fuzzy quantifier assigns a gradual result to each choice of .
An example of a fuzzy quantifier could be . A reasonable fuzzy definition of the fuzzy quantifier is:
Let us consider the evaluation of the sentence “all big houses are overvaluated” in a referential set . Let us assume that properties “big” and “overvaluated” are respectively defined as: , . If we use expression (2) then: .
Although a certain consensus may be achieved to accept this previous expression as a suitable definition for this is not the unique one. The problem of establishing consistent fuzzy definitions for quantifiers (e.g., “at least eighty percent”) is faced in  by introducing the concept of semi-fuzzy quantifiers. A semi-fuzzy quantifier represents a medium point between classic quantifiers and fuzzy quantifiers, and it is close but is far more general than the idea of Zadeh’s linguistic quantifiers . A semi-fuzzy quantifier only accepts crisp arguments, as classic quantifiers, but lets the result range on the truth grade scale , as for fuzzy quantifiers333An interesting classification of semi-fuzzy quantifiers is shown in . In [17, chapter 4] an extended classification is defined..
Definition 5 (Semi-fuzzy quantifier)
[34, pag. 71] An n-ary semi-fuzzy quantifier on a base set is a mapping .
assigns a gradual result to each pair of crisp sets .
Examples of semi-fuzzy quantifiers are:
where and are
shown in figure (1)444Functions and
are defined as
Let us consider the evaluation of the sentence “about at least 80% the students are Spanish”. Let us assume that properties “students” and “Spanish” are respectively defined as: , then .
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 order to do so mechanisms are needed that enable us 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. Glockner names those mechanisms quantifier fuzzification mechanisms.
[34, pag. 74]A quantifier fuzzification mechanism (QFM) assigns to each semi-fuzzy quantifier a corresponding fuzzy quantifier of the same artity and on the same base set.
3 Some properties to guarantee the good behavior of QFMs
Before proceeding to explain the QFM we will introduce some of the properties that let us to guarantee a good behavior of the QFMs. For the sake of brevity, we have only selected some of the more important properties to characterize the behavior of quantification models. A complete and detailed exposition, showing the intuitions under those definitions can be found in [34, chapters three and four.].
The set of properties is organized in three sets. First set is composed of the most important properties that are consequence of the DFS axioms. Second group is composed of some properties that are not consequence of the DFS framework but are important to characterize the behavior of QFMs for different reasons. The last group includes two very important properties that the model and the probabilistic models defined on  fulfills555One of the models defined in  is a generalization of an original proposal of Delgado et al. [15, 16] to semi-fuzzy quantifiers..
In the appendix we show the proof of those properties for the QFM.
3.1 Some properties that are consequence of the DFS axiomatic framework
3.1.1 Correct generalization property (P.1)
Perhaps the most fundamental property to be fulfilled by a QFM is the correct generalization property. This property, defined independently by Glöckner  for QFMs and by Delgado et al. for models following the Zadeh’s framework [46, 16], requires that the behavior of a fuzzy quantifier on crisp arguments was the expected; that is, the results obtained with a fuzzy quantifier and with the corresponding semi-fuzzy quantifier must coincide on crisp arguments.
We show now the definition of the property:
Definition 8 (Property of correct generalization)
[34, pag. 112] Let be an n-ary semi-fuzzy quantifier. We say that a QFM fulfills the property of correct generalization if for all the crisp subsets , then it holds .
For a detailed explanation of this property [34, Sections 3.2. and 4.2.] can be consulted.
For example, given crisp sets , , , then this property guarantees that
In the DFS axiomatic framework it is sufficient to guarantee this property in the unary case.
3.1.2 Membership assessment (P.2)
This property is related with the evaluation of the membership grade of a particular element [34, section 3.3.], and belongs to the set of axioms that are used to characterize the DFSs.
In the classic case, we can define a crisp quantifier that test if the element belongs to the argument set. In the same way, in the fuzzy case, we can define a fuzzy quantifier that returns the membership grade of . It is natural to require that a reasonable QFM maps to .
The formal definitions of and are:
[34, pag. 88] Let a base set and . The projection quantifier is defined by for all , where denotes the crisp characteristic funtion of the set .
The corresponding fuzzy definition is:
[34, pag. 88] Let a base set be given and . The fuzzy projection quantifier is defined by for all .
Using these definitions the property that establishes that a QFM generalizes the quantifier in the correct way is defined:
Definition 11 (Projection quantifiers)
[34, pag. 89, pag. 112] Let a QFM. fulfills the property of projection quantifiers if it holds for and .
3.1.3 Induced operators (P3)
Glöckner explains that a QFM can be used to transform crisp logical operators into fuzzy operators. For example, logical “or” can be extended by using the following semi-fuzzy quantifier defined on a referential set composed by two elements ():
and in this way is possible to define the fuzzy logical function that is induced by the fuzzification mechanism as
To formally define this property the next bijection is needed:
for all . In the fuzzy case the analogous bijection is for all and .
These bijections are used to transform the fuzzy truth functions (i.e. mappings ) in semi-fuzzy quantifiers . In the same way fuzzy quantifieres can be transformed in fuzzy truth functions .
The definition that let us to transform semi-fuzzy truth function in fuzzy truth functions by means of a QFM is the following:
[34, pag. 90] Suppose is a QFM and is a mapping (i.e. a ‘semi-fuzzy truth funtion’) for some . The semi-fuzzy quantifier is defined by for all . In terms of , the induced fuzzy truth function is defined by
for all .
The construction allows us to transform the usual crisp logical operators (, , , ) into the analogous fuzzy operators (, , , ). For a reasonable QFM we should expect that the induced operators were fuzzy valid operators.
For a DFS the next property is guaranteed666This is a resume of the longer exposition maked in [34, section 4.3].:
Definition 13 (Property of the induced truth functions)
Truth operations induced by a
quantifier fuzzification mechanism must be coherent with fuzzy logic; i.e.,
the following must hold:
a. (where is the bivalued identity truth function) is the fuzzy identity truth function.
b. is a strong negation operator.
c. is a tnorm.
d. is a tconorm.
e. is an implication function.
In this manner it is guaranteed that the fuzzy operators that are generated are reasonable from the perspective of fuzzy logic. For example, for where and it is guaranteed we obtaine the result of using the induced on .
3.1.4 External negation property (P.4)
Now we are going to present a set of three very important properties from a linguistic point of view. The properties of external negation, internal negation and duality. We will begin defining the external negation property [34, section 3.5]:
Definition 14 (External negation)
[34, pag. 93]The external negation of a semi-fuzzy quantifier is defined by for all . The definition of in the case of fuzzy quantifiers is analogous777The reasonable choice of the fuzzy negation is the induced negation of the QFM..
From a linguistic point of view, the external negation of “all the students are spanish” is “not all the students are spanish”.
A QFM correctly generalizes the external negation property if it fulfills the next property:888The property of external negation is one of the initial axioms of the axiomatic framework presented in [28, pag. 22] to define the DFSs.
Definition 15 (External negation property.)
For example, the fulfillment of this property assures:
That is, the equivalence between the expressions “at most ten rich students are intelligent” and “no more than eleven rich students are intelligent” is assured in the fuzzy case.
3.1.5 Internal negation property (P.5)
The internal negation or antonym of a semi-fuzzy quantifier is defined as:
Definition 16 (Internal negation.)
[34, pag. 93] Let a semi-fuzzy quantifier of arity be given. The internal negation of is defined by
for all . The internal negation of a fuzzy quantifier is defined analogously, based on the given fuzzy complement .
For example, the internal negation of is because
The definition of the property of internal negation is:999The property of internal negation is one of the initial axioms of the axiomatic framework presented in [28, pag. 22] to define the DFSs.
Definition 17 (Internal negation property)
For example, this property assures
That is, the equivalence between the expressions “all big houses are overvaluated” and “no big houses are undervaluated” is assured in the fuzzy case.
3.1.6 Duality property (P.6)
This property is a consequence of the fulfillment of the external and internal negation properties. In  is one of the axioms used to define the DFSs.
Definition 18 (Dual quantifier.)
[33, pag. 99]The dual of a semi-fuzzy quantifier , is defined by
for all . The dual of a fuzzy quantifier is defined analogously.
For example, the dual of is
Using the axiom of duality [34, pag. 94-96] the duality property can be defined:
Definition 19 (Duality property)
We say that a QFM fulfills the property of duality if for all semi-fuzzy quantifiers of arity .
For example this property assures that
that is, the equivalence of the sentences “not all the expensives cars are not good” and “some expensive car is good” is assured in the fuzzy case.
3.1.7 Internal meets property (P.7)
In combination with negation properties, this property assures boolean combination of arguments are mapped to the fuzzy case.
First, we show the “union” and “intersection” quantifiers:
Definition 20 (Union quantifier)
[34, section 3.7] Let be a semi-fuzzy quantifier, , be given. We define the fuzzy quantifier as
for all . In the case of fuzzy quantifiers is defined analogously, based on a fuzzy definition of .
Definition 21 (Intersection quantifier)
Let a semi-fuzzy quantifier, , be given. We define the semi-fuzzy quantifier as
for all . In the case of fuzzy quantifiers is defined analogously, based on a fuzzy definition of .
Expressions like “all are or ” where are crisp can be evaluated by means of less arity quantifiers with these constructions:
The definition of the property is:
Definition 22 (Internal meets property)
[34, pag. 97] Let a semi-fuzzy quantifier, . We will say a QFM preserves the property of internal meets if:
As a consequence,
3.1.8 Monotonicity in arguments property (P.8)
In this section we present the property of monotonicity in arguments. This property is one of the axioms used to define the DFSs.
Definition 23 (Monotonicity)
[34, pag. 98] A semi-fuzzy quantifier is said to be nondecreasing in its i-th argument, if
whenever the involved arguments satisfy . is said to be nonincreasing in the i-th argument if under the same conditions, it always holds that
The corresponding definitions for fuzzy quantifiers are entirely analogous. In this case, the arguments range over , and ‘’ is the usual fuzzy inclusion relation ( if for all ).
For example, the semi-fuzzy quantifier is monotonic nondecreasing in both arguments.
The next property guarantees the extension of the monotonicity to fuzzy quantifiers:
Definition 24 (Monotonicity property)
[34, pag. 100]A QFM is said to preserve monotonicity in the arguments if semi-fuzzy quantifiers which are nondecreasing (nonincreasing) in their i-th argument are mapped to fuzzy quantifiers which are also nondecreasing (nonincreasing) in their -ih argument.
For example, if a QFM guarantees this property then is monotonic non-decreasing in both arguments.
3.1.9 Monotonicity in quantifiers property (P.9)
The property of monotonicity in quantifiers is a very important consequence of the DFS axioms [28, 34]. Independently, this property has also been defined in [46, pag. 73], for unary quantifiers with the name of property of inclusion of quantifiers.
This property establishes that if a semi-fuzzy quantifier is included in other semi-fuzzy quantifier (i.e., the results of are smaller than the results of for all the selections of crisp arguments ) then the fuzzy extension is also included in .
Definition 25 (Monotonicity in the quantifiers)
[34, pag. 128] Suppose are semi-fuzzy quantifiers. Let us write if for all , . On fuzzy quantifiers we define analogously, based on arguments in .
For example, for the following semi-fuzzy quantifiers
it holds that .
The next property is defined based on the Theorem 4.32 in [34, pag. 128].
Definition 26 (Property of monotonicity in quantifiers)
Suppose is a QFM, and are semi-fuzzy quantifiers. We say that fulfills the property of monotonicity in quantifiers if and only if .
This property guarantees that for the semi-fuzzy quantifiers defined on the expression 4.
3.1.10 Property of functional application (P.10)
The property of compatibility with functional application forms part of the axioms that are used to define the DFSs . This property requires that a QFM must be compatible with its induced extension principle.
Definition 27 (Extension of a function to sets)
Let us consider function. Function is defined in the following way: .
The extension principle induced by a QFM is defined as:
Definition 28 (Induced extension principle)
[34, pág. 101] All QFM induce an extension principle that to each function (where ) assigns a function defined by for all , .
It should be noted that in the function is the extension to sets of the function and then
is the characteristic function of this extension; that is,is a semi-fuzzy quantifier that for a set returns if and in other case.
The property of compatibility with functional application is defined as:
Proposition 29 (Compatibility with functional application)
[34, Pág. 104] Let a given QFM. We will say that is compatible with its induced extension principle if or equivalently
is valid for all semi-fuzzy quantifier and all the function with domain , .
That is, if a QFM fulfills the property of functional application, the same results are obtained when we first apply the induced extension principle to the argument sets and then we apply the quantifier , and when we first apply the semi-fuzzy quantifier (that to the crisp sets apply the function , and then evaluates ), and then we apply to compute the function on .
This propery is very important in union with the rest of the axioms used to define the QFMs because all toghether assures the fulfillment of a very important and intuitive set of properties.
3.2 The DFS axiomatic framework
We now present the DFS axiomatic framework. In  the author dedicates the whole 4 chapter to describe the properties that are consequence of the axiomatic framework. For the sake of brevity, we have only described the set of properties we have consider more relevant. Other important properties the author describes in  are argument permutations (the QFMs are compatible with the trasposition of arguments), cylindrical extensions (that guaratees vacuous arguments are irrelevant), quantitativity (QFMs guarantees that quantitative semi-fuzzy quantifiers are mapped to quantitative fuzzy quantifiers), etc.
The framework the author sets out in [34, section 3.9] is a refinement of the original framework defined on [28, pag. 22] that it was composed by 9 interdependent axioms. The two frameworks are equivalent. We present now the definition of the DFS framework:
A QFM is called a determiner fuzzification scheme (DFS) if the
following conditions are satisfied for all semi-fuzzy quantifiers
Correct generalisation if (Z-1) Projection quantifiers if for some (Z-2) Dualisation (Z-3) Internal joins (Z-4) Preservation of monotonicity If is nonincreasing in the -th arg, then is nonincreasing in -th arg, (Z-5) Functional application where (Z-6)
In the previous definition is the underlying semi-fuzzy quantifier [34, pag. 75]; that is, the semi-fuzzy quantifier defined as:
for all crisp . The axiom 1 is equivalent to the fulfillment of the correct generalization property in the unary case.
3.3 Some properties that are not a consequence of the DFS axioms
Now we will describe some adequacy properties that are not guaranteed by the DFS framework because they impose an excesive restriction on the class of plausible models. In [34, chapter 6] a detailed exposition considering these and other properties can be consulted.
3.3.1 Property of continuity in arguments (P.11)
Continuity properties are fundamental. Models that do not fulfil these properties generally will not be valid from a practical viewpoint. One reason is that it is impossible to avoid measure errors and, as a consequence, errors in data measures could cause completely different analysis. Other reason is that from a user viewpoint, it would be very difficult to understand why no significant differences produce different results. Continutiy is also necessary from an application view (for example, imagine we need to use fuzzy quantifiers in a control system).
In this section we will explain the continuity in arguments property [34, Section 6.2]. The definition of this property is based on the next metric to measure the difference between two pairs of fuzzy sets :
Definition 31 ()
[34, pag. 162] For all base sets and all the metric is defined by
for all .
Using this metric the property of continuity in arguments is defined:
Definition 32 (Continuity in arguments property)
[34, pag. 163] We say that a QFM is arg-continuous if and only if maps all semi-fuzzy quantifiers to continuous fuzzy quantifiers ; i.e. for all and there exists such that for all with
3.3.2 Property of continuity in quantifiers (P.12)
In the same way we require continuity on argument sets, we also require continuity in quantifiers. That is, we do not expect big differences in results when we modify slightly the quantifiers.
The distance between two semi-fuzzy quantifiers is defined as:
Definition 33 ()
[34, pag. 163] For all semi-fuzzy quantifiers the distance between and is defined as:
and similarity for all fuzzy quantifiers
-continuity is defined as:
Definition 34 (Continuity in quantifiers property)
[34, pag. 163] We say that a QFM is -continuous if and only if for each semi-fuzzy quantifier and all , there exists such that whenever satisfies .
3.3.3 Property of the fuzzy argument insertion (P.13)
The property of fuzzy argument insertion is the fuzzy generalization of the crisp argument insertion [34, section 4.10]. Let a semi-fuzzy quantifier , and . By we will denote the semi-fuzzy quantifier defined as
for all . As a consequence of the DFS axioms it is fulfilled that
for all semi-fuzzy quantifier of arity , and all crisp .
Fuzzy argument insertion cannot be modeled directly, because a semi-fuzzy quantifier only accepts crisp arguments; that is, for all fuzzy only is defined and no . But as is explained in [34, sección 6.8], a QFM and a semi-fuzzy quantifier we can study if there exists a semi-fuzzy quantifier fulfilling
for all .
The reasonable election is the following:
[34, pag. 172]Let a QFM, a semi-fuzzy quantifier and a fuzzy set. Then is defined as
that is, for all crisp sets .
The next property resumes the fulfillent of the fuzzy argument insertion in the fuzzy case:
[34, pag. 172]Let be a QFM. We will say fulfills fuzzy argument insertion if for all semi-fuzzy quantifier of artity and all fuzzy is fulfilled
This property has a very strong relation with nested quantification. Althoug the sufficiency of this property for a DFS to adequate model nested quantifiers, in [34, section 12.6] the author has state the necessity of fulfilling this property. Moreover, the fulfillment of this property for standard DFSs is only achieved by the , a paradigmatic example of good theoretical behavior.
3.4 Some probabilistic properties
3.4.1 Property of averaging for the identity quantifier (P.14)
The fulfillment of this property for a QFM assures that when we apply the model to the unary semi-fuzzy quantifier identity