Negation functions based semantics for ordered linguistic labels

Transcription

1 Negation functions based semantics for ordered linguistic labels Vicenç Torra Dpt. d'enginyeria informática Escola Tècnica Superior d'enginyeria (ETSE) Universitat Rovira i Virgili Carretera de Salou, s/n E Tarragona (Spain) vtorra@etse.urv.es Abstract After arguing that in the knowledge acquisition framework experts can not always supply a precise semantics for the linguistic labels they use, we show that negation functions over an ordered set of linguistics labels induce a semantics. We study the semantics induced by classical negation, functions from L to L, and also the one induced by negation functions from L to parts of L. I. Introduction The process of building a Knowledge Based System, i.e., the development of a model, is a difficult task because it involves the definition of new languages to model experience that has not been expressed before 1,2. To help in this process, there have been built several methods to build models, e.g., the methods based on Personal Construct Theory. These systems, like KSSn 3, ETS 4, AQUINAS 5,3 and EAR 6, allow one or more experts by means of defining concepts, relations and rules to define a model in an interactive way. They offer the possibility to analyse at any time the structures already defined, so the expert can reconsider decisions already made on the model, or develop the features of it still incomplete. In classical systems based on Personal Construct Theory the model is built from a matrix that takes quantitative values for each pair (object, attribute). In this case, the system encompasses several tools, e.g. tools to combine information from different sources 7, tools to generate rules from the estructures defined by the user 8,4 and also tools to analyse the information represented by the experts (e.g. regression analysis 4, classifiers 9,...). Regression tools are used to analyse similarities between the values corresponding to two objects or two attributes. Classifiers, instead, are used, on one hand, as tools to analyse the similarity between objects or, in ill-defined domains, to obtain a structure of the elements of the domain. On the other hand, classifiers appear also as a part of some of the tools included in knowledge acquisition systems. For instance, EGAC 7, a tool for synthesising data matrices corresponding to a set of experts uses in some circumstances a classifier.

2 As it has been said, in knowledge acquisition systems based on Personal Construct Theory the model is built from a matrix where each pair (object, attribute) has attached a quantitative value. However, this kind of values, as argued in 7, is not very appropriate when the work is done in the medicine field. In this case it is usually preferable to use qualitative values to qualify traits and information about the patient. In this case, to obtain a classification it is not enough to define a distance over quantitative values but also over qualitative ones. Usually when a system considers linguistic labels, it also considers their corresponding semantics defined by means of fuzzy sets 10,11. This is, for each linguistic label there is a fuzzy set over a reference set attached to it. See for instance the use of linguistic labels in fuzzy control 12,13,14, in KBS 15,16,17, and aggregation of opinions 18,19,20,21. In these cases, when a semantic is available, it is possible to define the distance between linguistic labels as the distance between their corresponding fuzzy sets (e.g., by means of the extension principle 22 ). However, in the framework of modelling it is not always possible that the expert defines a fuzzy set for each linguistic label because the definition of a fuzzy set for each label requires an excess of accuracy that the expert can not always supply. In this work we study negation functions over linguistic labels and we show how these functions induce a semantics for each label. From the semantics of each pair of labels it is straightforward to define a distance. Usually negation functions over a set of ordered linguistic labels L={x 0,..., x n } (with x 0 <...<x n ) are defined 23 as functions from L to L that satisfy: N1) if x<x' then Neg(x)>Neg(x') for all x, x' in L N2) Neg(Neg(x)) = x for all x in L The later condition, N2, can be equivalently expressed as N3) if x=neg(x') then x'=neg(x) When these conditions hold, as the following proposition shows, the ordered linguistic labels L completely determines the negation function. Proposition For each set of ordered linguistic labels L={x 0,..., x n } there exists only one negation function that satisfies N1 and N2. This negation function is defined by:

3 N(x i )=x n-i for all x i in L (1.1) However, a set of linguistic labels with a negation function that satisfies N1 and N2 is not always adequate. In fact, a negation function according to proposition 1.1 stands for situations where each label of the pair <x i,x n-i > is equally informative. It is possible, instead, that a subdomain of the reference set is more informative than the rest of the domain. In this case, the density of linguistic labels in the subdomain would be greater than the density in the rest of the domain. For instance, suppose that we require a temperature control system with a very precise behaviour when the temperature is low. In this case, the set of linguistic labels would have a distribution over the reference set similar to figure 1. (In figure 1 AN stands for Almost-Nil, VL for Very-Low, QL=Quite-Low and VH=Very-High). AN VL QL Low Medium Hi gh VH Figure 1 When the set of linguistic labels is defined in this way, the negation function can not be of the form of proposition 1.1. Notice that in the example given above, the negation of Almost-Nil, and also the one of Very-Low, should be Very-High and therefore proposition 1.1 is not satisfied. In the same way Neg(Quite-Low)=Neg(Low)=High. Besides of that, if we consider the condition if x=neg(x') then x'=neg(x), then it is not even possible to define Neg(High). To overcome these problems we have defined (see section 3) a negation function, Neg, from L to (L) (i.e., parts of L). It has been defined weakening conditions N1 and N2. First, as Neg(Quite-Low)=Neg(Low)=High condition N1 is replaced by: C1) if x<x' then Neg(x)=Neg(x') for all x,x' L Besides of that, as Neg is from L to (L) instead of being from L to L, condition N3, that is equivalent to N2, has been replaced by C2) if x Neg(x') then x' Neg(x) It can be seen that it is now possible to define a negation function according to figure 1: Neg(almost-nil) = Neg(very-low) = {very-high}

4 Neg(quite-low)=Neg(low)={high} Neg(medium)={medium} Neg(high)={quite-low, low} Neg(very-high)={almost-nil, very-low} In this work we show that a negation function of this form induces an interval for each linguistic label. Now, we finish this section with an overview of the notation used in this work. Section 2 defines classical negation functions and considers the intervals attached to linguistic labels in this case. Section 3 defines negation functions from L to (L) and studies some of their properties. In the following we consider L as the ordered set of n+1 linguistic labels {x 0,...,x n }. We use < as its relation function (x i <x j if and only if i<j). For each label x i of L we denote by I(x i ) its corresponding interval, its semantics, in the unit interval (i.e., x i L, I(x i ) [0,1]), and we indicate by m in, m ix, m im and m i (see figure 2) the infimum, the supremum, the medium point and the measure of the interval I(x i ), i.e., m in = inf I(x i ), m ix = sup I(x i ), m im =(m in +m ix )/2, m i =m ix -m in Notice that when an interval I(x i ) is observed as a fuzzy set with membership function µ xi, then the measure of the interval (m i ) corresponds to the measure of the imprecision 25,7.... x... i m in m im m i Figure 2 m ix In this paper we denote by P the set of intervals attached to the linguistic labels in L, i.e., P={I(x i ): x i L}. We suppose that the set of intervals P attached to L, P={[m 0n,m 0x ],...,[m nn,m nx ]} satisfies 0=m 0n, 1=m nx and m ix =m i+1n for all i {0,1,...,n-1}, and therefore xi L I(x i )=[0,1] and I(x i ) I(x i+1 ) = {m ix } for i {0,1,...,n-1}. Once there is a set of intervals P={[m 0n,m 0x ],...,[m nn,m nx ]} attached to the set of linguistic labels L we can define a distance over each pair of labels (x i,x j ) as: d(x i,x j )=abs(m im -m jm ) (1.2)

5 II. Classical Negation Function In this section we introduce classical negation functions and we define a measure of neutrality for a given set of intervals. We will show that when neutrality is maximized the set of intervals of a set of linguistic labels is determined. Definition 2.1. A function Neg: L->L is a classical negation function if it satisfies: N1) if x<x' then Neg(x)>Neg(x') for all x,x' L N2) Neg(Neg(x))=x for all x L We define next the neutrality measure of a set of intervals as an overall measure of the similarity of the imprecision of the intervals. It is defined, see proposition 2.1, so that it gets its maximum value when all the intervals are equal. The name of this measure is taken from synthesis of fuzzy values where neutrality corresponds 26 to the case when the synthesised value does not depend on permutation of values (F(x 1,x 2,...,x n )=F(x σ(1), x σ(2),..., x s(n) ) where {σ(1),σ(2),...,σ(n)} is a permutation of {1,2,...,n}). We suppose below that when there is no information in relation to the linguistic labels, their corresponding intervals have to be defined in a way that their neutrality is maximized. That is, we choose the semantics for each label so that all the intervals have the same imprecision. Definition 2.2. Given a set of n+1 intervals P={[m 0n,m 0x ],...,[m nn,m nx ]} we define its neutrality measure as the product of the measure of the intervals, i.e., Neutrality (P) =? i=0 n mi Proposition 2.1. Given a set of linguistic labels, the corresponding set of intervals P maximises its neutrality if and only if m i =1/(n+1) for all x i. This condition together with the fact that x i L I(x i )=[0,1] is equivalent to define the interval attached to x i as: I(x i )=[i/(n+1), (i+1)/(n+1)] When it is required maximal neutrality (i.e., proposition 2.1 holds) and that the distance between any pair of linguistic labels satisfies (1.2), the distance between the labels x i and x j is:

6 d(x i,x j )=abs((2i+1)/2(n+1) - (2j+1)/2(n+1)) = abs(i-j) / (n+1) III. Negation Functions In this section we consider negation functions from L to (L). After introducing some definitions that are used later on, we establish a negation function as a weakening of the conditions given in definition 2.1. We study some properties of these negation functions. Definition 3.1. A subset X of L is convex if and only if for all x,y,z L such that x<y<z and x,z X, then y X. convex. Definition 3.2. A function f from L to (L) is convex if and only if for all x L, f(x) is Definition 3.3. A function f from L to (L) is non empty if and only if for all x L, f(x)?ø. Definition 3.4. Given two subsets X,Y L we say that X=Y if and only if min X = max Y where min X (respectively, max X) stands for the x X such that x=x' for any other x' X (resp., x=x' for any other x' X). In the same way we define X>Y as true, if and only if, min X > max Y. linguistic labels. Notice that in this work the relation < is defined over linguistic labels and also over sets of We define now a negation function as a generalization of conditions N1 and N3 (the condition equivalent to N2) in definition 2.1.We consider here an extra condition, C0, about Neg related to the fact that Neg(x i ) (L). Condition C0 requires that the negation of a label is not an empty set (every label has a negation) and that the negation of a label is a convex set. Definition 3.5. A function Neg from L to (L) is a negation function if it satisfies: C0) Neg is a convex and a non empty function C1) if x<x' then Neg(x)=Neg(x') for all x,x' L C2) if x Neg(x') then x' Neg(x)

7 Definition 3.6. Given a negation function Neg, over a set of linguistic labels L we define the extended negation function Neg * from (L) to (L) as Neg * (X)= x X Neg(x) Considering Neg * together with the negation function Neg we can generalize condition (N2) of definition 2.1 (x=neg(neg(x)) for all x L) as: C3) x Neg * (Neg(x)) for all x L It can be proved that the following proposition holds: Proposition 3.1. Let Neg be a negation function over L and let Neg * be its correspoding extended negation function. In these conditions C3 holds. Proof. Let x be an element of L, and let x' be an arbitrary element of Neg(x), so x' Neg(x). x' exists as Neg is a non empty function, by condition C1. When Neg * (Neg(x)) is considered we have that, according to the definition of Neg *, the following condition holds: Neg * (Neg(x)) = x* Neg(x) Neg(x*) (3.1) As the set {x* x* Neg(x)} can be decomposed in {x* x* Neg(x)}-{x'} {x'}, because we know that x' Neg(x), then (3.1) is equivalent to Neg * (Neg(x)) = { x* Neg(x)-x' Neg(x*)} Neg(x') Now, due to condition C2, we know that if x' Neg(x) then x Neg(x'). Therefore, it is proven that x Neg * (Neg(x)). o Notice that it is not true that a function from L to (L) that satisfies C0, C1 and C3 satisfies C2. See for example: L={a,b,c} Neg(a)={c}, Neg(b)={b}, Neg(c)={a,b} We have that b Neg(c) but c Neg(b). The fact that a function satisfying C0, C1 and C2 satisfies also C3, but that if it satisfies C0, C1 and C3 it may not satisfy C2 has influenced us in the definition of the negation function. In fact, a

8 negation function that do not satisfy C2 does not lead to intervals that satisfy I(x i ) I(x i+1 )={m ix } for all i {0,1,...,n-1}. We introduce below some propositions that are needed later. Proposition 3.2. Let Neg be a negation function from L to (L) according to definition 3.5. In these conditions it holds: Neg(x)={x k :x Neg(x k )} for all x L Proof. We show that the elements in Neg(x) are the elements in {x k :x Neg(x k )}. 1) Suppose that there exists an x k, that x Neg(x k ) but x k Neg(x). However, this is not possible because, due to condition C2, when x Neg(x k ) then x k Neg(x). 2) Suppose that there exists an x k, that x k Neg(x) but x Neg(x k ). Again, this is not possible because, due also to condition C2, if x k Neg(x) then x Neg(x k ). Therefore, as the elements of both sets coincide, the proposition is proven. o Proposition 3.3. Let Neg be a negation function from L to (L) according to definition 3.5, let x i be an arbitrary linguistic label in L, and let the set Neg(x i ) be {x i 0,...,x in } where x i 0 =min {x : x Neg(x i )} and x in =max {x : x Neg(x i )}. In these conditions if x k <x i0 for any x k L and x Neg(x k ) then x=x i. Proof. To prove the proposition given above, we suppose first that x k <x i, in this case, due 0 to C1 we have that Neg(x k )=Neg(x i 0 ). We also know that x i 0 Neg(x i ) therefore, x i Neg(x i 0 ). From Neg(x k )=Neg(x i 0 ) and x i Neg(x i 0 ) and x Neg(x k ) we can conclude that x=x i. o Proposition 3.4. Let Neg be a negation function from L to (L) according to definition 3.5, let x i be an arbitrary linguistic label in L, and let the set Neg(x i ) be {x i 0,...,x in } where x i 0 =min {x : x Neg(x i )} and x in =max {x : x Neg(x i )}. In these conditions if x k >x in for any x k L and x Neg(x k ) then x=x i. Proof. The proof of this proposition is analogous to the proof of proposition 3.3.

9 Proposition 3.5. Let Neg be a negation function from L to (L) according to definition 3.5, let x i be an arbitrary linguistic label in L such that Neg(x i ) =2 and let Neg(x i ) be {x i 0,...,x in } where x i 0 =min {x : x Neg(x i )} and x in = max {x : x Neg(x i )}. In these conditions x i = min Neg(x i 0 ) Proof. To proof this equality see that if exists some x j Neg(x i 0 ) such that x j <x i then as x j Neg(x i 0 ) then x i0 Neg(x j ) and as x j <x i then Neg(x j )=Neg(x i ). But as x i 0 Neg(x j ) and x in =max Neg(x i ) we have that x i0 =x in. But this is not possible as Neg(x i ) >2 and x i 0 =min Neg(x i ) and x in =max Neg(x i ). Therefore, as we get a contradiction the proposition is proven. o In section 2, when the classical negation function was considered, the set of intervals P attached to the set of linguistic labels L was required to be in a way that neutrality was maximised. In this section neutrality is not considered, as we know now through the negation function that not all the linguistic labels have a similar behaviour. We define below the semantics of each label by means of its behaviour in relation to the negation function. We will see in proposition 3.8 that the intervals obtained through neutrality maximisation coincide with the ones defined in this section in the case that the negation function satisfies Neg(x i ) =1 for all x i in L. In fact, we define here the measure of the interval attached to a linguistic label x i as a function of Neg(x i ), instead of being constant for all labels as it results in proposition 2.1 Definition 3.7. Let Neg be a negation function from L to (L) according to definition 3.5, we define P Neg as the set P={[m 0n,m 0x ],...,[m nn,m nx ]} where the measure of the interval corresponding to the label x i in L is defined as: m i = Neg(x i ) /? x L Neg(x) where X stands for the cardinality of the set X. This definition as I P I should equal [0,1] corresponds to sets I(x i ) for all x i in L of the form: I(x i )=[m in,m ix ]=[? x<xi Neg(x) /? x X Neg(x),? x=xi Neg(x) /? x X Neg(x) ]. Notice that P Neg satisfies 0=m 0n, 1=m nx and m ix =m i+1n for all i {0,1,...,n-1}.

10 Now we consider the negation function Neg and its corresponding interval according to definition 3.7 in relation with the usual negation function defined over the unit interval 27 N(x)=1-x. We study if the set of intervals P Neg is consistent in relation to N(x). Definition 3.8. Let Neg be a negation function from L to (L) according to definition 3.5, let P be a set of intervals attached to L. In these conditions the set P is consistent with N(x)=1-x if and only if: (i) for all x i L it is satisfied: for all x [m in,m ix ] holds 1-x [m i 0n,m inx ] where Neg(x i )={x i 0,...,x in }, and m i0n =inf I(x i0 ) and m inx =sup I(x in ) (ii) for all x i L such that Neg(x i ) =2, it is satisfied: there exists a value x [m in,m ix ] such that 1-x [m i 0n,m i0x ) there exists a value x [m in,m ix ] such that 1-x (m i nn,m inx ] where Neg(x i )={x i 0,...,x in }, and m i 0n =inf I(x i0 ), m i0x =sup I(x i0 ) and m i nn =inf I(x in ), m inx =sup I(x in ) Condition (i) states that the negation of all the elements of the interval attached to x i belong to the intervals attached to the negation of x i. This condition is equivalent to for all x I(x i ) N(x) I * (Neg(x i )) where I * is an extended interval defined as I * (X) = x X I(x). Condition (ii) establishes that if Neg(x i )={x i 0,...,x in }, then the label x i0 nor the x in are "superfluous" in relation to the negation function. This means that there exists at least an element of the interval attached to x i such that its negation belongs to the interval attached to x i 0, and respectively, to x i. These conditions can be equivalently be rewritten as: n there exists a value x I(x i ) such that N(x) I R (x i 0 ) there exists a value x I(x i ) such that N(x) I L (x i n ) In these conditions I R and I L mean that the interval is open in the upper limit and, respectively, in the lower limit.

11 Now we show that when the set of intervals P attached to L is the one defined in definition 3.7, then the set of intervals is consistent with the negation N(x)=1-x. Proposition 3.6. Let Neg be a negation function from L to (L) according to definition 3.5, and let P Neg be the set of intervals defined according to definition 3.7. In these conditions, the set of intervals P Neg is consistent with N(x)=1-x. This proposition is proven by means of the lemmas 3.1 and 3.2 given below. The first lemma corresponds to condition (i) in definition 3.8 and the second one corresponds to condition (ii) in the same definition. Lemma 3.1. Let Neg be a negation function from L to (L) according to definition 3.5, and let P Neg be the set of intervals attached to L defined according to definition 3.7. In these conditions the following relation is satisfied for all x i L, for all x [m in,m ix ] holds 1-x [m i 0n,m inx ] where Neg(x i )={x i 0,...,x in }, and m i0n =inf I(x i0 ) and m inx =sup I(x in ) Proof. The proof is based on the following inequalities that are equivalent to the condition given above. 1-m ix = m i 0n (3.1.1) 1-m in = m i nx (3.1.2) We proof now 3.1.1, the proof of condition is similar and is left to the reader. To prove we begin with an arbitrary x i in L, we substitute m ix and m i 0n by their definitions (def 3.7), we obtain in this way: 1-? x k=xi m k =? m xk<xi 0 k Replacing m k by their definitions, and 1 by the equivalent expression? x L Neg(x) /? x L Neg(x) we obtain:? x L Neg(x) /? x L Neg(x) -? x=x i Neg(x) /? x L Neg(x) =? x<x Neg(x) /? i 0 x L Neg(x) that is equivalent to:? x L Neg(x) -? x=x i Neg(x) =? x<xi 0 Neg(x)

12 as: Now, due to the fact that the set L-{x:x=x i } equals the set {x:x>x i } we rewrite this equation? x>x i Neg(x) =? x<xi 0 Neg(x) (3.1.3) Let us consider the term on the right hand side. The elements x that are counted, one or more times, in this term are the ones that appear in some of the Neg(x k ) for x k <x i0. Therefore, these elements x satisfy: x Neg(x 0 ) Neg(x 1 )... Neg(x i0-1 ) = xk<xi 0 Neg(x k ) Each element in this set can appear one or more times. Let n x be the number of times that the element x appear in the summation. As for each x k the element x can only be considered once, n x corresponds to: n x = {x k : x Neg(x k ), x k <x i0 } According to this, inequality (3.1.3) can be rewriten as Neg(x) {x k : x Neg(x k ), x k <x i0 } x>x i x xk <x i0 Neg(x k ) Now, we substitute Neg(x) by {x k :x Neg(x k )} as they are equivalent according to proposition 3.2. We get: {x k :x Neg(x k ) {x k : x Neg(x k ), x k <x i0 } x>x i x xk <x i0 Neg(x k ) According to proposition 3.3, if x k <x i0 and x Neg(x k ) then x=x i holds. In fact, it is not only true that x=x i but also x>x i. Notice that it is not possible x=x i because if it were possible, x=x i would imply Neg(x)=Neg(x i ) but as x Neg(x k ) then x k Neg(x)=Neg(x i ). However, as x k <x i0 and x i0 is the minimum of Neg(x i ) we get a contradiction because x k also belongs to Neg(x i ). Due to this, all the x x k<xi 0 Neg(x k ) in the right hand side satisfy also x>x i and they are considered in the summation on the left hand side. Besides of that, as the sets considered on the right hand side are more restrictives than those on the left hand side the equality is satisfied. The proof of condition is analogous considering proposition 3.4 instead of 3.3. Therefore, the lemma 3.1 is proven. o

13 Lemma 3.2. Let Neg be a negation function from L to (L) according to definition 3.5, let P Neg be the set of intervals attached to L defined according to definition 3.7. In these conditions it holds for all x i in L such that Neg(x i ) =2: there exists a value x [m in,m ix ] such that 1-x [m i 0n,m i0x ) there exists a value x [m in,m ix ] such that 1-x (m i nn,m inx ] where Neg(x i )={x i 0,...,x in }, and m i 0n =inf I(x i0 ), m i0x =sup I(x i0 ) and m i nn =inf I(x in ), m inx =sup I(x in ) given above. Proof. The proof is based on the following inequalities that are equivalent to the condition 1-m ix <m i 0x (3.2.1) 1-m in >m i nn (3.2.2) To prove these relations we begin with an arbitrary x i of L such that Neg(x i ) =2. We prove only condition 3.2.1, the proof of is analogous. We begin substituting in m ix and m i 0x by their definitions: 1-? x k=xi m k <? xk=xi 0 m k After replacing m k by their definition we apply some manipulations similar to the ones in the proof of lemma 3.1 we get? x>x i Neg(x) <? xk=xi 0 Neg(x k ) This expression, in a way similar to the manipulations in the proof of lemma 3.1 (according to proposition 3.2 and the analysis of the elements that are counted in the right hand side) can be rewritten as: {x k :x Neg(x k ) < Neg( x k ) = n x x>x i x k x i0 x xk Šx i0 Neg(x k ) where n x = {x k :x Neg(x k ), x k =x i 0 }. Therefore, this expression is equivalent to: {x k :x Neg(x k ) < {x k : x Neg(x k ), x k Šx i0 } x>x i x xk <x i0 Neg(x k ) (3.2.3) Now, as if x>x i and x Neg(x k ) then x k =x i (the proof of this condition resembles the proof 0 of proposition 3.3) we have that all the x in the left hand side are also considered in the right hand side.

14 According to this it is clear that the following relation holds: {x k :x Neg(x k ) Š {x k : x Neg(x k ), x k Šx i0 } x>x i x xk Šx i0 Neg(x k ) This inequality is the same of but for the equality. To proof that equality is not possible notice that as Neg(x i ) =2 and x i 0 =min Neg(x i ) then according to proposition 3.5 we have that x i =min Neg(x i 0 ). We take on the right hand side, x k =x i 0 and x=x i. We have that x xk=xi 0 {x i 0 :x i Neg(x i 0 ), x i0 =x i0 } =1 as we have just said that x i =min Neg(x i0 ). Neg(x k ) and On the other hand this element is not counted on the left hand side because x=x i do not satisfy x>x i. Therefore, the (3.2.3) is proven and also the lemma. We finish this section with two results of the negation function from L to (L) that show that when the cardinality of Neg(x i ) equals one for all x i in L, then the function reduces to the classical negation function defined in section 2 and that maximal neutrality is satisfied. Proposition 3.7. Let Neg be a negation function from L to (L) defined according to definition 3.5. In these conditions if Neg(x i ) =1 for all x i in L then it is forced that the negation function is of the form: Neg(x i )={x n-i } Proposition 3.8. Let Neg be a negation function from L to (L) defined according to definition 3.5. In these conditions if Neg(x i ) =1 for all x i in L, then m i =1/(n+1) IV. Conclusions In this work we have first shown the need of defining a semantics for linguistic labels from negation functions. We have studied then the classical negation functions from L to L and introduced a new class of negation functions from L to (L). We have defined a semantic in the unit interval of the set of linguistic labels by means of the negation function. We have seen that the semantics of L is consistent with the classical negation function N(x)=1-x in the unit interval. Finally we have shown

15 that the semantics of L according to negation functions from L to (L) reduces to the classical case when the cardinality of Neg(x i ) equals 1 for all x i in L. The semantics for linguistic labels introduced in this paper is used by the General Classification System Sedàs 28. Acknowledgments. The help of David Riaño and Dr. Lluís Godo is gratefully acknowledged. I would thank Aïda Valls that has implemented the Sedàs system. I would also thank the support of the European Community, under the contract VIM: CHRX-CT that allowed me to discuss this work with some European partners. Part of this work was completed at the Institut d'investigació en Intel.ligència Artificial (IIIA, CSIC). References 1. K. M. Ford, J. M. Bradshaw, Special Issue on "Knowledge Acquisition as Modeling", Parts I and II, International Journal of Intelligent Systems, 8:1-2 (1993). 2. W. J. Clancey, "The Knowledge Level Reinterpreted: Modeling Socio-Technical Systems", International Journal of Intelligent Systems, Vol. 8, pp In Ford, K.M., Bradshaw, J.M., B. R. Gaines, M. L. G. Shaw, "Knowledge Acquisition Tools Based on Personal Construct Psychology", The Knowledge Engineering Review, 8:1, 49-85, (1993). 4. J. H. Boose, Expertise Transfer for Expert System Design, Elsevier Science Publishers B.V., J. H. Boose, J. M. Bradshaw, "Expertise transfer and complex problems: using AQUINAS as a knowledge-acquisition workbench for knowledge-based systems", International Journal of Man-Machine Studies, 26, 3-28, (1987). 6. E. Plaza, C. Alsina, R. López de Mántaras, J. Aguilar, J. Agustí, "Consensus and knowledge acquisition", en R.R. Yager i B. Bouchon (Eds) Uncertainty Management in Knowledge-Based Systems, LNCS, Springer-Verlag, 1986.

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17 18. F. Herrera, E., Herrera-Viedma, J.L., Verdegay, "A Sequential Selection Process in Group Decision Making with a Linguistic Assessment Approach", Int. J. Information Science, 85, (1995). 19. F. Herrera, E. Herrera-Viedma, J.L. Verdegay, "Basis for a Consensus Modeling Group Decision Making with linguistic Preferences", Proceedings EUFIT'95 (Aachen, Germany, August 28-31), (1995). 20. R.R. Yager, Families of OWA operators, Fuzzy Sets and Systems, 59:2, (1993). 21. J. Kacprzyk, M. Fedrizzi, "'Soft' measures of consensus in the setting of partial (fuzzy) preferences", European Journal of Operational Research, Vol. 34, pp (1988). 22. D. Dubois, H. Prade, Fuzzy Sets and Systems: Theory and Applications, Academic press, New York, G. Mayor, J. Torrens, "On a Class of Operators for Expert Systems", Int. J. of Intelligent Systems, 8, (1993). 24. J. Agustí Cullel, F. Esteve, P. García, L. Godo, C. Sierra., "Combining Multiple-valued Logics in Modular Expert Systems", Proc. 7th Conference on Uncertainty in AI, Los Angeles, (1991) 25. A. Bardossy, L. Duckstein, I. Bogardi, "Combination of fuzzy numbers representing expert opinions", Fuzzy sets ad systems, 57, (1993) 26. D. Dubois, J-L. Koning, "Social choice axioms for fuzzy set aggregation", Fuzzy sets and systems, 43, (1991). 27. E. Trillas, "Sobre funciones de negación en la teoria de conjuntos difusos", Stochastica, Vol. 3, pp (1979). 28. A. Valls, Sedàs. General Classification Module for ill structured domains, (in catalan), Master Thesis, ETSE, Universitat Rovira i Virgili, (1995).

18 FIGURES: AN VL QL Low Medium Hi gh VH Figure 1... x... i m in m im m Figure 2 i m ix