Connotation-differential Prints

Comparing What Is Connoted Through (Fuzzy) Evaluations

Marcelo Loor

1,2

and Guy De Tr´e

Dept. of Telecommunications and Information Processing,Ghent University,

Sint-Pietersnieuwstraat 41 B - 9000, Ghent, Belgium

Dept. of Electrical and Computer Engineering, ESPOL University,

Campus Gustavo Galindo V., Km. 30.5 Via Perimetral, Guayaquil, Ecuador

Keywords:

Meaningful Comparison, Semantic Richer Similarity, Qualitative Comparison.

Abstract:

To evaluate the level to which an object belongs (or not) to a particular set, say A, one could focus on some

object’s features according to what one understands by A. With this consideration, using the evaluations of a

group of objects given by two persons, we want to determine the level to which their individual understandings

of A match. Therefore, hypothesizing that a difference in understandings (or connotations) of A could be

marked by a difference in one or more of the evaluations, we propose a connotation-differential print (CDP).

A CDP is a representation of any difference in connotations of A between two persons in a form that makes

itself available to computation. Additionally, we study how to use a CDP to extend a similarity measure for

intuitionistic fuzzy sets in order to reach a meaningful comparison between two of them.

1 INTRODUCTION

Imagine the following situation: three cousins, Alice,

Bob and Chloe, are evaluating individually to which

degree a cookie could be seen or not as a Grandma’s

cookie. Each cousin has a memory of how looks a

Grandma’s cookie (see Figure 1), which is used as a

referent to evaluate all the cookies depicted in Fig-

ure 2. Using a unit interval scale where 1 represents

the highest level and 0 the lowest, the cousins have

given their evaluations as shown in Table 1. It can be

seen from the data in this table that, in some evalu-

ations, adding both a ‘yes’-value and a ‘no’-value is

not necessarily equal to 1. Recording evaluations in

this fashion allows the cousins to express any hesita-

tion about their judgments as will be explained in Sec-

tion 4.2. Within this situation, in this paper we study

the following problem: using the evaluations given by

two cousins, how can we determine the level to which

the referent cookies used by them match? This kind

of problem is of particular relevance to similarity-

related processes where, although two (or more) per-

sons understand the meaning of a request (e.g., is this

cookie like a Grandma’s cookie?), they may have dif-

ferent understandings of such (e.g., Alice vs. Bob’s

connotations of a Grandma’s cookie depicted in Fig-

ure 1) —as will be shown in Section 2, Zadeh (Zadeh,

2013) highlights some ideas about truth and meaning

that are related somehow to this problem. Hereafter

we assume that it is not feasible (nor practical) for

two (or more) persons clarifying their individual un-

derstandings of a request. Thus, e.g., if a compari-

son between evaluations given by two cousins is per-

formed, it is possible that they “match” although the

cousins have different understandings (or memories)

of a Grandma’s cookie. Here, our aim is to detect this

kind of pseudo-matching in order to achieve more re-

liable results in such comparisons.

Straightforwardly, someone may consider that

measuring the difference between the evaluation sets

given individually by two cousins is a direct measure

of the difference between the referent cookies used by

them. However, this consideration assumes implicitly

that if Alice’s referent cookie matches with a particu-

lar cookie (which is denoted by an Alice’s evaluation)

and this particular cookie matches with Chloe’s ref-

erent cookie (which is denoted by a Chloe’s evalua-

tion), then Alice’s referent cookie should match with

Chloe’s, i.e., the similarity between cookies is seen

as a transitive relation, and also, as a symmetric rela-

tion due to it is also assumed that if Alice’s referent

matches Chloe’s, then Chloe’s referent should match

Alice’s. As will shown in Section 3, following the

psychological view of similarity presented by Tver-

127

Loor M. and De Tré G..

Connotation-differential Prints - Comparing What Is Connoted Through (Fuzzy) Evaluations.

DOI: 10.5220/0005079101270136

In Proceedings of the International Conference on Fuzzy Computation Theory and Applications (FCTA-2014), pages 127-136

ISBN: 978-989-758-053-6

 2014 SCITEPRESS (Science and Technology Publications, Lda.)

(a) cookie a (Alice). (b) cookie b (Bob). (c) cookie c (Chloe).

Figure 1: How looks a Grandma’s cookie according to each relative (Grandma’s cookie example).

(a) cookie 1. (b) cookie 2. (c) cookie 3. (d) cookie 4.

Figure 2: Do these cookies look like a Grandma’s cookie? (Grandma’s cookie example).

sky (Tversky, 1977) as less stringent but more prac-

tical, we consider in this paper that those assump-

tions should be avoided because they could not reﬂect

the human behavior in comparison judgments. Deal-

ing with such comparison judgments is the theoretical

motivation for the paper.

To detect any difference between the referent

cookies used by two cousins, we propose using the

differences resulting from a comparison of their cor-

responding evaluation sets in order to obtain a kind of

footprint that hints how different (or similar) the ref-

erent cookies are; we call this footprint a connotation-

differential print (CDP). This is based on the follow-

ing observation. In the abstraction process carried out

to give her evaluations, Alice could pay more atten-

tion than Chloe to some cookie’s features according

to what she remembers as a Grandma’s cookie, caus-

ing a difference in their individual referent objects,

which are individual connotations of a Grandma’s

cookie. We hypothesize that a difference in connota-

tions could be marked by a difference in one or more

evaluations. For instance, studying Table 1, it is pos-

sible to think that the similarity between Alice and

Chloe’s connotations (i.e., Alice vs. Chloe) is larger

than the similarity between Alice and Bob’s conno-

tations (i.e., Alice vs. Bob). Indeed, the evaluations

given by Alice and Chloe for cookie 1, cookie 2 and

cookie 3 match perfectly; however, the big difference

in evaluations given to cookie 4 by Alice and Chloe

suggests a difference in their individual connotations,

in fact, looking into the memories of the cousins de-

picted in Figure 2, Alice vs. Bob seems more similar

than Alice vs. Chloe. This observation is the practical

motivation for the paper.

There are two interesting things about a CDP: 1) it

is a representation that could be used to determine if

any difference between two evaluation sets is caused

by a difference in the magnitude of the evaluations,

or by a difference in the connotation of the referent

objects; and 2) it makes itself available to computa-

tion. Thus, e.g., if an intuitionistic fuzzy set (IFS)

is used to model the evaluation sets, it is possible to

use a CDP to extend a similarity measure for IFS in

order to perform a meaningful similarity comparison

between two of them (see Section 6.2). As will be

shown in Section 7, this contribution could help to

overcome an anomaly that happens in some similarity

measures for IFSs. Also, the two characteristics could

be useful in the following situation. Imagine that Al-

ice needs some help to evaluate additional cookies.

Which cousin should be chosen to help her? Maybe

the cousin whose connotation of a Grandma’s cookie

is more similar to Alice’s connotation could perform

a better evaluation than the cousin with whom Alice

agrees exactly on many of the givenvalues. This gives

us an idea of a potential application.

Picture yourself as an expert in rating of scripts to

be used in language courses for children between 7

and 9 years. Suppose that you have received a huge

collection of scripts from several sitcoms to be rated.

You have decided to ask the online community for

contributions to rate all the scripts in such a huge col-

lection —i.e., you have decided to use a crowdsourc-

ing model. It concerns you that in this kind of model

you know nothing about the people who perform the

job (which consist in rating one or more scripts), as

well as the different understandings that they may

have about it. You know that in a crowd-sourced con-

text a contributor’sanswer could be, among others, af-

fected by personal views, experience or background.

Thus, you have written down a request explaining as

clearly as possible how each script should be rated,

considering that it is not feasible to clarify individu-

ally any doubt about the request. Also, you have con-

sidered as necessary to use a ﬂexible scale (which is

similar to that used in Grandma’s cookie example) in

order to allow the contributors express any hesitation

in their answers. Moreover, you have decided that

the contributors must qualify for the job, so they must

perform an analysis of some scripts that you have al-

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Table 1: To which degree each cookie in Figure 2 is seen or not as a Grandma’s cookie by each cousin (Grandma’s cookie

example).

yes no

cookie 1 0.6 0.3

cookie 2 0.7 0.3

cookie 3 0.2 0.8

cookie 4 0.9 0.1

yes no

cookie 1 0.4 0.3

cookie 2 0.5 0.3

cookie 3 0 0.9

cookie 4 0.7 0.2

yes no

cookie 1 0.6 0.3

cookie 2 0.7 0.3

cookie 3 0.2 0.8

cookie 4 0.1 0.9

(a) Alice. (b) Bob. (c) Chloe.

ready analyzed. It is quite important for you being

as sure as possible that the contributors perform an

analysis similar to yours. If you can ﬁnd anyone who

performs an analysis connoting what you connote in

yours, you could trust his or her jobs more than oth-

ers. As you may have already noticed, our proposal

could help you to choose your right contributors and

to assess the quality of the jobs that they will perform.

To describe how our proposal provides an answer

to the Grandma’s cookie example, this paper is struc-

tured as follows. Section 2 presents some related

works. Section 3 shows why the paper considers the

psychological view of similarity presented in (Tver-

sky, 1977). Section 4 shows how the intuitionistic

fuzzy set (IFS) concept (Atanassov, 1986; Atanassov,

2012) could be used to model each of the evaluation

sets given by the cousins. Section 5 uses the spot-

difference concept (Loor and De Tr´e, 2013) to ﬁgure

out the difference in evaluations given for each cookie

by two cousins. Section 6 introduces and explains the

novel concept of connotation-differential print (CDP)

and shows how it could be used to extend a similarity

measure for IFS in order to perform meaningful com-

parisons. Section 7 compares our meaningful simi-

larity measure with others. We conclude the paper in

Section 8.

2 RELATED WORKS AND

DISCUSSIONS

About detecting any difference between two referents

used by two persons, who are evaluating to which de-

gree an object could be considered to be similar to

such a referent, we found in (Zadeh, 2013) some ideas

of Zadeh about truth and meaning that are related

somehow to ours. First, we agree that it is necessary

for two persons, P

and P

to understand the meaning

of a proposition p to assess its truth value —e.g., Al-

ice and Bob should understand what is a Grandma’s

cookie to evaluate the level to which cookie i is seen

as such. Second, we agree that understanding the

meaning of p is not sufﬁcient for P

and P

to de-

termine to which level their individual assessments of

p match —e.g., although both Alice and Bob under-

stand individually what a Grandma’s cookie is, this

is not enough to determine if both cousins have the

same connotation of such a cookie. Finally, we agree

that what is needed is a representation of any differ-

ence about the meaning of p between P

and P

in a

form that lends itself to computation —this is why we

propose a connotation-differential print.

About the semantic interpretation of the elements

of an IFS (see Section 4), we found in the theoreti-

cal model proposed by Ekman in (Ekman, 1963) —

which is used to compare two perceptions from an

observer— some analogies that ﬁt with those used in

our vector based interpretation of the membership and

non-membership of such elements. In his model, Ek-

man considers that the perceptual intensity is depicted

by the magnitude of a vector, and the perceptual qual-

ity, by the vector’s direction. Analogically, the per-

ceptual intensity corresponds to the degree to which

one element, say x

, belongs or not to an IFS A, i.e.,

) and ν

) respectively; while the perceptual

quality corresponds to the connotation of belonging

or not to A.

3 CHOOSING ONE SIMILARITY

APPROACH

The aim of this section is to show why the paper con-

siders the psychological view of similarity presented

by Tversky in (Tversky, 1977) in order to measure

any difference between the referent objects used by

two cousins in their evaluations.

3.1 Similarity According to Tversky

In (Tversky, 1977), Tversky provides empirical evi-

dence for asymmetric similarities and argues that sim-

ilarity should not be treated as a symmetric relation.

Representing an object in terms of its qualitative fea-

tures, he considers that the assessment of similarity

may be better described as a comparison of features

rather than as the computation of a metric distance

between points —a metric distance function, d, is de-

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ﬁned as a scale that assigns to every pair of points,

(x,y), a nonnegative number in accord with the fol-

lowing three axioms: d(x,y) ≥ d(x,x) (minimality);

d(x,y) = d(y,x) (symmetry); d(x, y) + (y,z) ≥ d(x,z)

(the triangle inequality). He sustains that similarity

judgments can be regarded as extensions of similarity

statements, i.e., statements of the form “x is like y”.

Such a statement is directional —x is the subject, and

y is the referent— and, in general, it is not equivalent

to the converse similarity statement “y is like x”.

In one of his experiments, called similarity of

countries, a group of people was asked to choose

which of two phrases they preferred to use: ”coun-

try A is similar to country B,” or ”country B is simi-

lar to country A”. Tversky observed that most of the

people chose the phrase in which the more prominent

country served as the referent —e.g., most of the peo-

ple chose “North Korea is similar to Red China” in-

stead “Red China is similar to Korea.” According to

him, those results demonstrate the presence of marked

asymmetries in the choice of similarity statements.

Using what he observed, Tversky presents the follow-

ing example that casts some doubts on the psycholog-

ical validity of transitivity in similarity relations: if

“Jamaica is similar to Cuba” and “Cuba is similar to

Russia” then the phrase “Jamaica is similar to Rus-

sia” should be possible, but Jamaica and Russia are

not similar at all. He pointed out that in “Jamaica is

similar to Cuba” has been relevant the geographical

proximity, while in “Cuba is similar to Russia” has

been relevant their political afﬁnity. This human be-

havior in which each person considers as relevant any

particular feature in order to provide an answer, is a

key component of his approach.

As seen in Section 1, there are situations where

it is not feasible (nor practical) to clarify individually

all the constraints in a request. Thus, even though two

or more persons understand the meaning of a request,

they may have different understandings of it accord-

ing to what they consider as relevant. Due to this kind

of human behavior was considered by Tversky in his

approach of similarity, we take it into account in the

paper.

4 MODELING (FUZZY)

EVALUATIONS

For the purpose of modeling the evaluations given by

each cousin, this section introduces the intuitionistic

fuzzy set (IFS) concept and uses a semantic interpre-

tation of its components in order to model the compo-

nents of the Grandma’s cookie example.

4.1 IFS Concept

As an extension of a fuzzy set (Zadeh, 1965), an IFS

∗

in E (Atanassov, 1986; Atanassov, 2012) is deﬁned

as a collection such that

∗

= {hx

,µ

),ν

)i|x

∈ E} (1)

where sets E and A are considered to be ﬁxed, A ⊂ E,

functions µ

: E → [0,1] and ν

: E → [0,1] deﬁne

the degree of membership and the degree of non-

membership of x

∈ E to the set A respectively, and

for each element x

∈ E

0 ≤ µ

) + ν

) ≤ 1. (2)

The lack of knowledge about the membership (or

non-membership) of element x

∈ E to set A is ex-

pressed by

) = 1− µ

) − ν

) (3)

and it is deﬁned as the degree of non-determinacy

—also known as hesitation margin (Szmidt and

Kacprzyk, 2013).

4.2 A Semantic Interpretation of

Components of an IFS

Using a semantic interpretation of IFS’s components,

we model the components of the Grandma’s cookie

example as follows. The collection of all cookies de-

picted in Figure 2 corresponds to the set E, where

each cookie is denoted by x

and i = 1, 2,3,4. A group

of cookies considered by Alice to be a Grandma’s

cookie corresponds to the set A (i.e., A ⊂ E). The

degree to which cookie i is considered by Alice to

be a Grandma’s cookie is represented by µ

) —

here, µ

) is interpreted as a degree of similar-

ity between x

and r

, where r

is what Alice re-

members as a Grandma’s cookie (cf. the seman-

tics of the membership grades treated in (Dubois

et al., 2000)). The degree to which cookie i is con-

sidered by Alice not to be a Grandma’s cookie is

represented by ν

). The degree to which Alice

hesitates if cookie i is considered or not to be a

Grandma’s cookie is represented by π

). In this

way, the evaluations given by Alice correspond to the

IFS A

∗

= {hx

,µ

),ν

)i|x

∈ E}. Likewise, the

group of cookies considered by Bob and Chloe to be

a Grandma’s cookie correspond to the sets B and C

respectively (i.e., B ⊂ E andC ⊂ E), and their evalua-

tions, to the IFSs B

∗

= {hx

,µ

),ν

)i|x

∈ E} for

Bob and C

∗

= {hx

,µ

),ν

)i|x

∈ E} for Chloe.

It is important to note that, although the deﬁnition

shows the difference between the IFS A

∗

and the set

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A, for simplicity (Atanassov, 1986; Atanassov, 2012)

the expression (1) will be denoted by

A = { hx

,µ

),ν

)i|x

∈ E}, (4)

in the remainder of the paper. Hence, the evaluation

sets will be denoted by A = {hx

,µ

),ν

)i|x

∈

E} for Alice, B = {hx

,µ

),ν

)i|x

∈ E} for

Bob, and C = {hx

,µ

),ν

)i|x

∈ E} for Chloe.

5 COMPARING EVALUATIONS

VS. COMPARING WHAT THEY

COULD MEAN

We have already mentioned in the theoretical and

practical motivation for the paper (see Section 1) that

measuring the similarity of the evaluation sets given

individually by two cousins is not a good option to

measure the similarity of the referent cookies because

of the implicit assumption about transitivity and sym-

metry in such a comparison. Therefore, although we

have modeled the evaluation sets as IFSs, in this paper

we should avoid using any of the similarity measures

for IFSs that assume the similarity to be a dual no-

tion of a metric distance (cf. the similarity measures

presented in Section 7.1).

Considering what Tversky proposed in (Tversky,

1977) about directionality, asymmetry and not tran-

sitivity in comparison judgments, this section intro-

duces some concepts presented in (Loor and De Tr´e,

2013) in order to know not just the magnitude, but

also the meaning (or sense) behind a comparison be-

tween two IFSs.

5.1 Spot-difference Concept

As seen in Section 4.2, from a semantic point of view,

) and ν

) represent the extent (or magnitude)

to which cookie i is considered by Alice respectively

to be a Grandma’s cookie, and not to be so. This

reﬂects that µ

) and ν

) have different mean-

ings (or directions), therefore, such an evaluation of

cookie i given by Alice could be interpreted as a vec-

tor



) + α

· π

)

) + (1− α

) · π

)



, (5)

where α

∈ [0,1] is considered to be a hesitation split-

ter from Alice (Loor and De Tr´e, 2013) –hereby, the

hesitation splitter α

allows it to split any Alice’s hes-

itation about seen cookie i as a Grandma’s cookie be-

tween µ

) and ν

To compare the evaluations for cookie i given by

Alice and Bob, which are interpreted respectively as

vectors a

and b

, the area of the parallelogram formed

by both vectors could be used as a reference to mea-

sure the difference between them. Thus, the larger

this area, the larger the difference between a

and b

Within this approach, the largest area is given by the

vectors m





and n





Deﬁnition 1 (Spot-difference). Consider an element

∈ E. Let a

be a vector representing the membership

and non-membership of x

to the IFS A and b

a vector

representing the membership and non-membership of

to the IFS B. A measure of their differences is

known as spot-difference and is deﬁned by

dif(a

) =

× b

× n

, (6)

where × denotes the vector product (Loor and De Tr

2013).

An expression obtained from (6) is

dif(a

) = (µ

) − µ

))

+ (α

· π

) − α

· π

)), (7)

which could be semantically interpreted as follows.

The ﬁrst part of the expression, (µ

) − µ

)), de-

notes that the difference between Alice and Bob’s

evaluations of cookie i is determined partly by the

difference in levels to which cookie i is considered

by them to be a Grandma’s cookie. The second part,

(α

· π

) − α

· π

)), denotes that the difference

is also inﬂuenced by any doubt about considering

cookie i to be a Grandma’s cookie, and moreover,

this part could be affected by managing both Alice

(α

) and Bob (α

)’s hesitation splitters —a manag-

ing strategy could be applying the same rule for Alice

and Bob, thus, α

= α

= α, in which case (7) could

be expressed as

dif

) = (µ

) − µ

))

+ α(π

) − π

)). (8)

The (+/−) sign in a spot-difference result seman-

tically denotes the relative difference between Alice

and Bob’s evaluations (cf. Tversky’s consideration

about directionality for comparison judgments (Tver-

sky, 1977)). For example, dif

) = +0.2 means

that Alice’s evaluation considers cookie 1 to be a

Grandma’s cookie 0.2 more than Bob’s evaluation; on

the other hand, dif

) = −0.2 means that Bob’s

evaluation considers cookie 1 to be a Grandma’s

cookie 0.2 less than Alice’s evaluation.

5.2 Spot-differences Footprint and

Similarity

A visual representation of the relative notion of dif-

ference given by the spot-difference concept allows it

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131

to observe the internal composition of the difference

between two evaluation sets (i.e., two IFSs) given in-

dividually by two cousins. For instance, consider us-

ing (8) with α = 0. Figure 3(a) shows the differ-

ence between Alice’s evaluations and Bob’s evalua-

tions for each cookie, as seen from Alice’s point of

view. Figure 3(b) does so, comparing Chloe’s evalu-

ations. In a similar way, Figure 3(c) and Figure 3(d)

show the differences between Chloe’s evaluations and

respectively Alice’s and Bob’s evaluations, as seen

from Chloe’s point of view. In these ﬁgures, each

spot-difference is represented by a ruler of height

one marked of with “difference”-units. The black

region denotes the magnitude of the corresponding

spot-difference, and the position of the black region,

above or below the line that represents no-difference,

denotes its relative difference. This kind of represen-

tation is called a spot-differences footprint (Loor and

De Tr´e, 2013).

A spot-differences footprint allows it to obtain a

measure for the similarity of two IFSs following the

Tversky’s ratio model (Tversky, 1977). Hereby, the

similarity is expressed as a proportion between the

common and the distinctive features in a normalized

form. The measure presented in (Loor and De Tr´e,

2013) is deﬁned by

(A,B) = 1−

∑

i=1

|dif

)|, (9)

and denotes the degree of similarity S

(A,B) between

IFSs A and B. As it could be noted, this expression by

itself just represents the magnitude of the similarity

between two IFSs and, by using it, it is not possible to

conclude about the correspondence in appearance of

the two IFSs. For instance, using this expression, the

similarity between Alice and Bob is S

(A,B) = 0.8,

as well as, the similarity between Alice and Chloe

is S

(A,C) = 0.8, i.e., although they differ in their

spot-differences footprint (see Figure 3), both Alice

vs. Bob and Alice vs. Chloe have the same value as a

measure of similarity.

To distinguish between the “0.8-uniform” simi-

larity in Alice vs. Bob and the “0.8-with-a-peak”

similarity in Alice vs. Chloe, it has been proposed

in (Loor and De Tr´e, 2013) to make use of the cor-

responding spot-differences footprint to extend (9)

in order to capture the semantic meaning (or sense)

within comparisons. This is the motivation for a se-

mantic footprint that is proposed in the next Section.

6 CONNOTATION-

DIFFERENTIAL

At this point, the question is how could we use a spot-

differences footprint to compare what is connoted

through evaluations given by two cousins? In (Tver-

sky, 1977), using a set-theoretical approach, Tver-

sky described the similarity between two objects o

and o

, which are represented as a collection of fea-

tures O

and O

respectively, as a feature-matching

process. Considering this approach, each cookie in

Grandma’s cookie example could be treated as an

object with features such as a square shape, linear

icing, or with a square hole. Thus, in the abstrac-

tion process carried out to give her evaluation, Alice

could pay more attention than Bob or Chloe to some

cookie’s features according to what she remembers as

a Grandma’s cookie, causing a difference in their in-

dividual connotations (of a Grandma’s cookie). For

instance, let us take a look into each cousin’s mem-

ory of a Grandma’s cookie depicted in Figure 1: Al-

ice’s memory is a square cookie with linear icing and

a square hole; Bob’s memory is a square-round cookie

with linear icing and a square hole; and Chloe’s mem-

ory is a round cookie with no icing and a round hole.

When Alice evaluated to which degree cookie 1 (see

Figure 2(a)) could be seen as a Grandma’s cookie,

she judged it as 0.6 for ‘yes’ and 0.3 for ‘no’ (see

Table 1) —it seems that she paid more attention to

the square shape of the cookie (which is a common

feature between cookie 1 and the cookie in her mem-

ory) and not to the missing linear icing nor the miss-

ing square hole (cf. Tversky’s example in (Tver-

sky, 1977)). When Chloe did so, she also judged

it as 0.6 for ‘yes’ and 0.3 for ‘no’ (see Table 1) —

it seems that she paid more attention to the missing

icing and not to the square shape nor the missing

round hole. Looking at the spot-differences footprint

resulting from Alice-vs.-Chloe comparison (see Fig-

ure 3(b)), Alice could hint that, althoughthey agree on

evaluations for cookie 1, cookie 2 and cookie 3, Chloe

paid less attention to the square shape or square hole

of cookie 4, i.e., Alice could hint that her connotation

of a Grandma’s cookie differs from Chloe’s. This il-

lustrates how spot-differences footprints can help to

compare what is connoted through evaluations.

6.1 Connotation-differential Marker

Following the aforementioned idea, a way to de-

tect the difference between Alice’s connotation of a

Grandma’s cookie in comparison to Bob’s or Chloe’s

is by shifting the focus onto some cookies having fea-

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0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

−0.1

−0.2

−0.3

−0.4

−0.5

−0.6

−0.7

−0.8

−0.9

−1.0

(a) Alice vs. Bob.

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

−0.1

−0.2

−0.3

−0.4

−0.5

−0.6

−0.7

−0.8

−0.9

−1.0

(b) Alice vs. Chloe.

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

−0.1

−0.2

−0.3

−0.4

−0.5

−0.6

−0.7

−0.8

−0.9

−1.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

−0.1

−0.2

−0.3

−0.4

−0.5

−0.6

−0.7

−0.8

−0.9

−1.0

(d) Chloe vs. Bob.

Figure 3: Comparing spot-differences footprints.

tures that she considered to be (or not) representative

in a Grandma’s cookie. For example, Alice could

focus on cookie 4 (see Figure 2(d)) because it has

the same square shape, as well as the same square

hole, as her memory of a Grandma’s cookie (see Fig-

ure 1(a)), and cookie 3 (see Figure 2(c)) because it has

neither the square shape nor the square hole remem-

bered by her —following these assumptions, cookie 4

and cookie 3 have respectively the best and the worst

of Alice’s evaluations (see Table 1). Focusing to one

of these “representative” cookies, we propose to use

a marker that denotes how a spot-difference related

to this cookie should be managed, i.e., if a spot-

difference should be treated as matching on evalua-

tions or not. Recalling the semantic interpretation in

a spot-difference result, such a marker should reﬂect

the relative difference between two evaluations, there-

fore, we deﬁne it as follows.

Deﬁnition 2 (Connotation-Differential Marker).

Consider an element x

∈ E. Let a

be a vector

representing the membership and non-membership

of x

to the IFS A, b

a vector representing the mem-

bership and non-membership of x

to the IFS B, and

dif

) a spot-difference between a

and b

. Now

consider a set S = {⋄|,

⋄

|}. A connotation-differential

marker (CDM) is a symbol s ∈ S that denotes the

relative amount of difference between a

and b

according to the following conditions:

• if |dif

)| ≤ δ then s = ⋄| ,

• if dif

) > δ then s =

⋄

| ,

• if dif

) < δ then s =

⋄

| ,

where δ ∈ [0, 1].

In this way, for instance, from Alice’s point of view,

with δ = 0.2, the spot-difference for cookie 4 between

her and Bob, di f

) = 0.2, could be treated as a

matching on evaluations and denoted by ⋄|. By con-

trast, the spot-difference for cookie 4 between Alice

and Chloe, dif

) = 0.8, could be treated as a

difference in evaluations where Alice considers that

cookie 4 is more similar to a Grandma’s cookie and,

as seen from her point of view, it is denoted by

⋄

6.2 A Meaningful Comparison

Knowing what a CDM could denote individually, we

could put one or more CDMs together in order to ob-

tain a representation that hints if a cousin has paid at-

tention or not to the same cookie’s features that have

been focused by another during the evaluation pro-

cess. A way to do that is by placing one or more

CDMs in a sequence with a particular order. For ex-

ample, to perform the comparisons from Alice’s point

of view, i.e., considering as referent what Alice re-

members as a Grandma’s cookie, we could build se-

quences with two CDMs: the ﬁrst one related to her

best evaluated cookie, i.e., cookie 4 (see Table 1),

and the second one, to her worst evaluated cookie,

i.e., cookie 3; thus, with δ = 0.2, a sequence corre-

sponding to (the comparison) Alice vs. Bob (see Fig-

ure 3(a)) is “⋄|⋄|”, and one corresponding to Alice vs.

Chloe (see Figure 3(b)) is “

⋄

|⋄|”. Despite using only

two CDMs, looking at Alice-vs.-Bob sequence, Alice

can now distinguish that Bob seems to agree with her

about cookie 4 having one or more features to con-

sider it to be a Grandma’s cookie, as well as, cookie 3

having one or more features to consider it not to be

so. On the other hand, looking at the ﬁrst CDM in

Alice-vs.-Chloe sequence, Alice could become aware

that some features of cookie 4 make Chloe to consider

it not to be a Grandma’s cookie and, thus, Alice could

hint that she and Chloe have used different connota-

tions (of a Grandma’s cookie) as referents. Now, let us

perform the comparison from Chloe’s point of view,

i.e., considering as referent what Chloe remembers as

Grandma’s cookie. Using the same strategy to build

Connotation-differentialPrints-ComparingWhatIsConnotedThrough(Fuzzy)Evaluations

133

a sequence, the ﬁrst CDM is related to the Chloe’s

best evaluated cookie, i.e. cookie 2 (see Table 1), and

the second to her worst’s, i.e., cookie 4. Thus, a se-

quence corresponding to Chloe-vs.-Bob comparison

(see Figure 3(c)) is “⋄|

⋄

|”, and a sequence correspond-

ing to Chloe-vs.-Alice comparison (see Figure 3(d))

is “⋄|

⋄

|”. Looking at these sequences, Chloe could hint

that neither Bob nor Alice remember a Grandma’s

cookie with no icing as she does. Due to these se-

quences allow Alice or Chloe to hint about a differ-

ence in connotations of a Grandma’s cookie, we call

any of them a connotation-differential print (CDP).

As it could be noticed above, a CDP depends on

the individual point of view of each cousin. In fact,

Alice has chosen the CDMs corresponding to cookie 4

and cookie 3, while Chloe has chosen cookie 2 and

cookie 4. This is an example of directionality and

asymmetry in comparison judgments pointed out by

Tversky in (Tversky, 1977). Furthermore, Alice could

assign a weight to each CDP in order to determine

which cousin’s referent is more similar to hers. For

example, according to her strategy to build a CDP, ⋄|⋄|

denotes a good match, thus, she assigns 1.0 to it.

⋄

and

⋄

| denote a not too bad match (they could become

⋄|⋄| by increasing δ) therefore, she gives 0.75 to them.

⋄

|⋄|, ⋄|

⋄

|⋄| and ⋄|

⋄

| denote a big difference, so, she gives

0.25 to them. Finally,

⋄

| and

⋄

| denote a huge differ-

ence, so, she assigns 0 to them.

At this point, we could use a CDP to extend (9) to

perform a meaningful similarity comparison between

two IFSs and, therefore, to achieve better ﬁne-tuned

and more reliable results. As an analogy, if (9) tells

us about how far is A from B, the use of a CDP tells

us in which “direction” B is in relation to A. With

this analogy, Figure 4 shows the similarity between

evaluations given by Alice and Bob, S

(A,B) = 0.8,

and so between Alice and Chloe’s, S

(A,C) = 0.8. In

Figure 4(a), using just the magnitude, there is no dif-

ference between both similarities; in contrast, in Fig-

ure 4(b), using the magnitude plus a CDP, it is note-

worthy how the “direction” of Alice vs. Bob differs

from Alice vs. Chloe’s. Hence, from Alice’s point of

view, the extended similarity between her evaluations

and Bob’s is h0.8, ⋄|⋄|i, and the corresponding one in

comparison to Chloe’s evaluations is h0.8,

⋄

|⋄|i. More-

over, using the weights that Alice proposed earlier, we

could say that Alice-vs.-Bob’s h0.8, ⋄|⋄|i is 0.8·1= 0.8,

and Alice-vs.-Chloe’s h0.8,

⋄

|⋄|i is 0.8·0.25= 0.2. This

reﬂects that, considering the individual connotation of

a Grandma’s cookie, the similarity between Alice and

Bob’s evaluations is better.

Using the previousanalogy, it is also possible to il-

lustrate how using a CDP as supplement could denote

in a better way an observed similarity. In the example

0.8

(a)

0.8

(b)

Figure 4: Alice vs. Bob and Alice vs. Chloe similarities.

presented by Tversky in (Tversky, 1977) it is stated

that: “considering the similarity between countries,

Jamaica is similar to Cuba (because of geographical

proximity), Cuba is similar to Russia (because of their

political afﬁnity), but Jamaica and Russia are not sim-

ilar at all”, one may notice that both comparisons,

Jamaica vs. Cuba and Cuba vs. Russia, have simi-

lar magnitudes in their corresponding similarity mea-

sures, but different connotations: the ﬁrst one focus-

ing on a “geographical proximity”-feature, and the

second one, on a “political afﬁnity”-feature. Using

such a representation of the difference in connota-

tions, one may observe the reason why Jamaica and

Russia are not similar at all. Maybe, someone might

argue here that the same properties should be used

in both comparisons. However, as was already men-

tioned in Section 1, it is not feasible (nor practical) in

some contexts (e.g., when a crowdsourcing model is

used) clarifying all the properties that an object may

have. Indeed, this is why (fuzzy) human evaluations

are needed in those contexts.

7 AN ADVANTAGE

This section compares our meaningful similarity mea-

sure with others in order to show how it could over-

come some difﬁculties such as those pointed out in

(Szmidt and Kacprzyk, 2013).

7.1 Some Difﬁculties in Geometric

Similarity Measures for IFSs

Using a geometric interpretation of the three terms

in an IFS-element (i.e., the membership, non-

membership and hesitation margin), the similarity be-

tween two IFSs is usually assumed to be a dual notion

of a metric distance (Szmidt and Kacprzyk, 2013).

Thus, given two IFSs A,B ∈ X and a normalized met-

ric distance function l : X

7→ [0,1], the similarity S

between A and B is expressed as S(A, B) = 1−l(A,B),

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134

where l follows the axioms of minimality, symme-

try and the triangle inequality (see Section 3.1). In

(Szmidt and Kacprzyk, 2013), Szmidt and Kacprzyk

have examined the effects of this assumption. They

found some difﬁculties that, according to them, are

a result of, ﬁrst, the symmetry of the three terms,

and second, of the important role played by these

terms in the deﬁnition of the complement of IFSs,

which should be considered in such similarity mea-

sures. Among others, they have studied the following

similarity measures:

(A,B) = 1−

∑

i=1

(|µ

) − µ

+ |ν

) − ν

+ |π

) − π

)|) (10)

and

H2D

(A,B) = 1−

∑

i=1

(|µ

) − µ

+ |ν

) − ν

)|) (11)

based on Hamming distance (Szmidt and Kacprzyk,

2000);

(A,B) = 1−

∑

i=1



(µ

) − µ

))

+ (ν

) − ν

))

+ (π

) − π

))





(12)

and

e2D

(A,B) = 1−

∑

i=1



(µ

) − µ

))

+ (ν

) − ν

))





(13)

based on Euclidean distance (Szmidt and Kacprzyk,

2000); and, the cosine similarity measure

mult

(A,B) =

∑

i=1

(µ

)µ

) + ν

)ν

)

+ π

)π

))/



)

+ ν

)

+ π

)





)

+ ν

)

+ π

)



. (14)

One of the difﬁculties is exempliﬁed as follows.

Consider X = {x

} and the IFSs M = {hx

,1,0i},

N = {hx

,0,1i} and H = {hx

,0,0i}. Also consider

the similarity measure S. If S is (10), (12) or (14), it is

obtained that S(M, N) = 0 and S(M, H) = 0 though

N and H are different. This anomaly is general-

ized to IFSs such as K = {hx

,0.5,0.3i} and L =

{hx

,0.5,0.2i} where the exchange of “the places”

between the non-membership value and the hesita-

tion margin in K and L results in S(M,K) = S(M,L).

Due to this anomaly is caused by the symmetry be-

tween the non-membership value and the hesitation

margin, Szmidt and Kacprzyk also veriﬁed the results

using the “two terms”-distances (11) and (13). How-

ever, they found that the situation does not change

in the sense of the information obtained. As a solu-

tion to this kind of anomalies, in (Loor and De Tr´e,

2013), it was proposed (9). Thus, in this example,

(M,N) = 0 and S

(M,H) = α, which makes sense

according to the semantic interpretation given in Sec-

tion 5.1. Despite this, there is an anomaly that (9)

could not manage by itself and it is needed a CDP.

7.2 An Anomaly That a CDP Could

Help to Solve

Consider X = {x

} and the IFSs P = {hx

,0.5,0.5i},

Q = {hx

,0.9,0.1i} and R = {hx

,0.1,0.9i}. Also

consider the similarity measure S. If S is (10),

(11), (12), (13), (14) or even (9), it is obtained that

S(P,Q) = S(P,R) though Q and R are obviouslydiffer-

ent. This anomaly could be generalized to IFSs such

asV = {hx

,0.7,0.3i} and W = {hx

,0.3,0.7i} where

giving the values µ

) = ν

), µ

) = ν

)

and π

) = π

) = 0 results in S(P,V) = S(P,W).

To solve this, we use the extended version of (9) as

follows. From (5) the corresponding vector interpre-

tations for x

are p



0.5



, q



0.9

0.1



and



0.1

0.9



—notice that π

) = 0, π

) = 0

and π

) = 0, which means that a hesitation split-

ter is not necessary. Then, considering the point of

view of P and using (8), the spot-differences for x

are dif

) = −0.4 and di f

) = 0.4. With

δ = 0.2, the corresponding connotation-differential

marker for di f

) is

⋄

| and the corresponding

one for dif

) is

⋄

|. To build the CDPs for P-vs.-

Q and P-vs.-R comparisons, we use the connotation-

differential markers given for x

, thus, from P’s view,

⋄

| is a CDP for P-vs.-Q, and

⋄

| is a CDP for P-vs.-

R. Finally, from (9) we obtain S

(P,Q) = 0.6 and

(P,R) = 0.6, and, using the corresponding CDPs,

we extend them to h0.6,

⋄

|i and h0.6,

⋄

|i respectively.

As expected, h0.6,

⋄

|i and h0.6,

⋄

|i are different. One

might argue that if the same weight is assigned to

Connotation-differentialPrints-ComparingWhatIsConnotedThrough(Fuzzy)Evaluations

135

⋄

| and

⋄

|, then S(P,Q) = S(P,R); however, it will de-

pend on the semantic interpretation of who assigns

the weights, which could be important in a decision

making context.

8 CONCLUSIONS

Following the psychological view of similarity pre-

sented by Tversky in (Tversky, 1977), we have envis-

aged an object as a collection of features and consid-

ered that, when a person is evaluating to which level

an object belongs (or not) to a particular set, say A,

she or he could focus on some speciﬁc features ac-

cording to what she or he understands by A, i.e., by

focusing on such features, this person makes an per-

sonal representation of an object that is used as ref-

erent of A. Thus, even if two persons understand

what is A, they could have different connotations of

it. Then, we have hypothesized that, if two persons

have evaluated to which degree each object within a

group could be considered to be similar to A, a differ-

ence in connotations of A could be marked by a differ-

ence in one or more of their evaluations. Therefore,

modeling such evaluations using a semantic interpre-

tation of the concept of intuitionistic fuzzy set given

by Attanasov in (Atanassov, 1999; Atanassov, 2012),

we have presented a connotation-differential marker

to denote how a difference between two (fuzzy) eval-

uations for an object given by two persons could be

managed.

Choosing one of the points of view of the per-

sons who perform the evaluation, we have shown

how to build a sequence from connotation-differential

markers corresponding to one or more representative

objects for this evaluator. We call this sequence a

connotation-differential print, which is assumed to be

a representation of the difference in connotations of A

between the person whose point of view is chosen and

any other person. Furthermore, we have given an ex-

ample of how a person could assign a weight to each

of the possible connotation-differential prints accord-

ing to the strategy followed by this person to build

such sequences.

We have illustrated how to use a connotation-

differential print as a supplement of a similarity mea-

sure for intuitionistic fuzzy sets in order to perform

a meaningful comparison between two of them. Fi-

nally, we have compared our extended similarity mea-

sure with others in order to show one of its possible

advantages.

In our future work, we will further explore the

applicability of connotation-differential prints in or-

der to perform qualitative comparisons. In particu-

lar, it interests us to look into the following problems:

1) assessing the quality of data gathered in a crowd-

sourced context; and, 2) detecting any difference in

understandings of a given topic between two knowl-

edge databases that come from different sources and

need to be merged.

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