# Three-way Decisions with Evaluative Linguistic Expressions

Stefania Boffa  0000-0002-4171-3459 Davide Ciucci  0000000280837809

###### Abstract

We propose a linguistic interpretation of three-way decisions, where the
regions of acceptance, rejection, and non-commitment are constructed by using
the so-called evaluative linguistic expressions, which are expressions of
natural language such as small, medium, very short, quite roughly strong,
extremely good, etc. Our results highlight new connections between two
different research areas: three-way decisions and the theory of evaluative
linguistic expressions.

Keywords— Three-way decisions, Rough sets, Probabilistic rough sets,
Evaluative linguistic expressions, Explainable Artificial Intelligence

##  1 Introduction

The theory of three-way decisions (TWD) divides a finite and non-empty
universe into three disjoint sets, which are called positive, negative, and
boundary regions. These regions respectively induce positive, negative, and
boundary rules: a positive rule makes a decision of acceptance, a negative
rule makes a decision of rejection, and a boundary rule makes an abstained or
non-committed decision [1, 2]. The concept of three-way decisions was
originally introduced in Rough Set Theory [1, 3] and until today, it has been
widely studied and applied to many decision-making problems (see [4, 5, 6, 7]
for some examples). Thus, several approaches have been proposed to generate
the three regions; one of them is based on probabilistic rough sets, which
generalizes probabilistic rough sets [8, 9] where the three regions are
constructed using a pair of thresholds and the notion of conditional
probability (in this case, the regions are called probabilistic positive,
negative, and boundary regions).

The contribution of this article is to provide a linguistic interpretation of
the positive, negative, and boundary regions. So, we propose a three-way
decision method based on the concept of _evaluative linguistic expressions_ ,
which are expressions of natural language such as _small, medium, very short,
quite roughly strong, extremely good, etc_. These are already considered in
the majority of applications of fuzzy modelling. Since we use evaluative
linguistic expressions to evaluate the size of sets, we focus on the
expressions involving the adjectives _small_ , _medium_ , and _big_ that can
be preceded by an adverb; examples are _very small_ , _roughly medium_ , and
_extremely big_. Mathematically, an evaluative linguistic expression is
modelled by a function $Ev:[0,1]\to[0,1]$.The formal theory of evaluative
linguistic expressions is introduced and explained in [10, 11, 12, 13].

The positive, negative, and boundary regions of a non-empty and finite
universe $U$ are defined here starting from a subset $X$ of $U$, an
equivalence relation $\mathcal{R}$ on $U$ (i.e. $\mathcal{R}$ is reflexive,
symmetric and transitive), an evaluative linguistic expression $Ev$, and a
pair of thresholds $(\alpha,\beta)$ with $0\leq\beta<\alpha\leq 1$. Then, an
object $x$ belongs to the positive region when _the size of
$[x]_{\mathcal{R}}\cap X$ evaluated w.r.t. $Ev$ is at least $\alpha$_, where
$[x]_{\mathcal{R}}$ is the equivalence class of $x$ w.r.t. $\mathcal{R}$.
Analogously, $x$ belongs to the negative region when _the size of
$[x]_{\mathcal{R}}\cap X$ evaluated w.r.t. $Ev$ is at most $\beta$_. Finally,
the remaining elements form the boundary region. In order to obtain the three
regions, the size of $X\cap[x]_{\mathcal{R}}$ is quantified using a fuzzy
measure [14, 15].

The role of evaluative linguistic expressions in the context of three-way
decisions can be better understood with the following example.

######  Example 1.

Suppose that the number of buses between the University of Buenos Aires and
the rest of the city has to be increased from 7 am to 8 am. Thus, we intend to
understand which city areas need buses the most, as resources are limited. Let
us denote the areas of the city with $A_{1},\ldots,A_{n}$ and map each area
$A_{i}$ with the set $S_{A_{i}}$ made of all students of the university who
live in $A_{i}$. Thus, $S_{A_{1}},\ldots,S_{A_{n}}$ can be seen as the
equivalence classes w.r.t. the relation $\mathcal{R}$ on the set of all
students of the University of Buenos Aires living in the city: $x\mathcal{R}y$
if and only if $x$ and $y$ live in the same area. Based on a survey, we also
consider a set $X$ made of all students that usually take a bus to the
university in the slot time [7 am, 8 am]. We also choose
$(\alpha,\beta)=(0.3,0.6)$ and $Ev=extemely\ big$. We construct three regions
in the following way. The positive region is the union of
$S_{A^{\prime}_{1}},\ldots,S_{A^{\prime}_{k}}$ (with
$\\{A^{\prime}_{1},\ldots,A^{\prime}_{k}\\}\subseteq\\{A_{1},\ldots,A_{n}\\}$)
so that the amount of students of $S_{A_{i}}$ that take a bus from 7 am to 8
am is “extremely big” with a value of at least 0.6. Similarly, the negative
region is the union of $S_{A^{*}_{1}},\ldots,S_{A^{*}_{h}}$ (with
$\\{A^{*}_{1},\ldots,A^{*}_{h}\\}\subseteq\\{A_{1},\ldots,A_{n}\\}$) so that
the amount of students of $S_{A^{*}_{i}}$ that take a bus from 7 am to 8 am is
extremely big with a value of at most 0.3. All other students form the
boundary region. The final decision is immediate: the buses are certainly
increased for the areas $A^{\prime}_{1},\ldots,A^{\prime}_{k}$, but not for
$A^{*}_{1},\ldots,A^{*}_{h}$. Furthermore, the decision is postponed for the
remaining areas (thati is, for each
$A_{i}\notin\\{A^{\prime}_{1},\ldots,A^{\prime}_{k}\\}\cup\\{A^{*}_{1},\ldots,A^{*}_{h}\\}$).
In order to make a decision in those areas, for example, we could take into
account the workers (besides the students) that need a bus in the slot of time
[7 am, 8 am].

The choice of $Ev$ depends on the context where the three regions are used.
Indeed, in the previous example, we have chosen _extremely big_ in order to
select the areas where a large number of students catch the bus from 7 am to 8
am. However, if we focus on the inverse problem (namely we need to eliminate
some existing bus rides), then we should identify the areas where there are
fewer students taking the bus in the time slot [7 am, 8 am]. Therefore, in
this case, the evaluative linguistic expression _extremely small_ is more
appropriate to construct the three regions.

A significant contribution of this article is providing a linguistic and novel
interpretation of the positive, negative, and boundary regions already
determined with probabilistic rough sets. Consequently, the reasons for
decisions of acceptance, rejection, and non-commitment can be explained in
terms of expressions of natural language. Of course, the advantage is that
non-technical users dealing with TWD models can better understand the
reliability of the procedures related to the final decisions. This is in line
with the scope of _Explainable Artificial Intelligence (XAI)_ , which is a new
approach to AI emphasizing the ability of machines to give sound motivations
about their decisions and behaviour [16].

The article is organized as follows. The next section reviews some basic
notions regarding probabilistic three-way decisions and the concept of
evaluative linguistic expressions. Also, the notion of fuzzy measure is
recalled. Section 3 presents a new model of three-way decisions based on the
theory of evaluative linguistic expressions. As a consequence, a linguistic
generalization of Pawlak rough sets is introduced. Finally, Section 4 connects
the TWD models based on evaluative linguistic expressions and probabilistic
rough sets. In particular, confining to the evaluative linguistic expressions
modelled by increasing functions, we find the class of thresholds so that the
corresponding probabilistic positive, negative, and boundary regions are equal
to those generated by a given evaluative linguistic expression.

##  2 Preliminaries

In the following, we consider a finite universe $U$, a subset $X$ of $U$, and
an equivalence relation $\mathcal{R}$ on $U$ (i.e. $\mathcal{R}$ is reflexive,
symmetric, and transitive). Moreover, we indicate the equivalence class of
$x\in U$ w.r.t. $\mathcal{R}$ with $[x]_{\mathcal{R}}$.

###  2.1 Three-way decisions with probabilistic rough sets

This subsection recalls the fundamental notions of three-way decisions based
on probabilistic rough sets.

Viewing $X$ and $[x]_{\mathcal{R}}$ as events of $U$, the symbol
$Pr(X|[x]_{\mathcal{R}})$ denotes the _conditional probability_ of $X$ given
$[x]_{\mathcal{R}}$, i.e.

$Pr(X|[x]_{\mathcal{R}})=\frac{|[x]_{\mathcal{R}}\cap
X|}{|[x]_{\mathcal{R}}|}.$ (1)

Then, three special subsets of $U$ are determined by using (1) and a pair of
thresholds as shown by the next definition.

######  Definition 1.

Let $\alpha,\beta\in[0,1]$ such that $\beta<\alpha$, the
$(\alpha,\beta)$-_probabilistic positive, negative_ and _boundary regions_ are
respectively the following:

  1. (i)

$POS_{(\alpha,\beta)}(X)=\\{x\in U\ |\ Pr(X|[x]_{\mathcal{R}})\geq\alpha\\}$,

  2. (ii)

$NEG_{(\alpha,\beta)}(X)=\\{x\in U\ |\ Pr(X|[x]_{\mathcal{R}})\leq\beta\\}$,

  3. (iii)

$BND_{(\alpha,\beta)}(X)=\\{x\in U\ |\
\beta<Pr(X|[x]_{\mathcal{R}})<\alpha\\}$.

We put

$\mathcal{T}_{(\alpha,\beta)}(X)=\\{POS_{(\alpha,\beta)}(X),NEG_{(\alpha,\beta)}(X),BND_{(\alpha,\beta)}(X)\\}$
(2)

and we say that $\mathcal{T}_{(\alpha,\beta)}(X)$ is a _tri-partition_ of $U$
due to the following remark 111By a tri-partition, we mean a partition of $U$
made of three equivalence classes. On the other hand,
$\\{POS_{(\alpha,\beta)}(X),NEG_{(\alpha,\beta)}(X),BND_{(\alpha,\beta)}(X)\\}$
could collapse into a bi-partition or the whole universe when one or two of
its sets are empty..

######  Remark 1.

The three regions of of $\mathcal{T}_{(\alpha,\beta)}(X)$ are mutually
disjoint, i.e. $A\cap B=\emptyset$ for each
$A,B\in\\{POS_{(\alpha,\beta)}(X),NEG_{(\alpha,\beta)}(X),BND_{(\alpha,\beta)}(X)\\}$
with $A\neq B$, and they cover the universe $U$, i.e.

$POS_{(\alpha,\beta)}(X)\cup NEG_{(\alpha,\beta)}(X)\cup
BND_{(\alpha,\beta)}(X)=U.$ (3)

In the context of three-way decisions, the following rules are considered: let
$x\in U$,

  * •

if $x\in POS_{(\alpha,\beta)}(X)$, then $x$ is accepted;

  * •

if $x\in NEG_{(\alpha,\beta)}(X)$, then $x$ is rejected;

  * •

if $x\in BND_{(\alpha,\beta)}(X)$, then we abstain on $x$.

The values $Pr(X|[x]_{\mathcal{R}})$ represents the _accuracy_ or _confidence_
of the rules:

  * •

the higher $Pr(X|[x]_{\mathcal{R}})$ is, the more confident we are that $x\in
POS_{(\alpha,\beta)}(X)$ is correctly accepted,

  * •

the lower $Pr(X|[x]_{\mathcal{R}})$ is, the more confident we are that $x\in
NEG_{(\alpha,\beta)}(X)$ is correctly rejected.

Definition 1 is strictly related to the notion of probabilistic rough sets.

######  Definition 2.

The _$(\alpha,\beta)$ -probabilistic rough set_ of $X$ is the pair

$(\mathcal{L}_{(\alpha,\beta)}(X),\mathcal{U}_{(\alpha,\beta)}(X)),$

where

$\mathcal{L}_{(\alpha,\beta)}(X)=POS_{(\alpha,\beta)}(X)$ and
$\mathcal{U}_{(\alpha,\beta)}(X)=POS_{(\alpha,\beta)}(X)\cup
BND_{(\alpha,\beta)}(X)$,

which are respectively called $(\alpha,\beta)-$ _lower and upper
approximations_ of $X$.

######  Remark 2.

When $\alpha=1$ and $\beta=0$,
$(\mathcal{L}_{(\alpha,\beta)}(X),\mathcal{U}_{(\alpha,\beta)}(X))$ is the
rough set $(\mathcal{L}(X),\mathcal{U}(X))$ of $X$ defined by Pawlak in [3],
namely

$(\mathcal{L}(X),\mathcal{U}(X))=(\\{x\in U\ |\ [x]_{R}\subseteq X\\},\\{x\in
U\ |\ [x]_{R}\cap X\neq\emptyset\\}).$ (4)

The sets $\mathcal{L}(X)$ and $\mathcal{U}(X)$ are respectively called _lower_
and _upper approximations_ of $X$ w.r.t. $\mathcal{R}$.

###  2.2 Evaluative Linguistic Expressions

This subsection reviews concepts and results that are found in [10, 17] and it
recalls the notion of fuzzy measure.

_Evaluative linguistic expressions_ are special expressions of natural
language, which people commonly employ to evaluate, judge, estimate, and in
many other situations. Examples of evaluative linguistic expressions are
_small, medium, big, about twenty-five, roughly one hundred, very short, more
or less deep, not very tall, roughly warm or medium-hot_ , etc. For
convenience, we will often omit the adjective “linguistic” and use only the
term “evaluative expressions”. The simplest evaluative expressions are called
_pure evaluative expressions_ and have the following structure:

_$\langle\mbox{linguistic hedge}\rangle\langle\mbox{TE-adjective}\rangle,$_

where

  * •

a linguistic hedge is an adverbial modification such as _very, roughly,
approximately, significantly_ and

  * •

a TE-adjective is an adjective such us _good, medium_ , _big_ , _short_ , etc.
TE stands for _trichotomous evaluative_ , indeed TE-adjectives typically form
pairs of antonyms like _small_ and _big_ completed by a middle member, which
is _medium_ in the case of _small_ and _big_. Other examples are “ _weak,
medium-strong,_ and strong” and “ _soft, medium-hard,_ and _hard_ ”.

The _empty linguistic hedge_ is employed to deal with evaluative expressions
made of only a TE-adjective; hence, _small_ , _medium_ , and _big_ are
considered evaluative expressions. Other pure evaluative expressions are the
fuzzy numbers like _about twenty-five_. Two or more pure evaluative
expressions can be connected to form _negative evaluative expressions_ like
_“NOT very small”_ and _compound evaluative expressions_ like _“very expensive
AND extremely small”_ and _“very expensive OR extremely small”_.

The semantics of evaluative expressions is based on the essential concepts of
_context_ , _intension_ , and _extension_.

  * •

The _context_ is a state of the world at a given time and place in which an
evaluative expression appears. Each context is represented by a linearly
ordered scale, which is bounded by $s$ and $b$. Moreover, a context is given
by a triple $\omega=\langle s,m,b\rangle$, where $s$ is the “most typical”
small value, $m$ is the “most typical” medium value, and $b$ is the “most
typical” big value. For example, suppose that evaluative expressions are used
to evaluate the size of apartments. If we are thinking of apartments for one
person, then we could choose $\omega_{1}=\langle 40,70,100\rangle$ as context,
which means that flats measuring $40\ m^{2}$, $70\ m^{2}$, and $100\ m^{2}$
are typically small, medium and big, respectively. On the other hand, when
changing context and thinking of apartments for a family of 5 people, the
context $\omega_{5}=\langle 70,120,160\rangle$ is more appropriate.

  * •

The _intension_ of an evaluative expression is a function mapping each context
into a fuzzy set of a given universe. Taking up the previous example, we
consider a universe made of four apartments $\\{a_{1},a_{2},a_{3},a_{4}\\}$,
then the intension of $small$ is the map $Int_{small}$ that assigns to the
context $\omega_{5}$ the fuzzy set $A_{\omega_{5}}$ so that
$A_{\omega_{5}}(a_{i})$ is the degree to which _$a_{i}$ is small in the
context $\omega_{5}$_ (namely, _$a_{i}$ is small for 5 people_).

  * •

The _extension_ of an evaluative expression $Ev$ is a fuzzy set determined by
the intension of $Ev$, given a context $\omega$. Concerning the previous
example, $Int_{small}(\omega_{5})=A_{\omega_{5}}$ is an example of an
extension of _small_.

In this article, we confine to the TE adjectives _small_ , _medium_ , and
_big_ because we use evaluative expressions to evaluate the size of sets. So,
let $X$ be a subset of a universe $U$, we will say that the size of $X$ w.r.t.
to the size of $U$ is _very small_ , _extremely big_ , etc. Furthermore, we
confine to the _standard context_ , which is $\langle 0,0.5,1\rangle$.
Finally, since sizes are expressed by means of a fuzzy measure (by Example 2,
the measure of the size of a set $X$ is a value of $[0,1]$), the extensions of
our evaluative expressions are functions from $[0,1]$ to $[0,1]$, which have a
specific formula. The extension of an evaluative expression like $\langle
linguist\ hedge\rangle\langle TE-adjective\rangle$ with $TE-
adjective\in\\{small,medium,big\\}$ is obtained by composing two functions,
one models the linguistic hedge and the other models the TE-adjective. The
function describing a linguistic hedge depends on three parameters, which are
experimentally estimated (see [17] for more details).

In what follows, we provide the formula of $\neg Sm:[0,1]\to[0,1]$,
$BiVe:[0,1]\to[0,1]$, and $BiEx:[0,1]\to[0,1]$, which are the extensions of
the evaluative expressions _not small_ , _very big_ , and _extremely big_ ,
where the context $\langle 0,0.5,1\rangle$ is fixed 222We have got the
formulas of $BiVe$ and $BiEx$ using the function
$\nu_{a,b,c}(LH(\omega^{-1}))$ and Table 5.1 given in [17]. Concerning the
formula of $\neg Sm$, we have considered that $\neg Sm(x)=1-Sm(x)$. After
that, we have found the formula of $Sm$ using the function
$\nu_{a,b,c}(RH(\omega^{-1}))$ and Table 5.1 of [17]..

$\neg Sm(x)=\begin{dcases}1&\mbox{ if }x\in[0.275,1],\\\
1-\frac{(0.275-x)^{2}}{0.02305}&\mbox{ if }x\in(0.16,0.275)\\\
\frac{(x-0.0745)^{2}}{0.01714}&\mbox{ if }x\in(0.0745,0.16]\\\ 0&\mbox{ if
}x\in[0,0.0745]\end{dcases}$ (5) $BiVe(x)=\begin{dcases}1&\mbox{ if
}x\in[0.9575,1],\\\ 1-\frac{(0.9575-x)^{2}}{0.00796}&\mbox{ if
}x\in[0.895,0.9575),\\\ \frac{(x-0.83)^{2}}{0.00828}&\mbox{ if
}x\in(0.83,0.895),\\\ 0&\mbox{ if }x\in[0,0.83].\end{dcases}$ (6)
$BiEx(x)=\begin{dcases}1&\mbox{ if }x\in[0.995,1],\\\
1-\frac{(0.995-x)^{2}}{0.00495}&\mbox{ if }x\in[0.95,0.995),\\\
\frac{(x-0.885)^{2}}{0.00715}&\mbox{ if }x\in(0.885,0.95),\\\ 0&\mbox{ if
}x\in[0,0.885].\end{dcases}$ (7)

######  Remark 3.

The evaluative expressions $\neg Sm$, $BiVe$, and $BiEx$ have a special role:
they are respectively used to construct the formula of fuzzy quantifiers
_many_ , _most_ , and _almost all_ [18].

A further class of linguistic expressions is $\\{\Delta_{t}:[0,1]\to\\{0,1\\}\
|\ t\in[0,1]\\}$, where given $t\in[0,1]$ the formula of $\Delta_{t}$ is the
following: let $a\in[0,1]$,

$\Delta_{t}(a)=\begin{cases}1&\mbox{ if }a\geq t\\\ 0&\mbox{ otherwise.
}\end{cases}$ (8)

In the sequel, we need the notion of fuzzy measure [14, 15].

######  Definition 3.

Let $U$ be a finite universe, a mapping $\varphi:2^{U}\to\mathbb{R}$ is called
_fuzzy measure_ if and only if

  1. (a)

$\varphi(\emptyset)=0$;

  2. (b)

if $X\subseteq Y$ then $\varphi(X)\leq\varphi(Y)$, for each $X,Y\subseteq U$
(monotonicity).

A fuzzy measure $\varphi$ is called _normalized_ or _regular_ if
$\varphi(U)=1$.

In this paper, we focus on the normalized fuzzy measure given by the next
example.

######  Example 2.

Let $U$ be a finite universe, the function $f:2^{U}\to\mathbb{R}$ that assigns
$\dfrac{|Y|}{|U|}$ to each $Y\subseteq U$ is a fuzzy measure.

The value $\dfrac{|Y|}{|U|}$ belongs to [0,1] and measures _“how much $Y$ is
large with respect to $U$ in the scale $[0,1]$”_.

Let us observe that in Probability theory $\dfrac{|Y|}{|U|}$ represents _“how
likely the event $Y$ is to occur”_.

##  3 Three-way decisions with linguistic expressions

This subsection proposes a novel model for three-way decisions, which is based
on the concept of evaluative linguistic expressions previously described.

In the sequel, we use the symbol $\mathcal{E}$ to denote the collection of the
extensions of all evaluative expressions in the context $\langle
0,0.5,1\rangle$.

Therefore, let $Ev\in\mathcal{E}$, let $X\subseteq U$, and let
$\alpha,\beta\in[0,1]$ with $\beta<\alpha$, three regions of $U$ are
determined. In particular, the region of a given element $x\in U$ is found by
taking into account the following steps:

  1. 1.

computing
$Ev\left(\dfrac{|X\cap[x]_{\mathcal{R}}|}{|[x]|_{\mathcal{R}}}\right)$, which
is the evaluation of _the size of $X\cap[x]_{\mathcal{R}}$ w.r.t. the size of
$[x]_{\mathcal{R}}$_ by using $Ev$;

  2. 2.

comparing
$Ev\left(\dfrac{|X\cap[x]_{\mathcal{R}}|}{|[x]|_{\mathcal{R}}}\right)$ with
the thresholds $\alpha$ and $\beta$.

For example, regarding point 1, if _Ev_ models the evaluative expression “
_significantly big_ ”, then
$Ev\left(\dfrac{|X\cap[x]_{\mathcal{R}}|}{|[x]_{\mathcal{R}}|}\right)$
measures

_“how much the size of $X\cap[x]_{\mathcal{R}}$ is significantly big w.r.t.
the size of $[x]_{\mathcal{R}}$”._

Equivalently, we are saying that

_“the size of the set of the elements of $[x]_{\mathcal{R}}$ that also belong
to $X$ is significantly big with the truth degree
$Ev\left(\dfrac{|X\cap[x]_{\mathcal{R}}|}{|[x]_{\mathcal{R}}|}\right)$”._

Observe that $\dfrac{|X\cap[x]_{\mathcal{R}}|}{|[x]_{\mathcal{R}}|}$
syntactically coincides with the conditional probability (see (1)), but here
it has a different interpretation: it is the fuzzy measure specified by
Example 2.

Formally, the three regions of $U$ determined by an evaluative expression are
given by the following definition.

######  Definition 4.

Let $Ev\in\mathcal{E}$, the $(\alpha,\beta)$-linguistic positive, negative,
and boundary regions induced by $Ev$ are respectively the following:

  1. (i)

$POS^{Ev}_{(\alpha,\beta)}(X)=\left\\{x\in U\ |\
Ev\left(\dfrac{|[x]_{\mathcal{R}}\cap
X|}{|[x]_{\mathcal{R}}|}\right)\geq\alpha\right\\}$;

  2. (ii)

$NEG^{Ev}_{(\alpha,\beta)}(X)=\left\\{x\in U\ |\
Ev\left(\dfrac{|[x]_{\mathcal{R}}\cap
X|}{|[x]_{\mathcal{R}}|}\right)\leq\beta\right\\}$;

  3. (iii)

$BND^{Ev}_{(\alpha,\beta)}(X)=\left\\{x\in U\ |\
\beta<Ev\left(\dfrac{|[x]_{\mathcal{R}}\cap
X|}{|[x]_{\mathcal{R}}|}\right)<\alpha\right\\}$.

We put

$\mathcal{T}_{(\alpha,\beta)}^{Ev}(X)=\\{POS_{(\alpha,\beta)}^{Ev}(X),NEG_{(\alpha,\beta)}^{Ev}(X),BND_{(\alpha,\beta)}^{Ev}(X)\\}$
(9)

and we say that $\mathcal{T}_{(\alpha,\beta)}^{Ev}(X)$ is a tri-partition of
$U$.

######  Remark 4.

The three regions of $\mathcal{T}_{(\alpha,\beta)}^{Ev}(X)$ are mutually
disjoint, i.e. $A\cap B=\emptyset$ for each
$A,B\in\\{POS_{(\alpha,\beta)}^{Ev}(X),NEG_{(\alpha,\beta)}^{Ev}(X),BND_{(\alpha,\beta)}^{Ev}(X)\\}$
with $A\neq B$, and they cover the universe $U$, i.e.

$POS_{(\alpha,\beta)}^{Ev}(X)\cup NEG_{(\alpha,\beta)}^{Ev}(X)\cup
BND_{(\alpha,\beta)}^{Ev}(X)=U.$ (10)

######  Remark 5.

Let us focus on the evaluative expressions _not small_ , _very big_ ,
_extremely big_ , and _utmost_. The first three expressions are respectively
modelled by (5), (6), and (7). According to Remark 3, $\neg Sm$, $BiVe$, and
$BiEx$ respectively appear into the formula of quantifiers _many_ , _most_ ,
and _almost all_. Moreover, as explained in [19] (see Lemma 4.5), considering
that $X$ and $[x]_{\mathcal{R}}$ are crisp set, $\neg
Sm\left(\dfrac{|X\cap[x]_{\mathcal{R}}|}{|[x]_{\mathcal{R}}|}\right)$,
$BiVe\left(\dfrac{|X\cap[x]_{\mathcal{R}}|}{|[x]_{\mathcal{R}}|}\right)$, and
$BiEx\left(\dfrac{|X\cap[x]_{\mathcal{R}}|}{|[x]_{\mathcal{R}}|}\right)$
exactly coincide with the formula of quantifiers _many_ , _most_ , and _almost
all_. Hence, they have the following meaning:

  * •

$\neg Sm\left(\dfrac{|X\cap[x]_{\mathcal{R}}|}{|[x]_{\mathcal{R}}|}\right)$ is
degree to which “ _many_ objects of $[x]_{\mathcal{R}}$ are in $X$”,

  * •

$BiVe\left(\dfrac{|X\cap[x]_{\mathcal{R}}|}{|[x]_{\mathcal{R}}|}\right)$ is
degree to which “ _most_ objects of $[x]_{\mathcal{R}}$ are in $X$”,

  * •

$BiEx\left(\dfrac{|X\cap[x]_{\mathcal{R}}|}{|[x]_{\mathcal{R}}|}\right)$ is
degree to which “ _almost all_ objects of $[x]_{\mathcal{R}}$ are in $X$”.

Moreover, the function $\Delta_{1}$ that is obtained by (8) and putting $t=1$,
models the evaluative expression _utmost_ and corresponds to the quantifier
_all_. Therefore,
$\Delta^{1}\left(\dfrac{|X\cap[x]_{\mathcal{R}}|}{|[x]_{\mathcal{R}}|}\right)$
is understood as the degree to which “ _all_ objects of $[x]_{\mathcal{R}}$
are in $X$”.

Observe that here the universe of quantification coincides with
$[x]_{\mathcal{R}}$, which is always non-empty, considering that $\mathcal{R}$
is reflexive, hence $\\{x\\}\subseteq[x]_{\mathcal{R}}$. In mathematical
logic, the assumption expressing that the universe of quantification must be
non-empty is called _existential import_ (or _presupposition_) [20].

######  Remark 6.

Consider the evaluative expressions represented by (8). Let $t\in[0,1]$, then
$\Delta_{t}\left(\dfrac{|X\cap[x]_{\mathcal{R}}|}{|[x]_{\mathcal{R}}|}\right)$
is the degree to which “ _the size of the set of elements of
$[x]_{\mathcal{R}}$ that also belong to $X$ is at least as large as $t$ (in
the scale [0,1])_”.

###  3.1 An illustrative example

In this subsection, we provide an example of how to use linguistic three-way
decision to provide recommendations based on users’s profile.

We consider a universe $U=\\{u_{1},\ldots,u_{32}\\}$ made of users of online
communities and the following equivalence relation $\mathcal{R}$ on $U$: let
$x,y\in U$, $x\mathcal{R}y$ if and only if $x$ and $y$ belong to the same
community. Therefore, $\mathcal{R}$ corresponds to the partition
$\\{C_{1},\ldots,C_{6}\\}$ of $U$, where $C_{i}$ is the set of users of $U$
belonging to the community $i$. In particular, we suppose that

$C_{1}=\\{u_{1},\ldots,u_{5}\\},\ C_{2}=\\{u_{6},\ldots,u_{10}\\},\
C_{3}=\\{u_{11},\ldots,u_{15}\\},\\\ C_{4}=\\{u_{16},\ldots,u_{20}\\},\
C_{5}=\\{u_{21},\ldots,u_{25}\\},\mbox{ and
}C_{6}=\\{u_{26},\ldots,u_{32}\\}.$

We use the symbol $X_{T}$ to denote the set of users of $U$ interested in a
specific topic $T$.

For example,

$X_{Sport}=\\{u_{10},u_{11},u_{12},u_{18},u_{19},u_{20},u_{21},u_{22},u_{23},u_{24},u_{26}\\}$

is the set of all users of $U$ interested in the topic _Sport_.

Using three-way decisions based on evaluative expressions, we intend to select
the most appropriate communities among $C_{1},\ldots,C_{6}$ to which propose
news related to the topic $T$.

If we choose $(\alpha,\beta)=(0.8,0.2)$ and the evaluative expression $\neg
Sm$ corresponding to the fuzzy quantifier _many_ , then we decide to assign
the news about the topic $T$ to the communities of $POS_{(0.8,0.2)}^{\neg
Sm}(X_{T})$. Indeed, enough users of $POS_{(0.8,0.2)}^{\neg Sm}(X_{T})$ are
interested in $T$: $x\in POS_{(0.8,0.2)}^{\neg Sm}$ if and only if the degree
to which

_“many users of the community of $x$ are interested in $T$”_

is greater than or equal to 0.8.

In the sequel, we determine the communities to which provide the news about
$Sport$. To do this, we firstly compute the value
$\dfrac{|X_{Sport}\cap[x]_{\mathcal{R}}|}{|[x]_{\mathcal{R}}|}$ for each $x\in
U$:

$\dfrac{|X_{Sport}\cap[x]_{\mathcal{R}}|}{|[x]_{\mathcal{R}}|}=\begin{cases}0&\mbox{
if }x\in C_{1},\\\ 0.14&\mbox{ if }x\in C_{6},\\\ 0.2&\mbox{ if }x\in
C_{2},\\\ 0.4&\mbox{ if }x\in C_{3},\\\ 0.6&\mbox{ if }x\in C_{4},\\\
0.8&\mbox{ if }x\in C_{5}.\end{cases}$ (11)

According to the definition of $\neg Sm$ that is given by (5), we get $\neg
Sm(0)=0$, $\neg Sm(0.14)=0.25$, $\neg Sm(0.2)=0.75$ and $\neg Sm(0.4)=\neg
Sm(0.6)=\neg Sm(0.8)=1$.

Consequently,

$\neg
Sm\left(\frac{|X_{Sport}\cap[x]_{\mathcal{R}}|}{|[x]_{\mathcal{R}}|}\right)=\begin{cases}0&\mbox{
if }x\in C_{1},\\\ 0.25&\mbox{ if }x\in C_{6},\\\ 0.75&\mbox{ if }x\in
C_{2}\\\ 1&\mbox{ if }x\in C_{3}\cup C_{4}\cup C_{5}.\end{cases}$ (12)

Then, the positive, negative and boundary regions induced by $(0.8,0.2)$ and
$\neg Sm$ are the following:

$POS^{\neg Sm}_{(0.8,0.2)}(X_{Sport})=\left\\{x\in U\ |\ \neg
Sm\left(\dfrac{|X_{Sport}\cap[x]_{\mathcal{R}}|}{|[x]_{\mathcal{R}}|}\right)\geq
0.8\right\\}=C_{3}\cup C_{4}\cup C_{5}$,

$NEG^{\neg Sm}_{(0.8,0.2)}(X_{Sport})=\left\\{x\in U\ |\ \neg
Sm\left(\dfrac{|X_{Sport}\cap[x]_{\mathcal{R}}|}{|[x]_{\mathcal{R}}|}\right)\leq
0.2\right\\}=C_{1}$,

$BND^{\neg Sm}_{(0.8,0.2)}(X_{Sport})=\left\\{x\in U\ |\ 0.2<\neg
Sm\left(\dfrac{|X_{Sport}\cap[x]_{\mathcal{R}}|}{|[x]_{\mathcal{R}}|}\right)<0.8\right\\}=C_{2}\cup
C_{6}$.

The three regions lead the following decisions. Firstly, we choose to provide
the news about the sport to the communities $C_{3}$, $C_{4}$, and $C_{5}$ that
form the positive region, considering that _these contains many users
interested in sport_ with a degree that we think is enough high ($\geq 0.8$).
Moreover, we require further analysis on the communities $C_{2}$ and $C_{6}$
forming the boundary region, before providing news about sports. For example,
we could evaluate the interests of their users in the future or once new users
join them. Finally, we surely do not provide sports news to $C_{1}$ because we
think that not enough of its users are interested in sports topics, indeed we
consider the degree to which _many users of $C_{1}$ are interested in sports_
low ($\leq 0.2$).

###  3.2 Linguistic Rough Sets

Definition 4 also leads to a novel generalization of Pawlak rough sets.

######  Definition 5.

Let $Ev\in\mathcal{E}$, the $(\alpha,\beta)$-linguistic rough set of $X$
determined by $\mathcal{R}$ and $Ev$ is the pair
$(\mathcal{L}_{(\alpha,\beta)}^{Ev}(X),\mathcal{U}_{(\alpha,\beta)}^{Ev}(X)),$
where

$\mathcal{L}_{(\alpha,\beta)}^{Ev}(X)=\left\\{x\in X\ |\
Ev\left(\dfrac{|[x]_{\mathcal{R}}\cap
X|}{|[x]_{\mathcal{R}}|}\right)\geq\alpha\right\\}\ \mbox{ and }\ \\\
\mathcal{U}_{(\alpha,\beta)}^{Ev}(X)=\left\\{x\in X\ |\
Ev\left(\dfrac{|[x]_{\mathcal{R}}\cap
X|}{|[x]_{\mathcal{R}}|}\right)>\beta\right\\}.$

$\mathcal{L}_{(\alpha,\beta)}^{Ev}(X)$ and
$\mathcal{U}_{(\alpha,\beta)}^{Ev}(X)$ are respectively called
$(\alpha,\beta)$-_linguistic lower and upper approximation_ of $X$ determined
by $\mathcal{R}$ and $Ev$.

Equivalently, by Definition 4, we get

$\mathcal{L}^{Ev}_{(\alpha,\beta)}(X)=POS_{(\alpha,\beta)}^{Ev}(X)$ and
$\mathcal{U}^{Ev}_{(\alpha,\beta)}(X)=POS_{(\alpha,\beta)}^{Ev}(X)\cup
BND_{(\alpha,\beta)}^{Ev}(X)$.

Let $x\in U$, the value $Ev\left(\dfrac{|[x]_{\mathcal{R}}\cap
X|}{|[x]_{\mathcal{R}}|}\right)$ is viewed as the degree of confidence
expressing _how much we can trust that $x$ belongs to $X$_.

The following is an illustrative example.

######  Example 3.

Consider Example 3.1. In terms of generalized rough sets, $X_{Sport}$ can be
approximated by the $(0.8,0.2)$-linguistic rough set

$(\mathcal{L}_{(0.8,0.2)}^{\neg Sm}(X_{Sport}),\mathcal{U}_{(0.8,0.2)}^{\neg
Sm}(X_{Sport}))=(C_{3}\cup C_{4}\cup C_{5},C_{2}\cup C_{3}\cup C_{4}\cup
C_{5}\cup C_{6})=\\\
(\\{u_{11},\ldots,u_{25}\\},\\{u_{6},\ldots,u_{32}\\})\\}.$

Each element $x\in U$ is associated with the value $\neg
Sm\left(\dfrac{|X_{Sport}\cap[x]_{\mathcal{R}}}{[x]_{\mathcal{R}}}\right)$,
which is understood as the degree of confidence expressing _how much we can
trust that $x$ is interested in sports_.

##  4 Connection with TWD methods

In this section, we find a link between the TWD methods based on probabilistic
rough sets and evaluative expressions. In particular, we fix a finite universe
$U$, a subset $X$ of $U$, an equivalence relation $\mathcal{R}$ on $U$, and a
pair of thresholds $(\alpha,\beta)$ such that $0\leq\alpha<\beta\leq 1$, and
we aim to determine for each evaluative expression $Ev$, the class of all
pairs of thresholds like $(\alpha^{\prime},\beta^{\prime})$ so that
$\mathcal{T}_{(\alpha^{\prime},\beta^{\prime})}(X)$ coincides with
$\mathcal{T}_{(\alpha,\beta)}^{Ev}(X)$.

In this paper, we confine to the class $\mathcal{E}^{+}\subset\mathcal{E}$,
which is made of all extensions that are increasing functions, i.e. let
$Ev\in\mathcal{E}$, $Ev\in\mathcal{E}^{+}$ if and only if “$Ev(x)\leq Ev(y)$
for each $x,y\in[0,1]$ such that $x\leq y$”. Examples of evaluative
expressions so that their extension is an increasing function, are _not small_
, _very big_ , and _extremely big_ (see (5), (6), and (7)). However, there
exist evaluative expressions like _small_ that are represented by a decreasing
function and others like _medium_ that are represented by a non-monotonic
function.

In order to obtain the results of this section, we need to define the values
$\alpha^{Ev}_{1}$, $\alpha^{Ev}_{2}$, $\beta^{Ev}_{1}$, and $\beta^{Ev}_{2}$,
which are associated with $\mathcal{T}^{Ev}_{(\alpha,\beta)}(X)$, where
$Ev\in\mathcal{E}$.

######  Definition 6.

Let $Ev\in\mathcal{E}$. If
$POS_{(\alpha,\beta)}^{Ev}(X),NEG_{(\alpha,\beta)}^{Ev}(X)$,
$BND_{(\alpha,\beta)}^{Ev}(X)\neq\emptyset$, then we put

  1. (i)

$\alpha^{Ev}_{1}=\max\left\\{\dfrac{|X\cap[x]_{\mathcal{R}}|}{|[x]_{\mathcal{R}}|}\
|\ x\in BND^{Ev}_{(\alpha,\beta)}(X)\right\\}$,

  2. (ii)

$\alpha^{Ev}_{2}=\min\left\\{\dfrac{|X\cap[x]_{\mathcal{R}}|}{|[x]_{\mathcal{R}}|}\
|\ x\in POS^{Ev}_{(\alpha,\beta)}(X)\right\\}$,

  3. (iii)

$\beta^{Ev}_{1}=\max\left\\{\dfrac{|X\cap[x]_{\mathcal{R}}|}{|[x]_{\mathcal{R}}|}\
|\ x\in NEG^{Ev}_{(\alpha,\beta)}(X)\right\\},$

  4. (iv)

$\beta^{Ev}_{2}=\min\left\\{\dfrac{|X\cap[x]_{\mathcal{R}}|}{|[x]_{\mathcal{R}}|}\
|\ x\in BND^{Ev}_{(\alpha,\beta)}(X)\right\\}.$

######  Example 4.

Consider the universe $U$, its subset $X_{Sport}$, and the pair of thresholds
$(\alpha,\beta)$ that are defined by Example 3.1. Then, the corresponding
positive, negative, and boundary regions are the following:

$POS_{(0.8,0.2)}^{\neg Sm}(X_{Sport})=C_{3}\cup C_{4}\cup C_{5},\ NEG^{\neg
Sm}_{(0.8,0.2)}(X_{Sport})=C_{1},\ \mbox{ and }\\\ BND^{\neg
Sm}_{(0.8,0.2)}(X_{Sport})=C_{2}\cup C_{6}.$

Hence, by (11), we get 333Recall that the equivalence classes of
$\\{[x]_{\mathcal{R}}\ |\ x\in U\\}$ are the sets
$C_{1},C_{2},C_{3},C_{4},C_{5}$, and $C_{6}$.

$\left\\{\dfrac{|X_{Sport}\cap[x]_{\mathcal{R}}|}{|[x]_{\mathcal{R}}|}\ |\
x\in POS^{Ev}_{(\alpha,\beta)}(X_{Sport})\right\\}=\\\
\left\\{\dfrac{|X_{Sport}\cap C_{3}|}{|C_{3}|},\dfrac{|X_{Sport}\cap
C_{4}|}{|C_{4}|},\dfrac{|X_{Sport}\cap
C_{5}|}{|C_{5}|}\right\\}=\\{0.4,0.6,0.8\\};$
$\left\\{\dfrac{|X_{Sport}\cap[x]_{\mathcal{R}}|}{|[x]_{\mathcal{R}}|}\ |\
x\in
NEG^{Ev}_{(\alpha,\beta)}(X_{Sport})\right\\}=\left\\{\dfrac{|X_{Sport}\cap
C_{1}|}{|C_{1}|}\right\\}=\\{0\\};$
$\left\\{\dfrac{|X_{Sport}\cap[x]_{\mathcal{R}}|}{|[x]_{\mathcal{R}}|}\ |\
x\in
BND^{Ev}_{(\alpha,\beta)}(X_{Sport})\right\\}=\left\\{\dfrac{|X_{Sport}\cap
C_{2}|}{|C_{2}|},\dfrac{|X_{Sport}\cap C_{6}|}{|C_{6}|}\right\\}\\\
=\\{0.2,0.14\\}.$

Finally, by Definition 6, $\alpha_{1}^{\neg Sm}=\max\\{0.14,0.2\\}=0.2$,
$\alpha_{2}^{\neg Sm}=\min\\{0.4,0.6,0.8\\}=0.4$, $\beta_{1}^{\neg
Sm}=\max\\{0\\}=0$, and $\beta_{2}^{\neg Sm}=\min\\{0.14,0.2\\}\\\ =0.14$.

If $Ev$ is an increasing function, namely $Ev\in\mathcal{E}^{+}$, then we can
order $\beta_{1}^{Ev}$, $\beta_{2}^{Ev}$, $\alpha_{1}^{Ev}$, and
$\alpha_{2}^{Ev}$ as shown in the following proposition.

######  Proposition 1.

Let $Ev\in\mathcal{E}^{+}$. If
$POS_{(\alpha,\beta)}^{Ev}(X),NEG_{(\alpha,\beta)}^{Ev}(X)$,
$BND_{(\alpha,\beta)}^{Ev}(X)$ $\neq\emptyset$, then
$0\leq\beta_{1}^{Ev}<\beta_{2}^{Ev}\leq\alpha_{1}^{Ev}<\alpha_{2}^{Ev}\leq 1$.

###### Proof.

By Definition 6, it is trivial that
$0\leq\beta_{1}^{Ev},\beta^{Ev}_{2},\alpha^{Ev}_{1},\alpha^{Ev}_{2}\leq 1$.

$(\beta_{1}^{Ev}<\beta_{2}^{Ev})$.

By Definition 6 ((iii) and (iv)),
$\beta_{1}^{Ev}=\dfrac{|X\cap[x_{1}]_{\mathcal{R}}|}{|[x_{1}]_{\mathcal{R}}|}$
with $x_{1}\in NEG_{(\alpha,\beta)}^{Ev}(X)$ and
$\beta^{Ev}_{2}=\dfrac{|X\cap[x_{2}]_{\mathcal{R}}|}{|[x_{2}]_{\mathcal{R}}|}$
with $x_{2}\in BND_{(\alpha,\beta)}^{Ev}(X)$. Then, by Definition 4 ((ii) and
(iii)), $Ev(\beta^{Ev}_{1})\leq\beta$ and $\beta<Ev(\beta_{2}^{Ev})<\alpha$.
Hence, $Ev(\beta^{Ev}_{1})<Ev(\beta^{Ev}_{2})$. Thus, considering that $Ev$ is
an increasing function, we can conclude that $\beta^{Ev}_{1}<\beta^{Ev}_{2}$.

($\alpha^{Ev}_{1}<\alpha^{Ev}_{2}$).

By Definition 6 ((i) and (ii)),
$\alpha_{1}^{Ev}=\dfrac{|X\cap[x_{1}]_{\mathcal{R}}|}{|[x_{1}]_{\mathcal{R}}|}$
with $x_{1}\in BND_{(\alpha,\beta)}^{Ev}(X)$ and
$\alpha^{Ev}_{2}=\dfrac{|X\cap[x_{2}]_{\mathcal{R}}|}{|[x_{2}]_{\mathcal{R}}|}$
with $x_{2}\in POS_{(\alpha,\beta)}^{Ev}(X)$. Thus, by Definition 4 ((iii) and
(i)), $\beta<Ev(\alpha^{Ev}_{1})<\alpha$ and $Ev(\alpha^{Ev}_{2})\geq\alpha$.
Thus, $Ev(\alpha^{Ev}_{1})<Ev(\alpha^{Ev}_{2})$. Hence, considering that $Ev$
is an increasing function, $\alpha^{Ev}_{1}<\alpha^{Ev}_{2}$.

($\beta^{Ev}_{2}\leq\alpha^{Ev}_{1}$).

By Definition 6 ((i) and (iii)),
$\alpha_{1}^{Ev}=\dfrac{|X\cap[x_{1}]_{\mathcal{R}}|}{|[x_{1}]_{\mathcal{R}}|}$
with $x_{1}\in BND_{(\alpha,\beta)}^{Ev}(X)$ and
$\beta_{2}^{Ev}=\dfrac{|X\cap[x_{2}]_{\mathcal{R}}|}{|[x_{2}]_{\mathcal{R}}|}$
with $x_{2}\in BND_{(\alpha,\beta)}^{Ev}(X)$. Therefore, since $x_{1},x_{2}\in
BND_{(\alpha,\beta)}^{Ev}(X)$, $\beta^{Ev}_{2}\leq\alpha^{Ev}_{1}$ clearly
holds.

∎

######  Example 5.

In Example 4, we have shown that $\alpha_{1}^{\neg Sm}=0.2$, $\alpha_{2}^{\neg
Sm}=0.4$, $\beta_{1}^{\neg Sm}=0$, and $\beta_{2}^{\neg Sm}=0.14$. Then,
according to Proposition 1,
$0\leq\beta_{1}^{Ev}<\beta_{2}^{Ev}\leq\alpha_{1}^{Ev}<\alpha_{2}^{Ev}\leq 1$.

The next theorems show that the three regions generated by
$Ev\in\mathcal{E}^{+}$ can be also obtained by using the probabilistic
approach and changing the initial thresholds. We separately analyze the
following cases: all three regions are non-empty (Theorem 1) and only one of
the three regions is empty (Theorems 2-4). The remaining case where only one
region is non-empty (namely, one of the three regions coincides with the
universe) is omitted because not significant.

######  Theorem 1.

Let $Ev\in\mathcal{E}^{+}$ such that
$POS_{(\alpha,\beta)}^{Ev}(X),NEG_{(\alpha,\beta)}^{Ev}(X)$,
$BND_{(\alpha,\beta)}^{Ev}(X)$ $\neq\emptyset$ and let
$\alpha^{\prime},\beta^{\prime}\in[0,1]$ with
$\beta^{\prime}<\alpha^{\prime}$. Then,

$\mathcal{T}_{(\alpha,\beta)}^{Ev}(X)=\mathcal{T}_{(\alpha^{\prime},\beta^{\prime})}(X)$
if and only if $\alpha^{\prime}\in(\alpha^{Ev}_{1},\alpha^{Ev}_{2}]$ and
$\beta^{\prime}\in[\beta^{Ev}_{1},\beta^{Ev}_{2})$.

######  Remark 7.

Before to prove Theorem 1, let us represent the intervals that contain
$\alpha^{\prime}$ and $\beta^{\prime}$ (i.e. the values for generating
$\mathcal{T}_{(\alpha,\beta)}^{Ev}(X)$ with probabilistic rough sets) by
Figure 1. By Definition 6, these intervals separates
$POS_{(\alpha,\beta)}^{Ev}(X)$ from $BND_{(\alpha,\beta)}^{Ev}(X)$ and
$NEG_{(\alpha,\beta)}^{Ev}(X)$ from $BND_{(\alpha,\beta)}^{Ev}(X)$. More
precisely, the value
$Ev\left(\dfrac{|X\cap[x]_{\mathcal{R}}|}{|[x]_{\mathcal{R}}|}\right)$ belongs
to $[0,\beta^{Ev}_{1}]$ when $x\in NEG_{(\alpha,\beta)}^{Ev}(X)$, to
$[\beta^{Ev}_{2},\alpha^{Ev}_{1}]$ when $x\in BND^{Ev}_{(\alpha,\beta)}(X)$,
and to $[\alpha^{Ev}_{2},1]$ when $x\in POS_{(\alpha,\beta)}^{Ev}(X)$.

$\bullet$$0$$[$$\beta_{1}^{Ev}$$)$$\beta^{Ev}_{2}$$($$\alpha^{Ev}_{1}$$]$$\alpha^{Ev}_{2}$$\bullet$$1$

Figure 1: Intervals $[\beta_{1}^{Ev},\beta_{2}^{Ev})$ and
$(\alpha_{1}^{Ev},\alpha_{2}^{Ev}]$.

###### Proof.

$(\Leftarrow).$ Let $\alpha^{\prime}\in(\alpha^{Ev}_{1},\alpha^{Ev}_{2}]$ and
let $\beta^{\prime}\in[\beta^{Ev}_{1},\beta^{Ev}_{2})$, we need to prove that
$POS_{(\alpha,\beta)}^{Ev}(X)=POS_{(\alpha^{\prime},\beta^{\prime})}(X)$,
$NEG_{(\alpha,\beta)}^{Ev}(X)=NEG_{(\alpha^{\prime},\beta^{\prime})}(X)$, and
$BND_{(\alpha,\beta)}^{Ev}$ $(X)=BND_{(\alpha^{\prime},\beta^{\prime})}(X)$.

($POS_{(\alpha,\beta)}^{Ev}(X)=POS_{(\alpha^{\prime},\beta^{\prime})}(X)$).

Let $\bar{x}\in POS^{Ev}_{(\alpha,\beta)}(X)$, then
$\dfrac{|X\cap[\bar{x}]_{\mathcal{R}}|}{|[\bar{x}]_{\mathcal{R}}|}\geq\alpha^{Ev}_{2}$
from Definition 6 (ii). Moreover, $\alpha^{\prime}\leq\alpha^{Ev}_{2}$ because
$\alpha^{\prime}\in(\alpha^{Ev}_{1},\alpha^{Ev}_{2}]$. Consequently,
$\dfrac{|X\cap[\bar{x}]_{\mathcal{R}}|}{|[\bar{x}]_{\mathcal{R}}|}\geq\alpha^{\prime}$.
Finally, $\bar{x}\in POS_{(\alpha^{\prime},\beta^{\prime})}(X)$ from
Definition 1 (i).

Let $\bar{x}\in POS_{(\alpha^{\prime},\beta^{\prime})}(X)$, then
$\dfrac{|X\cap[\bar{x}]_{\mathcal{R}}|}{|[\bar{x}]_{\mathcal{R}}|}\geq\alpha^{\prime}$
from Definition 1 (i). By the previous inequality and
$\alpha^{\prime}>\alpha^{Ev}_{1}$, we get
$\dfrac{|X\cap[\bar{x}]_{\mathcal{R}}|}{|[\bar{x}]_{\mathcal{R}}|}>\alpha^{Ev}_{1}$.
Hence, considering that $\alpha^{Ev}_{1}$ is the maximum of
$\left\\{\dfrac{|X\cap[x]_{\mathcal{R}}|}{|[x]_{\mathcal{R}}|}\ |\ x\in
BND_{(\alpha,\beta)}^{Ev}(X)\right\\}$ (see Definition 6(i)), we are sure that
$\bar{x}\notin BND_{(\alpha,\beta)}^{Ev}(X)$. Furthermore,
$\dfrac{|X\cap[\bar{x}]_{\mathcal{R}}|}{|[\bar{x}]_{\mathcal{R}}|}>\alpha^{Ev}_{1}$
and $\beta_{1}^{Ev}<\alpha^{Ev}_{1}$ (see Proposition 1) imply that
$\dfrac{|X\cap[\bar{x}]_{\mathcal{R}}|}{|[\bar{x}]_{\mathcal{R}}|}>\beta^{Ev}_{1}$.
Thus, considering that $\beta^{Ev}_{1}$ is the maximum of
$\left\\{\dfrac{|X\cap[x]_{\mathcal{R}}|}{|[x]_{\mathcal{R}}|}\ |\ x\in
NEG_{(\alpha,\beta)}^{Ev}(X)\right\\}$ (see Definition 6(iii)), we have
$\bar{x}\notin NEG_{(\alpha,\beta)}^{Ev}(X)$. Ultimately, by (10), $\bar{x}\in
POS_{(\alpha,\beta)}^{Ev}(X)$.

($BND_{(\alpha,\beta)}^{Ev}(X)=BND_{(\alpha^{\prime},\beta^{\prime})}(X)$).

Let $\bar{x}\in BND_{(\alpha,\beta)}^{Ev}(X)$. By Definition 6 ((i) and (iv)),
$\beta_{2}^{Ev}\leq\dfrac{|X\cap[\bar{x}]_{\mathcal{R}}|}{|[\bar{x}]_{\mathcal{R}}|}\leq\alpha_{1}^{Ev}$.
Moreover, by hypothesis, $\beta^{\prime}<\beta_{2}^{Ev}$ and
$\alpha^{\prime}>\alpha_{1}^{Ev}$. Thus, we can conclude that
$\beta^{\prime}<\dfrac{|X\cap[\bar{x}]_{\mathcal{R}}|}{|[\bar{x}]_{\mathcal{R}}|}<\alpha^{\prime}$,
namely $\bar{x}\in BND_{(\alpha^{\prime},\beta^{\prime})}(X)$ from Definition
1 (iii).

Let $\bar{x}\in BND_{(\alpha^{\prime},\beta^{\prime})}(X)$, then
$\beta^{\prime}<\dfrac{|X\cap[\bar{x}]_{\mathcal{R}}|}{|[\bar{x}]_{\mathcal{R}}|}<\alpha^{\prime}$
from Definition 1 (iii). By hypothesis, $\beta^{Ev}_{1}\leq\beta^{\prime}$ and
$\alpha^{\prime}\leq\alpha^{Ev}_{2}$. Hence, we know that
$\beta^{Ev}_{1}<\dfrac{|X\cap[\bar{x}]_{\mathcal{R}}|}{|[\bar{x}]_{\mathcal{R}}|}<\alpha^{Ev}_{2}$.
By Definition 6 (iii),
$\beta^{Ev}_{1}<\dfrac{|X\cap[\bar{x}]_{\mathcal{R}}|}{|[\bar{x}]_{\mathcal{R}}|}$
implies that $\bar{x}\notin NEG_{(\alpha,\beta)}^{Ev}(X)$. Furthermore, by
Definition 6 (ii),
$\dfrac{|X\cap[\bar{x}]_{\mathcal{R}}|}{|[\bar{x}]_{\mathcal{R}}|}<\alpha^{Ev}_{2}$
implies that $\bar{x}\notin POS^{Ev}_{(\alpha,\beta)}(X)$. Ultimately, by
(10), $\bar{x}\in BND^{Ev}_{(\alpha,\beta)}(X)$.

($NEG_{(\alpha,\beta)}^{Ev}(X)=NEG_{(\alpha^{\prime},\beta^{\prime})}(X)$).

We have previously shown that $POS_{(\alpha,\beta)}^{Ev}(X)$
$=POS_{(\alpha^{\prime},\beta^{\prime})}(X)$ and
$BND_{(\alpha,\beta)}^{Ev}(X)=BND_{(\alpha^{\prime},\beta^{\prime})}(X)$. So,
by (3) and (10), it is clear that
$NEG_{(\alpha,\beta)}^{Ev}(X)=NEG_{(\alpha^{\prime},\beta^{\prime})}(X)$.

$(\Rightarrow).$ Let
$\mathcal{T}_{(\alpha,\beta)}^{Ev}(X)=\mathcal{T}_{(\alpha^{\prime},\beta^{\prime})}(X)$,
we intend to prove that $\beta^{Ev}_{1}\leq\beta^{\prime}<\beta^{Ev}_{2}$ and
$\alpha^{Ev}_{1}<\alpha^{\prime}\leq\alpha^{Ev}_{2}$.

($\alpha^{\prime}\leq\alpha^{Ev}_{2}$).

Let $x_{2}\in U$ such that
$\alpha^{Ev}_{2}=\dfrac{|X\cap[x_{2}]_{\mathcal{R}}|}{|[x_{2}]_{\mathcal{R}}|}$.
By Definition 6 (ii), $x_{2}\in POS_{(\alpha,\beta)}^{Ev}(X)$. Hence,
$\alpha^{\prime}>\alpha^{Ev}_{2}$ means that
$\dfrac{|X\cap[x_{2}]_{\mathcal{R}}|}{|[x_{2}]_{\mathcal{R}}|}<\alpha^{\prime}$.
Thus, $x_{2}\notin POS_{(\alpha^{\prime},\beta^{\prime})}(X)$ from Definition
1 (i). This contradicts that
$POS^{Ev}_{(\alpha,\beta)}(X)=POS_{(\alpha^{\prime},\beta^{\prime})}(X)$.
Thus, it must be true that $\alpha^{\prime}\leq\alpha^{Ev}_{2}$.

($\alpha^{Ev}_{1}<\alpha^{\prime}$).

Let $x_{1}\in U$ such that
$\alpha^{Ev}_{1}=\dfrac{|X\cap[x_{1}]_{\mathcal{R}}|}{|[x_{1}]_{\mathcal{R}}|}$.
By Definition 6 (i), $x_{1}\in BND^{Ev}_{(\alpha,\beta)}(X)$. If
$\alpha^{Ev}_{1}\geq\alpha^{\prime}$, then
$\dfrac{|X\cap[x_{1}]_{\mathcal{R}}|}{|[x_{1}]_{\mathcal{R}}|}\geq\alpha^{\prime}$.
So, $x_{1}\in POS_{(\alpha^{\prime},\beta^{\prime})}(X)$ from Definition 1
(i). This contradicts that
$POS^{Ev}_{(\alpha,\beta)}(X)=POS_{(\alpha^{\prime},\beta^{\prime})}(X)$.
Thus, it must be true that $\alpha^{Ev}_{1}<\alpha^{\prime}$.

$(\beta_{1}^{Ev}\leq\beta^{\prime})$.

Let $x_{1}\in U$ such that
$\beta_{1}^{Ev}=\dfrac{|X\cap[x_{1}]_{\mathcal{R}}|}{|[x_{1}]_{\mathcal{R}}|}$.
By Definition 6(iii), $x_{1}\in NEG^{Ev}_{(\alpha,\beta)}(X)$. If
$\beta_{1}^{Ev}>\beta^{\prime}$, then
$\dfrac{|X\cap[x_{1}]_{\mathcal{R}}|}{|[x_{1}]_{\mathcal{R}}|}>\beta^{\prime}$,
which implies that $x_{1}\notin NEG_{(\alpha^{\prime},\beta^{\prime})}(X)$
from Definition 1(ii). This contradicts that
$NEG^{Ev}_{(\alpha,\beta)}(X)=NEG_{(\alpha^{\prime},\beta^{\prime})}(X)$.
Thus, it must be true that $\beta_{1}^{Ev}\leq\beta^{\prime}$.

$(\beta^{\prime}<\beta^{Ev}_{2})$.

Let $x_{2}\in U$ such that
$\beta^{Ev}_{2}=\dfrac{|X\cap[x_{2}]_{\mathcal{R}}|}{|[x_{2}]_{\mathcal{R}}|}$.
By Definition 6(iv), $x_{2}\in BND^{Ev}_{(\alpha,\beta)}(X)$. Also, if
$\beta^{\prime}\geq\beta^{Ev}_{2}$, then
$\dfrac{|X\cap[x_{2}]_{\mathcal{R}}|}{|[x_{2}]_{\mathcal{R}}|}\leq\beta^{\prime}$,
which implies that $x_{2}\in NEG_{(\alpha^{\prime},\beta^{\prime})}(X)$ from
Definition 1 (ii). This contradicts that
$BND^{Ev}_{(\alpha,\beta)}(X)=BND_{(\alpha^{\prime},\beta^{\prime})}(X)$.
Thus, it must be true that $\beta^{\prime}<\beta^{Ev}_{2}$.

∎

######  Example 6.

Consider Example 3.1, $\neg Sm$ is an increasing function and all the three
regions of $\mathcal{T}_{(0.8,0.2)}^{\neg Sm}(X_{Sport})$ are non-empty. In
Example 4, we have found that $\alpha_{1}^{\neg Sm}=0.2$, $\alpha_{2}^{\neg
Sm}=0.4$, $\beta_{1}^{\neg Sm}=0$, and $\beta_{2}^{\neg Sm}=0.14$. Therefore,
according to Theorem 1, we get $\mathcal{T}^{\neg
Sm}_{(0.8,0.2)}(X_{Sport})=\mathcal{T}_{(\alpha^{\prime},\beta^{\prime})}(X_{Sport})$
for each $(\alpha^{\prime},\beta^{\prime})$ such that
$\alpha^{\prime}\in(0.2,0.4]$ and $\beta^{\prime}\in[0,0.14)$

For example, we can easily verify that $\mathcal{T}^{\neg
Sm}_{(0.8,0.2)}(X_{Sport})=\mathcal{T}_{(0.3,0.1)}(X_{Sport})$. Indeed, by
(11) and by Definition 1,

  * •

$POS_{(0.3,0.1)}(X_{Sport})=\left\\{x\in U\ |\
\dfrac{|X\cap[x]_{\mathcal{R}}|}{|[x]_{\mathcal{R}}|}\geq
0.3\right\\}=C_{3}\cup C_{4}\cup C_{5}$,

  * •

$NEG_{(0.3,0.1)}(X_{Sport})=\left\\{x\in U\ |\
\dfrac{|X\cap[x]_{\mathcal{R}}|}{|[x]_{\mathcal{R}}|}\leq 0.1\right\\}=C_{1}$,
and

  * •

$BND_{(0.3,0.1)}(X_{Sport})=\left\\{x\in U\ |\
0.1<\dfrac{|X\cap[x]_{\mathcal{R}}|}{|[x]_{\mathcal{R}}|}<0.3\right\\}=C_{2}\cup
C_{6}$.

By Theorem 1, we can connect linguistic rough sets with classical rough sets.
More precisely, the following corollary holds.

######  Corollary 1.

Let $Ev\in\mathcal{E}^{+}$ with
$POS_{(\alpha,\beta)}^{Ev}(X),NEG_{(\alpha,\beta)}^{Ev}(X)$,
$BND_{(\alpha,\beta)}^{Ev}(X)$ $\neq\emptyset$. Then,

$(\mathcal{L}_{(\alpha,\beta)}^{Ev}(X),\mathcal{U}_{(\alpha,\beta)}^{Ev}(X))=(\mathcal{L}(X),\mathcal{U}(X))$
444Recall that
$(\mathcal{L}_{(\alpha,\beta)}^{Ev}(X),\mathcal{U}_{(\alpha,\beta)}^{Ev}(X))$
is the linguistic rough set of $X$ given by Definition 5 and
$(\mathcal{L}(X),\mathcal{U}(X))$ is the classical rough set of $X$ given by
Eq. (4). if and only if $\beta^{Ev}_{1}=0$ and $\alpha^{Ev}_{2}=1$.

###### Proof.

$(\Rightarrow).$ Suppose that
$(\mathcal{L}_{(\alpha,\beta)}^{Ev}(X),\mathcal{U}_{(\alpha,\beta)}^{Ev}(X))$
is the rough set of $X$. Then, by (4), let $x\in U$, $x\in
POS_{(\alpha,\beta)}^{Ev}(X)$ if and only if $[x]_{\mathcal{R}}\subseteq X$.
The latter means that
$\dfrac{|X\cap[x]_{\mathcal{R}}|}{|[x]_{\mathcal{R}}|}=1$ for each $x\in
POS_{(\alpha,\beta)}^{Ev}(X)$. Hence, By Definition 6 (ii),
$\alpha_{2}^{Ev}=1$.

By (4), let $x\in U$, $x\in POS_{(\alpha,\beta)}^{Ev}(X)\cup
BND_{(\alpha,\beta)}^{Ev}(X)$ if and only if $[x]_{\mathcal{R}}\cap
X\neq\emptyset$. Since $POS_{(\alpha,\beta)}^{Ev}(X)\cup
BND_{(\alpha,\beta)}^{Ev}(X)=U\setminus NEG_{(\alpha,\beta)}^{Ev}(X)$, we know
that $\dfrac{|X\cap[x]_{\mathcal{R}}|}{|[x]_{\mathcal{R}}|}=0$ for each $x\in
NEG_{(\alpha,\beta)}^{Ev}(X)$. Finally, by Definition 6 (iii),
$\beta^{Ev}_{1}=0$.

$(\Leftarrow)$. Suppose that $\alpha^{Ev}_{2}=1$ and $\beta^{Ev}_{1}=0$.
Trivially, $\alpha^{Ev}_{2}\in(\alpha^{Ev}_{1},\alpha^{Ev}_{2}]$ and
$\beta^{Ev}_{1}\in[\beta^{Ev}_{1},\beta^{Ev}_{2})$. Then, by Theorem 1,
$(POS_{(\alpha,\beta)}^{Ev}(X),$ $POS_{(\alpha,\beta)}^{Ev}(X)\cup
BND_{(\alpha,\beta)}^{Ev}(X))=(POS_{(1,0)}(X),POS_{(1,0)}(X)\cup
BND_{(1,0)}(X))$. Moreover, by Remark 2,
$(\mathcal{L}_{(1,0)}(X),\mathcal{U}_{(1,0)}(X))=(POS_{(1,0)}(X),$
$POS_{(1,0)}(X)\cup BND_{(1,0)}(X))$ coincides with the rough set
$(\mathcal{L}(X),\mathcal{U}(X))$ given by (4). ∎

######  Example 7.

Consider the universe $U=\\{u_{1},\ldots,u_{20}\\}$ and the evaluative
expression _Very big_ , which is modelled by (6). We suppose that $U$ is
partitionated into three equivalence classes:
$C_{1}=\\{u_{1},\ldots,u_{5}\\}$, $C_{2}=\\{u_{6},\ldots,u_{10}\\}$, and
$C_{3}=\\{u_{11},\ldots,u_{20}\\}$. If $X=\\{u_{6},\ldots,u_{19}\\}$, we can
simply prove that
$(\mathcal{L}^{BiVe}_{(0.7,0.3)}(X),\mathcal{E}^{BiVe}_{(0.7,0.3)}(X))=(\mathcal{L}(X),\mathcal{U}(X))$.
Indeed, we get

$\dfrac{|X\cap[x]_{\mathcal{R}}|}{|[x]_{\mathcal{R}}|}=\begin{cases}0&\mbox{
if }x\in C_{1},\\\ 0.9&\mbox{ if }x\in C_{3},\\\ 1&\mbox{ if }x\in
C_{2}.\end{cases}\mbox{ and
}BiVe\left(\frac{|X\cap[x]_{\mathcal{R}}|}{|[x]_{\mathcal{R}}|}\right)=\begin{cases}0&\mbox{
if }x\in C_{1},\\\ 0.59&\mbox{ if }x\in C_{3},\\\ 1&\mbox{ if }x\in
C_{2}.\end{cases}$

Thus,

$POS_{(0.7,0.3)}^{BiVe}(X)=\\{C_{i}\ |\ i\in\\{1,2,3\\}\mbox{ and
}BiVe\left(\dfrac{|X\cap C_{i}|}{|C_{i}|}\right)\geq 0.7\\}=C_{2}$;

$NEG_{(0.7,0.3)}^{BiVe}(X)=\\{C_{i}\ |\ i\in\\{1,2,3\\}\mbox{ and
}BiVe\left(\dfrac{|X\cap C_{i}|}{|C_{i}|}\right)\leq 0.3\\}=C_{1}$;

$BND_{(0.7,0.3)}^{BiVe}(X)=\\{C_{i}\ |\ i\in\\{1,2,3\\}\mbox{ and
}0.3<BiVe\left(\dfrac{|X\cap C_{i}|}{|C_{i}|}\right)<0.7\\}=C_{3}$.

Also, by Definition 6, $\beta_{1}^{BiVe}=0$, $\alpha_{2}^{BiVe}=1$,
$\alpha_{1}^{BiVe}=\beta_{2}^{BiVe}=0.9$. Since the hypothesis of the previous
corollary is satisfied, we expect that
$(\mathcal{L}(X),\mathcal{U}(X))=(\mathcal{L}^{BiVe}_{(0.7,0.3)}(X),\mathcal{U}^{BiVe}_{(0.7,0.3)}(X))=(C_{2},C_{2}\cup
C_{3})$. We can immediately verify that this is true: $\mathcal{L}(X)=C_{2}$
because $C_{2}$ is the unique class among $C_{1},C_{2}$, and $C_{3}$ that is
included in $X$; moreover, $\mathcal{U}(X)=C_{2}\cup C_{3}$ because $X\cap
C_{2}\neq\emptyset$ and $X\cap C_{3}\neq\emptyset$, while $X\cap
C_{1}=\emptyset$.

We are now going to deal with the cases where one of
$BND^{Ev}_{(\alpha,\beta)},POS^{Ev}_{(\alpha,\beta)},$
$NEG^{Ev}_{(\alpha,\beta)}$ is empty.

######  Theorem 2.

Let $Ev\in\mathcal{E}^{+}$ such that $BND_{(\alpha,\beta)}^{Ev}=\emptyset$ and
$POS_{(\alpha,\beta)}^{Ev},NEG_{(\alpha,\beta)}^{Ev}\neq\emptyset$. Let
$\alpha^{\prime},\beta^{\prime}\in[0,1]$ such that
$\beta^{\prime}<\alpha^{\prime}$. Then,

$\mathcal{T}_{(\alpha,\beta)}^{Ev}(X)=\mathcal{T}_{(\alpha^{\prime},\beta^{\prime})}(X)$
if and only if
$\beta^{Ev}_{1}\leq\beta^{\prime}<\alpha^{\prime}\leq\alpha^{Ev}_{2}$.

###### Proof.

$(\Leftarrow).$ Let $\alpha^{\prime},\beta^{\prime}\in[0,1]$ such that
$\beta^{Ev}_{1}\leq\beta^{\prime}<\alpha^{\prime}\leq\alpha^{Ev}_{2}$, we need
to prove that
$POS_{(\alpha,\beta)}^{Ev}(X)=POS_{(\alpha^{\prime},\beta^{\prime})}(X)$,
$NEG_{(\alpha,\beta)}^{Ev}(X)=NEG_{(\alpha^{\prime},\beta^{\prime})}(X)$, and
$BND_{(\alpha,\beta)}^{Ev}$ $(X)=BND_{(\alpha^{\prime},\beta^{\prime})}(X)$.

$(POS_{(\alpha,\beta)}^{Ev}(X)=POS_{(\alpha^{\prime},\beta^{\prime})}(X))$.

Let $\bar{x}\in POS_{(\alpha,\beta)}^{Ev}(X)$. Then,
$\dfrac{|X\cap[\bar{x}]_{\mathcal{R}}|}{|[\bar{x}]_{\mathcal{R}}|}\geq\alpha^{Ev}_{1}$
from Definition 6 (i). By hypothesis, $\alpha^{\prime}\leq\alpha_{2}^{Ev}$.
Finally, by the previous two inequalities, we obtain that
$\dfrac{|X\cap[\bar{x}]_{\mathcal{R}}|}{|[\bar{x}]_{\mathcal{R}}|}\geq\alpha^{\prime}$.
Namely, $\bar{x}\in POS_{(\alpha^{\prime},\beta^{\prime})}(X)$ from Definition
1 (i).

Let $\bar{x}\in POS_{(\alpha^{\prime},\beta^{\prime})}(X)$. Then,
$\dfrac{|X\cap[\bar{x}]_{\mathcal{R}}|}{|[\bar{x}]_{\mathcal{R}}|}\geq\alpha^{\prime}$
by Definition 1 (i). Let $x_{1}\in U$ such that
$\beta^{Ev}_{1}=\dfrac{|X\cap[x_{1}]_{\mathcal{R}}|}{|[x_{1}]_{\mathcal{R}}|}$.
Moreover, by hypothesis $\beta^{Ev}_{1}<\alpha^{\prime}$. So, by the previous
two inequalities, we get
$\dfrac{|X\cap[\bar{x}]_{\mathcal{R}}|}{|[\bar{x}]_{\mathcal{R}}|}>\beta^{Ev}_{1}$.
Then, by Definition 6 (iii), $\bar{x}\notin NEG_{(\alpha,\beta)}^{Ev}(X)$.
Lastly, by (10) and by the hypothesis
$BND_{(\alpha,\beta)}^{Ev}(X)=\emptyset$, we can conclude that $\bar{x}\in
POS_{(\alpha,\beta)}^{Ev}(X)$.

$(NEG_{(\alpha,\beta)}^{Ev}(X)=NEG_{(\alpha^{\prime},\beta^{\prime})}(X))$.

Let $\bar{x}\in NEG_{(\alpha,\beta)}^{Ev}(X)$. Then, by Definition 6 (iii),
$\dfrac{|X\cap[\bar{x}]_{\mathcal{R}}|}{|[\bar{x}]_{\mathcal{R}}|}\leq\beta^{Ev}_{1}$.
Additionally, we know that $\beta^{Ev}_{1}\leq\beta^{\prime}$ from hypothesis.
Then,
$\dfrac{|X\cap[\bar{x}]_{\mathcal{R}}|}{|[\bar{x}]_{\mathcal{R}}|}\leq\beta^{\prime}$,
namely $\bar{x}\in NEG_{(\alpha^{\prime},\beta^{\prime})}(X)$ from Definition
1 (ii).

Let $\bar{x}\in NEG_{(\alpha^{\prime},\beta^{\prime})}(X)$, then
$\dfrac{|X\cap[\bar{x}]_{\mathcal{R}}|}{|[\bar{x}]_{\mathcal{R}}|}\leq\beta^{\prime}$
from Definition 1 (ii). By hypothesis, $\beta^{\prime}<\alpha^{Ev}_{2}$. Thus,
by Definition 6 (ii), $\bar{x}$ cannot belong to
$POS_{(\alpha,\beta)}^{Ev}(X)$. So, by (10) and the hypothesis
$BND_{(\alpha,\beta)}^{Ev}(X)=\emptyset$, we can deduce that $\bar{x}\in
NEG_{(\alpha,\beta)}^{Ev}(X).$

$(BND_{(\alpha,\beta)}^{Ev}(X)=BND_{(\alpha^{\prime},\beta^{\prime})}(X))$.

This equality follows from
$POS_{(\alpha,\beta)}^{Ev}(X)=POS_{(\alpha^{\prime},\beta^{\prime})}(X)$ and
$NEG_{(\alpha,\beta)}^{Ev}(X)=NEG_{(\alpha^{\prime},\beta^{\prime})}(X)$,
considering that the sets
$POS_{(\alpha,\beta)}^{Ev}(X),NEG_{(\alpha,\beta)}^{Ev}(X)$, and
$BND_{(\alpha,\beta)}^{Ev}(X)$ (as well as
$POS_{(\alpha^{\prime},\beta^{\prime})}(X)$,
$NEG_{(\alpha^{\prime},\beta^{\prime})}(X)$, and
$BND_{(\alpha^{\prime},\beta^{\prime})}(X)$) cover the universe $U$ (see (3)
and (10)).

$(\Leftarrow).$ Let
$\mathcal{T}_{(\alpha,\beta)}^{Ev}(X)=\mathcal{T}_{(\alpha^{\prime},\beta^{\prime})}(X)$,
we intend to prove that $\beta^{Ev}_{1}\leq\beta^{\prime}$ and
$\alpha^{\prime}\leq\alpha^{Ev}_{2}$.

$(\beta^{Ev}_{1}\leq\beta^{\prime})$.

Let $x_{1}\in U$ such that
$\beta^{Ev}_{1}=\dfrac{|X\cap[x_{1}]_{\mathcal{R}}|}{|[x_{1}]_{\mathcal{R}}|}$.
Of course, $x_{1}\in NEG^{Ev}_{(\alpha,\beta)}(X)$ from Definition 6 (iii).

It is clear that the inequality $\beta^{Ev}_{1}>\beta^{\prime}$ leads to a
contradiction:

  * •

if $\beta^{Ev}_{1}>\beta^{\prime}$, then $\bar{x}\notin
NEG_{(\alpha^{\prime},\beta^{\prime})}(X)$ from Definition 6 (iii);

  * •

but, this contradicts that
$NEG^{Ev}_{(\alpha,\beta)}(X)=NEG_{(\alpha^{\prime},\beta^{\prime})}(X)$.

So, $\beta^{Ev}_{1}\leq\beta^{\prime}$ must hold.

$(\alpha^{\prime}\leq\alpha^{Ev}_{2})$.

Let $x_{2}\in U$ such that
$\alpha^{Ev}_{2}=\dfrac{|X\cap[x_{2}]_{\mathcal{R}}|}{|[x_{2}]_{\mathcal{R}}|}$.
Then, $x_{2}\in POS^{Ev}_{(\alpha,\beta)}(X)$ from Definition 6 (ii).

It is clear that the inequality $\alpha^{\prime}>\alpha^{Ev}_{2}$ leads to a
contradiction:

  * •

if $\alpha^{\prime}>\alpha^{Ev}_{2}$, then $x_{2}\notin
POS_{(\alpha^{\prime},\beta^{\prime})}(X)$ from Definition 6 (ii);

  * •

but, this contradicts that
$POS^{Ev}_{(\alpha,\beta)}(X)=POS_{(\alpha^{\prime},\beta^{\prime})}(X)$.

Finally, $\alpha^{\prime}\leq\alpha^{Ev}_{2}$ must hold.

∎

Examples of evaluative expressions satisfying the hypothesis of Theorem 2 can
be obtained from the class defined by (8). Indeed, let $t\in[0,1]$,
$\Delta_{t}$ is trivially an increasing function (i.e.
$\Delta_{t}\in\mathcal{E}^{+}$) and the boundary region determined by
$\Delta_{t}$ is always empty as shown by the following proposition. In
addition, in Proposition 2, the formula of the three regions that are related
to $\Delta_{t}$ is rewritten so that the thresholds $\alpha$ and $\beta$ do
not appear in it.

######  Proposition 2.

Let $t\in[0,1]$ and let $\alpha,\beta\in[0,1]$ such that $\beta<\alpha$, then

  1. (a)

$POS_{(\alpha,\beta)}^{\Delta_{t}}(X)=\left\\{x\in U\ |\
\dfrac{|X\cap[x]_{\mathcal{R}}|}{|[x]_{\mathcal{R}}|}\geq t\right\\}$;

  2. (b)

$NEG_{(\alpha,\beta)}^{\Delta_{t}}(X)=\left\\{x\in U\ |\
\dfrac{|X\cap[x]_{\mathcal{R}}|}{|[x]_{\mathcal{R}}|}<t\right\\}$;

  3. (c)

$BND_{(\alpha,\beta)}^{\Delta_{t}}(X)=\emptyset$.

###### Proof.

(a). Let $\bar{x}\in U$. Thus, $\bar{x}\in POS_{(\alpha,\beta)}^{Ev}(X)$ if
and only if

$\Delta_{t}\left(\dfrac{|X\cap[\bar{x}]_{\mathcal{R}}|}{|[\bar{x}]_{\mathcal{R}}|}\right)\geq\alpha$
(13)

from Definition 4 (i).

By (8), the inequality (13) is true if and only if
$\dfrac{|X\cap[\bar{x}]_{\mathcal{R}}|}{|[\bar{x}]_{\mathcal{R}}|}\geq t$.

(b). Let $\bar{x}\in U$. Then, $\bar{x}\in NEG_{(\alpha,\beta)}^{Ev}(X)$ if
and only if

$\Delta_{t}\left(\dfrac{|X\cap[\bar{x}]_{\mathcal{R}}|}{|[\bar{x}]_{\mathcal{R}}|}\right)\leq\beta$
(14)

from Definition 4 (ii).

By (8), the inequality (14) is true if and only if
$\dfrac{|X\cap[x]_{\mathcal{R}}|}{|[x]_{\mathcal{R}}|}<t$.

(c). Notice that $\\{x\in U\ |\
\dfrac{|X\cap[x]_{\mathcal{R}}|}{|[x]_{\mathcal{R}}|}\geq t\\}\cup\\{x\in U\
|\ \dfrac{|X\cap[x]_{\mathcal{R}}|}{|[x]_{\mathcal{R}}|}<t\\}=U$. Moreover, we
have proved that $POS_{(\alpha,\beta)}^{Ev}(X)=\\{x\in U\ |\
\dfrac{|X\cap[x]_{\mathcal{R}}|}{|[x]_{\mathcal{R}}|}\geq t\\}$ and
$NEG_{(\alpha,\beta)}^{Ev}(X)=\\{x\in U\ |\
\dfrac{|X\cap[x]_{\mathcal{R}}|}{|[x]_{\mathcal{R}}|}<t\\}$. Hence, according
to (10), $BND_{(\alpha,\beta)}^{Ev}(X)$ must be empty. ∎

######  Example 8.

Let us focus on $\mathcal{T}^{\Delta_{0.5}}_{(\alpha,\beta)}(X)$, where $U$,
$X$, and $\mathcal{R}$ are defined in Example 7. By Proposition 2, it is easy
to verify that $POS_{(\alpha,\beta)}^{\Delta_{0.5}}(X)=C_{2}\cup C_{3}$,
$NEG_{(\alpha,\beta)}^{\Delta_{0.5}}(X)=C_{1}$, and
$BND_{(\alpha,\beta)}^{\Delta_{0.5}}(X)=\emptyset$ for each
$\alpha,\beta\in[0,1]$ with $\beta<\alpha$. Furthermore, according to Theorem
2, $POS_{(\alpha^{\prime},\beta^{\prime})}(X)=C_{2}\cup C_{3}$,
$NEG_{(\alpha^{\prime},\beta^{\prime})}(X)$ $=C_{1}$, and
$BND_{(\alpha^{\prime},\beta^{\prime})}(X)=\emptyset$, for each
$\alpha^{\prime},\beta^{\prime}\in[0,1]$ such that
$\beta_{1}^{\Delta_{0.5}}\leq\beta^{\prime}<\alpha^{\prime}\leq\alpha_{2}^{\Delta_{0.5}}$,
where $\beta_{1}^{\Delta_{0.5}}=0$, $\alpha_{2}^{\Delta_{0.5}}=0.9$. For
example, if we choose $\alpha^{\prime}=0.2$ and $\beta^{\prime}=0.7$, we
obtain $POS_{(0.7,0.2)}(X)=C_{2}\cup C_{3}$, $NEG_{(0.7,0.2)}(X)=C_{1}$, and
$BND_{(0.7,0.2)}(X)=\emptyset$.

Now, let us suppose that the negative region is empty.

######  Theorem 3.

Let $Ev\in\mathcal{E}^{+}$ such that $NEG_{(\alpha,\beta)}^{Ev}=\emptyset$ and
$POS_{(\alpha,\beta)}^{Ev},BND_{(\alpha,\beta)}^{Ev}\neq\emptyset$. Let
$\alpha^{\prime},\beta^{\prime}\in[0,1]$ such that
$\beta^{\prime}<\alpha^{\prime}$. Then,

$\mathcal{T}_{(\alpha,\beta)}^{Ev}(X)=\mathcal{T}_{(\alpha^{\prime},\beta^{\prime})}(X)$
if and only if $\beta^{\prime}\in[0,\beta^{Ev}_{2})$ and
$\alpha^{\prime}\in(\alpha^{Ev}_{1},\alpha^{Ev}_{2}]$.

###### Proof.

The proof is similar to that of Theorems 1 and 2. So, it is omitted. ∎

######  Example 9.

Let $U=\\{u_{1},\ldots,u_{30}\\}$. We supposed that $U$ is divided into the
following equivalence classes: $C_{1}=\\{u_{1},\ldots,u_{5}\\}$,
$C_{2}=\\{u_{6},\ldots,u_{10}\\}$, and $C_{3}=\\{u_{11},\ldots,u_{30}\\}$.

Also, let $X=\\{u_{1},\ldots u_{28}\\}$, we are interested in
$\mathcal{T}^{BiVe}_{(0.8,0.4)}(X)$. Then,

$\dfrac{|X\cap[x]_{\mathcal{R}}|}{|[x]_{\mathcal{R}}|}=\begin{cases}0.9&\mbox{
if }x\in C_{3},\\\ 1&\mbox{ if }x\in C_{1}\cup C_{2}.\end{cases}$

and

$BiVe\left(\frac{|X\cap[x]_{\mathcal{R}}|}{|[x]_{\mathcal{R}}|}\right)=\begin{cases}0.58&\mbox{
if }x\in C_{3},\\\ 1&\mbox{ if }x\in C_{1}\cup C_{2}.\end{cases}$

Thus, by Definition 4, $POS_{(0.8,0.4)}^{BiVe}(X)=C_{1}\cup C_{2}$,
$BND_{(0.8,0.4)}^{BiVe}(X)=C_{3}$, and $NEG_{(0.8,0.4)}^{BiVe}(X)=\emptyset$.
Moreover, $\beta^{Ev}_{2}=0.9$, $\alpha_{1}^{BiVe}=0.9$, and
$\alpha_{2}^{BiVe}=1$. So, according to Theorem 3,
$POS_{(\alpha^{\prime},\beta^{\prime})}(X)=C_{1}\cup C_{2}$,
$BND_{(\alpha^{\prime},\beta^{\prime})}(X)=C_{3}$, and
$NEG_{(\alpha^{\prime},\beta^{\prime})}(X)=C_{3}$ for each
$\beta^{\prime}\in[0,0.9)$ and $\alpha^{\prime}\in(0.9,1]$. For example, if
$\alpha^{\prime}=0.95$ and $\beta^{\prime}=0.8$, then
$POS_{(0.8,0.95)}(X)=C_{1}\cup C_{2}$, $BND_{(0.8,0.95)}(X)=C_{3}$, and
$NEG_{(0.8,0.95)}(X)=\emptyset$.

Finally, the case of an empty positive region.

######  Theorem 4.

Let $Ev\in\mathcal{E}^{+}$ such that $POS_{(\alpha,\beta)}^{Ev}=\emptyset$ and
$NEG_{(\alpha,\beta)}^{Ev},BND_{(\alpha,\beta)}^{Ev}\neq\emptyset$. Let
$\alpha^{\prime},\beta^{\prime}\in[0,1]$ such that
$\beta^{\prime}<\alpha^{\prime}$. Then,

$\mathcal{T}_{(\alpha,\beta)}^{Ev}(X)=\mathcal{T}_{(\alpha^{\prime},\beta^{\prime})}(X)$
if and only if $\beta^{\prime}\in[\beta^{Ev}_{1},\beta^{Ev}_{2})$ and
$\alpha^{\prime}\in(\alpha^{Ev}_{1},1]$.

###### Proof.

The proof is similar to that of Theorems 1 and 2. So, it is omitted. ∎

######  Example 10.

Consider the universe $U$ and the equivalence classes $C_{1},C_{2}$, and
$C_{3}$, which are defined by Example 9. Let $X=\\{u_{1},u_{6},u_{11},\ldots
u_{28}\\}$, we focus on $\mathcal{T}_{(0.7,0.2)}^{BiVe}(X)$. Then,

$\dfrac{|X\cap[x]_{\mathcal{R}}|}{|[x]_{\mathcal{R}}|}=\begin{cases}0.5&\mbox{
if }x\in C_{1}\cup C_{2},\\\ 0.9&\mbox{ if }x\in C_{3}.\end{cases}$

and

$BiVe\left(\frac{|X\cap[x]_{\mathcal{R}}|}{|[x]_{\mathcal{R}}|}\right)=\begin{cases}0.58&\mbox{
if }x\in C_{3},\\\ 0&\mbox{ if }x\in C_{1}\cup C_{2}.\end{cases}$

By Definition 4, $NEG_{(0.7,0.2)}^{BiVe}(X)=C_{1}\cup C_{2}$,
$BND_{(0.7,0.2)}^{BiVe}(X)=C_{3}$, and
$POS_{(0.7,0.2)}^{BiVe}(X)=\emptyset$. Also, $\beta^{Ev}_{1}=0.5$,
$\beta^{Ev}_{2}=0.9$, $\alpha_{1}^{Ev}=0.9$. Thus, according to Theorem 4,
$NEG_{(\alpha^{\prime},\beta^{\prime})}(X)=C_{1}\cup C_{2}$,
$BND_{(\alpha^{\prime},\beta^{\prime})}(X)=C_{3}$, and
$POS_{(\alpha^{\prime},\beta^{\prime})}(X)=\emptyset$ for each
$\beta^{\prime}\in[0.5,0.9)$ and $\alpha^{\prime}\in(0.9,1]$. For example, let
$(\alpha^{\prime},\beta^{\prime})=(0.95,0.6)$, we can easily verify that
$NEG_{(0.95,0.6)}(X)=C_{1}\cup C_{2}$, $BND_{(0.95,0.6)}(X)=C_{3}$, and
$POS_{(0.95,0.6)}(X)=\emptyset$.

##  5 Conclusions and future directions

This work proposes a novel model for three-way decisions based on the concept
of evaluative linguistic expressions. Thus, a new way is provided to divide
the initial universe into three regions with the corresponding decisions
rules. Moreover, our results allow decision-makers to give a linguistic
interpretation to the regions already obtained using the probabilistic
approach. Let us indicate some possible directions to continue this work.
Firstly, we need to extend the results of Section 4 to the evaluative
expressions that are not necessarily represented by increasing functions.
Then, we want to deepen the study of linguistic-regions by comparing our
methods with those presented in [21]. In addition, we intend to understand how
the decisions about the elements change using different evaluative
expressions. Finally, we could analyze the logical relations between the
linguistic regions determined by a given evaluative expression and investigate
their consequences in terms of decisions by constructing an hexagon of
opposition.

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