This section describes the mathematical entities needed to better understand the arguments used in this article.

#### 2.2. Numerical and Linguistic Truth-Value or Degree of Belief of a Feature

Let

p(

F_{i},

S_{j}) be a proposition of the form

F_{i} is

S_{j} where

F_{i} is the

i-th feature of a product and

S_{j} ∈

State. Let

T be a process as follows:

The process T defined in Equation (1) determines the Degree of Belief (DoB) of each proposition p(F_{i},S_{j}), i = 1,2,…, j = 1,...,4, DoB(.) ∈ [0,1]. This means that each proposition p(F_{i},S_{j}) has a truth-value (or DoB) in the interval [0,1], and the process denoted as T determines it.

The

DoB of a compound proposition can be determined as follows:

In Equation (2), S_{k} is a state drawn from State, and h is a hedge called “more or less” or “somewhat.”

However, the truth-value or DoB of the above-mentioned propositions can be assigned either numerically or linguistically. The description is as follows.

Let F_{i} be sedan (a feature of a car), i.e., F_{i} = sedan. Using the states defined in State, the following four propositions can be considered: p_{1}(sedan, must-be feature), p_{2}(sedan, should-be feature), p_{3}(sedan, could-be feature), and p_{4}(sedan, unreliable feature). A numerical value that lies in the interval [0, 1] can be assigned to each proposition subjectively or following a computation approach as its truth-value or DoB. Let, for instance, DoBs of the propositions be DoB(sedan, must-be feature) = 0.2, DoB(sedan, should-be feature) = 0.7, DoB(sedan, could-be feature) = 0.95, and DoB(sedan, unreliable feature) = 0.05. Linguistically, DoB = 0.2 means that “it is quite false that sedan is a must-be feature of a car,” i.e., DoB = 0.2 refers to a linguistic truth-value “quite false.” Similarly, DoB = 0.7 means that “it is somewhat true that sedan is a should-be feature of a car,” i.e., DoB = 0.7 refers to a linguistic truth-value “somewhat true.” Similarly, DoB = 0.95 means that “it is mostly true that sedan is a could-be feature of a car,” i.e., DoB = 0.95 refers to a linguistic truth-value “mostly true.” Finally, DoB = 0.05 means that “it is mostly false that the opinions obtained on the car feature called sedan is unreliable,” i.e., DoB = 0.05 refers to a linguistic truth-value “mostly false.”

Thus, a crisp

DoB,

i.e., a numerical value in the interval [0, 1], can be interpreted in terms of a linguistic expression (e.g., mostly false, somewhat true, and alike), which is referred to as linguistic truth-value. This means that a linguistic truth-value of a crisp

DoB is its linguistic interpretation or counterpart. The linguistic counterpart (

L(

c)) of a crisp

DoB (

c =

DoB(

F_{i},.)) is given as

Using a set of fuzzy numbers defined in the universe of discourse [0, 1], the linguistic truth-values

LT_{i},

i = 1, 2, ... can be defined. In this study, a set of seven linguistic truth-values are considered that are given by the membership functions (or

DoBs) of the seven fuzzy numbers [

16,

19,

20,

21,

22] labeled “mostly false (

mf),” “quite false (

qf),” “somewhat false (

sf),” “neither true nor false (

tf),” “somewhat true (

st),” “quite true (

qt),” and “mostly true (

mt).” The membership functions are illustrated in

Figure 1.

The definitions of the membership functions shown in

Figure 1 are as follows:

In Equations (4)–(10), a numerical truth-value is denoted as

c,

i.e.,

c ∈ [0, 1]. Let the linguistic counterpart of

c be

L(

c). If the condition underlying Equation (1) is applied, then the linguistic counterpart of

c ∈ [0, 0.05] is mostly false (

mf). Similarly, the linguistic counterpart of

c ∈ (0.05, 0.2] is quite false (

qf). The linguistic counterpart of

c ∈ (0.2, 0.4] is somewhat false (

sf). The linguistic counterpart of

c ∈ (0.4, 0.6] is neither true nor false (

tf). The linguistic counterpart of

c ∈ (0.6, 0.8] is somewhat true (

st). The linguistic counterpart of

c ∈ (0.8, 0.95] is quite true (

qt). Finally, the linguistic counterpart of

c ∈ (0.95, 1] is mostly true (

mt). A linguistic counterpart of

c, as described above, has an expected (crisp) value denoted as

E(

L(

c)) that is often calculated by the centroid method. The expected values of the linguistic true-values shown in

Figure 1 are as follows:

E(

mf(

c)) = 0.033,

E(

qf(

c)) = 0.133,

E(

sf(

c)) = 0.3,

E(

tf(

c)) = 0.5,

E(

st(

c)) = 0.7,

E(

qt(

c)) = 0.867, and

E(

mt(

c)) = 0.967, according to the centroid method.