Linguistic quantifiers modeled by interval-valued intuitionistic Sugeno integrals 1

Abstract

Ying’s model of linguistic quantifiers based on Sugeno integral is generalized to interval-valued intuitionistic Sugeno integral, the truth value of a quantified proposition is evaluated by using interval-valued intuitionistic Sugeno integral. Some logical properties of linguistic quantifiers in this model are discussed, and some application examples in uncertainty decision making and linguistic summarization of data are presented.

Keywords

Fuzzy set Intuitionistic fuzzy set fuzzy measure Sugeno integral interval-valued intuitionistic fuzzy quantifier

1 Introduction

The theory of fuzzy measure (non-additive measure) and fuzzy integrals was introduced by Sugeno [18], has been developed by many researchers and applied to various fields (see [1,15,16,19,20,26,27,31,38,39 , 1,15,16,19,20,26,27,31,38,39]). Linguistic quantification is a very important topic in the field of high level knowledge representation and reasoning (see [34]). For linguistic quantifiers (or fuzzy quantifiers), the first fuzzy set theoretic approach was described by Zadeh [35]. A novel approach to model linguistic quantifiers was proposed by Ying [33,34, 33,34] based on fuzzy measures and Sugeno integral. Cui and Li [5] generalized Yingąŕs model to Choquet integral. Cui, Li and Zhang [6] investigated intuitionistic fuzzy linguistic quantifier based on intuitionistic fuzzy Sugeno integral. In the recent paper [10], intuitionistic fuzzy Choquet integral is introduced and applied to intuitionistic fuzzy linguistic quantifier. In fact, intuitionistic fuzzy set is also applied to rough set theory (see [37]).

Atanassov and Gargov [2] first introduced the concept of interval-valued intuitionistic fuzzy set. Wu, Zhang and Sang [29] extended the concept and theory of Sugeno integral and fuzzy measure, introduced the notions of interval-valued intuitionistic fuzzy measures and interval intuitionistic fuzzy-valued Sugeno integrals.

This paper generalizes Ying’s model of linguistic quantifiers to interval-valued intuitionistic linguistic quantifiers based on interval intuitionistic fuzzy-valued Sugeno integrals. An interval-valued intuitionistic fuzzy quantifier is represented by a family of interval-valued intuitionistic fuzzy measures and the interval-valued intuitionistic truth value of a quantified proposition is calculated by using interval intuitionistic fuzzy-valued Sugeno integral. Some logical properties of interval-valued intuitionistic linguistic quantifiers are derived in new model, and some application examples in uncertainty decision making, linguistic summarization of data and fuzzy queries of relational database are presented.

Note that, this paper once had submitted to ICCCI 2010 and GCSE 2011, but it has not been officially published (there are only informal proceedings of these conferences).

2 Preliminaries

Definition 2.1. [2,30, 2,30] Let X be a nonempty set. An interval-valued intuitionistic fuzzy set (IVIFS) on a universe X is an object of the form $A = {(x, [a^{(1)} (x), a^{(2)} (x)], [a^{(3)} (x), a^{(4)} (x)]) ∣ x \in X},$ where interval [a ⁽¹⁾ (x) , a ⁽²⁾ (x)] ⊆ [0, 1] and [a ⁽³⁾ (x) , a ⁽⁴⁾ (x)] ⊆ [0, 1] are intervals, and for any x ∈ X, a ⁽²⁾ (x) + a ⁽⁴⁾ (x) ≤1.

For an IVIFS A, the pair ([a ⁽¹⁾ (x) , a ⁽²⁾ (x)] , [a ⁽³⁾ (x) , a ⁽⁴⁾ (x)]) is called an interval-valued intuitionistic fuzzy number IVIFN ([30,31, 30,31]). For convenience we denote an IVIFN by A = ([a ⁽¹⁾, a ⁽²⁾] , [a ⁽³⁾, a ⁽⁴⁾]), where [a ⁽¹⁾, a ⁽²⁾] ⊆ [0, 1] , [a ⁽³⁾, a ⁽⁴⁾] ⊆ [0, 1] , a ⁽²⁾ + a ⁽⁴⁾ ≤ 1.

Definition 2.2. [2,29,31 , 2,29,31] Let A = ([a ⁽¹⁾, a ⁽²⁾] , [a ⁽³⁾, a ⁽⁴⁾]) and B = ([b ⁽¹⁾, b ⁽²⁾] , [b ⁽³⁾, b ⁽⁴⁾]) be two IVIFNs on the set X, then

A≤ B ⇔ a ⁽¹⁾ ≤ b ⁽¹⁾, a ⁽²⁾ ≤ b ⁽²⁾, a ⁽³⁾ ≥ b ⁽³⁾, a ⁽⁴⁾ ≥ b ⁽⁴⁾ ;

A≥ B ⇔ B ≤ A ;

A = B ⇔ a ⁽¹⁾ = b ⁽¹⁾, a ⁽²⁾ = b ⁽²⁾, a ⁽³⁾ = b ⁽³⁾, a ⁽⁴⁾ = b ⁽⁴⁾ ;

$\bar{A} = ([a^{(3)}, a^{(4)}], [a^{(1)}, a^{(2)}]);$

A ∧ B = ([min {a ⁽¹⁾, b ⁽¹⁾} , min {a ⁽²⁾, b ⁽²⁾}] , [max {a ⁽³⁾, b ⁽³⁾} , max {a ⁽⁴⁾, b ⁽⁴⁾}]) ;

A ∨ B = ([max {a ⁽¹⁾, b ⁽¹⁾} , max {a ⁽²⁾, b ⁽²⁾}] , [min {a ⁽³⁾, b ⁽³⁾} , min {a ⁽⁴⁾, b ⁽⁴⁾}]) .

The set of interval-valued intuitionistic fuzzy number IVIFN is denoted by $L$ , that is, $\begin{matrix} L = {([a^{(1)}, a^{(2)}], [a^{(3)}, a^{(4)}]) ∣ a^{(1)}, a^{(2)}, a^{(3)}, a^{(4)} \\ \in [0, 1], a^{(1)} \leq a^{(2)}, a^{(3)} \leq a^{(4)}, a^{(2)} + a^{(4)} \leq 1} . \end{matrix}$ It is easy to prove that $(L; \land, \lor, 0_{L}, 1_{L})$ is a completely, complete distributive lattice, where $0_{L} = ([0, 0], [1, 1]), 1_{L} = ([1, 1], [0, 0])$ .

Definition 2.3. [18,21,22 , 18,21,22]. Let X be a nonempty set. A Borel field over X is a subset $A$ of 2^X (power set) satisfying the following conditions:

$\emptyset \in A;$

If $E \in A$ , then $X - E \in A;$

If $E_{n} \in A$ for 1≤ n < ∞, then $⋃_{n = 1}^{\infty} E_{n} \in A .$

Definition 2.4. [18,21,22 , 18,21,22] If X is a nonempty set and $A$ a Borel field over X, then $(X, A)$ is called a measurable space.

Definition 2.5. [29] Let X be a nonempty set. An interval-valued intuitionistic fuzzy measure over a measurable space $(X, A)$ is a mapping $π : A \to L$ with the following properties:

$π (\emptyset) = 0_{L};$

$π (X) = 1_{L};$

If $E, F \in A$ and E ⊆ F, then π (E) ≤ π (F) .

For simplicity, we mainly consider the special measurable space $(X, A)$ in which $A = 2^{X}$ (power set).

For any measurable space $(X, A)$ , we write IVIFM $(X, A)$ for the set of all interval-valued intuitionistic fuzzy measure on $(X, A)$ .

Remark 1. (1) In the above definition, denote the mapping $π : A \to L$ by π = ([π ⁽¹⁾, π ⁽²⁾] , [π ⁽³⁾, π ⁽⁴⁾]), if the mapping π satisfies π ⁽¹⁾ = π ⁽²⁾, π ⁽³⁾ = π ⁽⁴⁾ and π ⁽³⁾ (·) =1 - π ⁽¹⁾ (·), then the interval-valued intuitionistic fuzzy measure is reduced to fuzzy measure. This means that the notion of interval-valued intuitionistic fuzzy measure is a natural generalization of fuzzy measure in [18,34, 18,34].

(2) As a generalization of fuzzy measure ([18]), lattice-valued fuzzy measure is introduced (see [11,36, 11,36]). The above concept of interval-valued intuitionistic fuzzy measure can be regarded as a special kind of lattice-valued fuzzy measure based on the lattice $L$ .

Definition 2.6. [29] Let $(X, A)$ be a measurable space and $\tilde{f} : A \to L$ be an interval intuitionistic fuzzy-valued mapping. The function $\tilde{f}$ is called $A$ -measurable if $(\forall α \in L)$ $\begin{matrix} {\tilde{f}}_{\leq α} = {x \in X ∣ \tilde{f} (x) \leq α} \in A, \\ {\tilde{f}}_{\geq α} = {x \in X ∣ \tilde{f} (x) \geq α} \in A . \end{matrix}$

Lemma 2.7. [29] Let $A$ be a Borel field on X and $\tilde{f} : X \to L$ , ∀x ∈ X, $\begin{matrix} \tilde{f} (x) = ([{\tilde{f}}^{(1)} (x), {\tilde{f}}^{(2)} (x)], [{\tilde{f}}^{(3)} (x), {\tilde{f}}^{(4)} (x)]) . \end{matrix}$

The interval intuitionistic fuzzy-valued mapping $\tilde{f}$ is $A$ -measurable if and only if ${\tilde{f}}^{(1)} (x), {\tilde{f}}^{(2)} (x), {\tilde{f}}^{(3)} (x),$ ${\tilde{f}}^{(4)} (x)$ are $A$ -measurable.

Definition 2.8. [24]. Let $(X, A)$ be a measurable space, $π : A \to L$ be an interval-valued intuitionistic fuzzy measure and $\tilde{f} : X \to L$ be an interval intuitionistic fuzzy-valued mapping. The interval intuitionistic fuzzy-valued Sugeno integral of $\tilde{f}$ on X is defined by $\begin{matrix} (\tilde{s}) \int_{X} \tilde{f} \circ π = ⋁_{α \in L} (α \land π ({\tilde{f}}_{\geq α})), \end{matrix}$ where ${\tilde{f}}_{\geq α} = {x \in X ∣ \tilde{f} (x) \geq α}$ for each $α \in L$ .

Remark 2. In the above definition, denote the mapping $π : A \to L$ by π = ([π ⁽¹⁾, π ⁽²⁾] , [π ⁽³⁾, π ⁽⁴⁾]), and denote the interval intuitionistic fuzzy-valued mapping $\tilde{f} : X \to L$ by $\tilde{f} = ([{\tilde{f}}^{(1)}, {\tilde{f}}^{(2)}], [{\tilde{f}}^{(3)}, {\tilde{f}}^{(4)}])$ , if the mapping π and $\tilde{f}$ satisfy π ⁽¹⁾ = π ⁽²⁾, π ⁽³⁾ = π ⁽⁴⁾, π ⁽³⁾ (·) =1 - π ⁽¹⁾ (·) and ${\tilde{f}}^{(1)} = {\tilde{f}}^{(2)}, {\tilde{f}}^{(3)} = {\tilde{f}}^{(4)}, {\tilde{f}}^{(3)} (\cdot) = 1 - {\tilde{f}}^{(1)} (\cdot)$ , then interval intuitionistic fuzzy-valued Sugeno integral is reduced to fuzzy Sugeno integral (see [18,34, 18,34]).

Theorem 2.9. [29] Let $(X, A)$ be a measurable space, $π : A \to L, π = ([π^{(1)}, π^{(2)}], [π^{(3)}, π^{(4)}])$ be an interval-valued intuitionistic fuzzy measure and $\tilde{f} : X \to L, \tilde{f} = ([{\tilde{f}}^{(1)}, {\tilde{f}}^{(2)}], [{\tilde{f}}^{(3)}, {\tilde{f}}^{(4)}])$ be an interval intuitionistic fuzzy-valued mapping. Then $\begin{matrix} (\tilde{s}) \int_{X} \tilde{f} \circ π \\ = ([⋁_{α \in [0, 1]} (α \land π^{(1)} ({\tilde{f}}_{\geq α}^{(1)})), ⋁_{α \in [0, 1]} (α \land π^{(2)} ({\tilde{f}}_{\geq α}^{(2)}))], \\ [⋀_{α \in [0, 1]} (α \lor π^{(3)} ({\tilde{f}}_{\geq α}^{(3)})), ⋀_{α \in [0, 1]} (α \lor π^{(4)} ({\tilde{f}}_{\geq α}^{(4)}))]) . \end{matrix}$

Theorem 2.10. [29] Let $π : A \to L, π = ([π^{(1)},$ π⁽²⁾] , [π⁽³⁾, π⁽⁴⁾]) be an interval-valued intuitionistic fuzzy measure and $\tilde{f} : X \to L, \tilde{f} = ([{\tilde{f}}^{(1)}, {\tilde{f}}^{(2)}],$ $[{\tilde{f}}^{(3)}, {\tilde{f}}^{(4)}])$ be an interval intuitionistic fuzzy-valued mapping. Then $\begin{matrix} (\tilde{s}) \int_{X} \tilde{f} \circ π \\ = ([(s) \int_{X} {\tilde{f}}^{(1)} \circ π^{(1)}, (s) \int_{X} {\tilde{f}}^{(2)} \circ π^{(2)}], \\ [\bar{(s) \int_{X} \bar{{\tilde{f}}^{(3)}} \circ \bar{π^{(3)}}}, \bar{(s) \int_{X} \bar{{\tilde{f}}^{(4)}} \circ \bar{π^{(4)}}}]), \end{matrix}$ where $(s) \int_{X} {\tilde{f}}^{(1)} \circ π^{(1)}$ is the usual Sugeno integral of ${\tilde{f}}^{(1)}$ with respect to π ⁽¹⁾, so is ${\tilde{f}}^{(2)}$ ; $\bar{{\tilde{f}}^{(3)}}, \bar{{\tilde{f}}^{(4)}} : X \to [0, 1]$ is defined by $\bar{{\tilde{f}}^{(3)}} (x) = 1 - {\tilde{f}}^{(3)} (x), \bar{{\tilde{f}}^{(4)}} (x) = 1 - {\tilde{f}}^{(4)} (x)$ ; $\bar{π^{(3)}}, \bar{π^{(4)}} : A \to [0, 1]$ is defined by $\bar{π^{(3)}} (E) = 1 - π^{(3)} (E), \bar{π^{(4)}} (E) = 1 - π^{(4)} (E)$ .

Remark 3. In [3], the componentwise decomposition theorem of lattice-valued fuzzy Sugeno integral is proved (see Theorem 11 in [3]). This result is useful to calculate in a simple way the values of some integrals from the values of the components. In fact, interval intuitionistic fuzzy-valued Sugeno integral is a special kind of lattice-valued fuzzy Sugeno integral, therefore, Theorem 2.10 can be regarded as an application of Theorem 11 in [3].

Theorem 2.11. Let $(X, A)$ be a measurable space, $π, π_{(1)}, π_{(2)} : A \to L$ be an interval-valued intuitionistic fuzzy measure and $\tilde{f} : X \to L$ be an $A$ -measurable interval intuitionistic fuzzy-valued mapping. Then

If π ₍₁₎ ≤ π ₍₂₎, that is, π ₍₁₎ (E) ≤ π ₍₂₎ (E) for all $E \in A$ , then $\begin{matrix} (\tilde{s}) \int_{X} \tilde{f} \circ π^{(1)} = (\tilde{s}) \int_{X} \tilde{f} \circ π^{(2)} . \end{matrix}$

If the Borel field $A$ is the power set 2^X of X, then $\begin{matrix} (\tilde{s}) \int_{X} \tilde{f} \circ π = ([⋁_{F \in 2^{X}} (⋀_{x \in F} {\tilde{f}}^{(1)} (x) \land π_{1} (F)), \\ ⋁_{F \in 2^{X}} (⋀_{x \in F} {\tilde{f}}^{(2)} (x) \land π_{2} (F))], \\ [⋀_{F \in 2^{X}} (⋁_{x \in F} {\tilde{f}}^{(3)} (x) \lor π_{3} (F)), \\ ⋀_{F \in 2^{X}} (⋁_{x \in F} {\tilde{f}}^{(4)} (x) \lor π_{4} (F))]) . \end{matrix}$

If $α \in L$ , then $\begin{matrix} (\tilde{s}) \int_{X} (α \lor \tilde{f}) \circ π = α \lor (\tilde{s}) \int_{X} \tilde{f} \circ π, \\ (\tilde{s}) \int_{X} (α \land \tilde{f}) \circ π = α \land (\tilde{s}) \int_{X} \tilde{f} \circ π . \end{matrix}$

If X = {x ₁, x ₂, . . . , x _n} is a finite set, then $(\tilde{s}) \int_{X} \tilde{f} \circ π = ([⋁_{i = 1}^{n} ({\tilde{f}}^{(1)} (x_{i}^{(1)}) \land π^{(1)} (X_{i}^{(1)})),$ $⋁_{i = 1}^{n} ({\tilde{f}}^{(2)} (x_{i}^{(2)}) \land π^{(2)} (X_{i}^{(2)}))],$ $[⋀_{i = 1}^{n} ({\tilde{f}}^{(3)} (x_{i}^{(3)}) \lor π^{(3)} (X_{i}^{(3)})),$ $⋀_{i = 1}^{n} ({\tilde{f}}^{(4)} (x_{i}^{(4)}) \lor π^{(4)} (X_{i}^{(4)}))]) .$

where

{\tilde{f}}^{(m)} : X \to [0, 1]

is such that

{\tilde{f}}^{(m)} (x_{i}^{(m)}) \leq {\tilde{f}}^{(m)} (x_{i + 1}^{(m)})

for 1 ≤ i ≤ n - 1,

X_{i}^{(m)} = {x_{i}^{(m)}, x_{i + 1}^{(m)},

. . ., x_{n}^{(m)}},

and

{x_{1}^{(m)}, x_{2}^{(m)}, . . ., x_{n}^{(m)}}

is a permutation of {x ₁, x ₂, . . . , x _n} (rearrange according to function valued in

{\tilde{f}}^{(m)}

), m = 1, 2; and

{\tilde{f}}^{(m)} : X \to [0, 1]

is such that

{\tilde{f}}^{(m)} (x_{i}^{(m)}) \geq {\tilde{f}}^{(m)} (x_{i + 1}^{(m)})

for 1 ≤ i ≤ n - 1,

X_{i}^{(m)} = {x_{i}^{(m)}, x_{i + 1}^{(m)}, . . ., x_{n}^{(m)}},

and

{x_{1}^{(m)}, x_{2}^{(m)}, . . ., x_{n}^{(m)}}

is a permutation of {x ₁, x ₂, . . . , x _n}, m = 3, 4 .

Proof. The proof is similar to Lemma 5 in [6].

Remark 4. Fuzzy integrals and ordered weighted average (OWA) operators are closely related. In [12], OWA operator is extended to any complete lattice, and the relationship between Sugeno integral and OWA operators are analyzed. Since interval intuitionistic fuzzy-valued Sugeno integral is a special kind of lattice-valued fuzzy Sugeno integral, therefore, we can investigate interval intuitionistic OWA operators. This topic will be discussed in another paper.

3 A first order language with interval-valued intuitionistic fuzzy quantifiers

Definition 3.1. An interval-valued intuitionistic fuzzy quantifier consists of the following two items:

each nonempty set X is endowed with a Borel field $A_{X}$ ;

a choice function $\begin{matrix} Q : (X, A_{X}) \to Q_{(X, A_{X})} \in IVIFM (X, A_{X}) \end{matrix}$ of the (proper) class ${IVIFM (X, A_{X}) : (X, A_{X})$ is a measurable space}.

Example 3.2. The simplest interval-valued intuitionistic fuzzy quantifiers are the interval-valued intuitionistic universal quantifier ∀=“all” and the interval-valued intuitionistic existential quantifier ∃=“some” which could be defined as follows, respectively: for any set X and E ⊆ X, $\begin{matrix} \forall_{X} (E) = {\begin{matrix} ([1, 1], [0, 0]), & if E = X, \\ ([0, 0], [1, 1]), & otherwise . \end{matrix} \end{matrix}$

$\begin{matrix} \exists_{X} (E) = {\begin{matrix} ([1, 1], [0, 0]), & if E \neq \emptyset, \\ ([0, 0], [1, 1]), & otherwise . \end{matrix} \end{matrix}$

Example 3.3. The interval-valued intuitionistic fuzzy quantifier Q =“many” is defined as $\begin{matrix} {many}_{X} (E) & = & ([0.9 \frac{μ (E)}{μ (X)}, \frac{μ (E)}{μ (X)}], \\ [0.85 \frac{μ (X - E)}{μ (X)}, \frac{μ (X - E)}{μ (X)}]) . \end{matrix}$ on X for each E ⊆ X, where X is a nonempty finite set, μ (E) is the cardinality of E.

The fuzzy quantifier “almost all” may be given as $\begin{matrix} almost {all}_{X} (E) \\ = ([0.80 (\frac{μ (E)}{μ (X)})^{2}, 0.90 (\frac{μ (E)}{μ (X)})^{2}], \\ [0.05 (\frac{μ (E)}{μ (X)})^{2}, 0.10 (\frac{μ (E)}{μ (X)})^{2}]) . \end{matrix}$

Definition 3.4. Let Q, Q ₁ and Q ₂ be interval-valued intuitionistic fuzzy quantifiers. Then the dual Q ^* of Q, the meet Q ₁ ∩ Q ₂ and the union Q ₁ ∪ Q ₂ of Q ₁ and Q ₂ are defined respectively as follows: For any nonempty set X and for any $E \in A_{X}$ ,

$Q_{X}^{*} (E) =^{def} \bar{Q_{X} (X - E)}$

$= ([Q_{X}^{(3)} (X - E), Q_{X}^{(4)} (X - E)],$

$[Q_{X}^{(1)} (X - E), Q_{X}^{(2)} (X - E)]);$

(Q ₁ ∩ Q ₂) _X (E) = ^def Q _1X (E) ∧ Q _2X (E)

$= ([Q_{1 X}^{(1)} (E) \land Q_{2 X}^{(1)} (E), Q_{1 X}^{(2)} (E) \land Q_{2 X}^{(2)} (E)],$

$[Q_{1 X}^{(3)} (E) \lor Q_{2 X}^{(3)} (E), Q_{1 X}^{(4)} (E) \lor Q_{2 X}^{(4)} (E)]);$

(Q ₁ ∪ Q ₂) _X (E) = ^def Q _1X (E) ∨ Q _2X (E)

$= ([Q_{1 X}^{(1)} (E) \lor Q_{2 X}^{(1)} (E), Q_{1 X}^{(2)} (E) \lor Q_{2 X}^{(2)} (E)],$

$[Q_{1 X}^{(3)} (E) \land Q_{2 X}^{(3)} (E), Q_{1 X}^{(4)} (E) \land Q_{2 X}^{(4)} (E)]) .$

Definition 3.5. The alphabet of L _q is given as follows:

A countable set of individual variables: x ₀, x ₁, x ₂, . . .;

A set $P = ⋃_{n = 0}^{\infty} P_{n}$ of predicate symbols, where P _n is the set of all n-place predicate symbols for each n ≥ 0. It is assumed that $⋃_{n = 0}^{\infty} P_{n} \neq \emptyset$ ;

Propositional connectives:¬, ∧;

Parentheses: (,).

The syntax of L _q is then presented by the next definition.

Definition 3.6. The set of well-formed formulas (or Wff for short) is the smallest set of symbol strings satisfying the following conditions:

If n ≥ 0, P ∈ P _n, and x ₁, . . . , x _n are individual variables, then P (x ₁, . . . , x _n)∈Wff;

If Q is an interval-valued intuitionistic fuzzy quantifier, x is an individual variable, and φ∈Wff, then (Qx) φ∈Wff; and

If φ, φ ₁, φ ₂∈Wff, then ¬φ, φ ₁ ∧ φ ₂Wff.

The semantics of L _q is given by the following two definitions.

Definition 3.7. An interval-valued intuitionistic interpretation I of L _q consists of the following items:

A measurable space $(X, A)$ , called the domain of I;

For any n ≥ 0, we associate the individual variable x _i with an element $x_{i}^{I}$ in X; and

For any n ≥ 0 and P ∈ P _n, there is a $A^{n}$ -measurable interval intuitionistic fuzzy-valued function $P^{I} : X^{n} \to L$ .

Definition 3.8. Let I be an interval-valued intuitionistic interpretation. Then the interval-valued intuitionistic truth value T _I (φ) of a formula φ under I is defined recursively as follows:

If φ = P (x ₁, . . . , x _n), then $\begin{matrix} T_{I} (φ) = P^{I} (x_{1}^{I}, . . ., x_{n}^{I}) . \end{matrix}$

If φ = (Qx) ψ, then

$T_{I} (φ) = (\tilde{s}) \int_{X} T_{I {. / x}} (ψ) \circ Q_{X}$

$= ([(s) \int_{X} T_{I {. / x}} (ψ)^{(1)} \circ Q_{X}^{(1)},$

$(s) \int_{X} T_{I {. / x}} (ψ)^{(1)} \circ Q_{X}^{(2)}],$

$[\bar{(s) \int_{X} \bar{T_{I {. / x}} (ψ)^{(3)}} \circ \bar{Q_{X}^{(3)}}},$

$\bar{(s) \int_{X} \bar{T_{I {. / x}} (ψ)^{(4)}} \circ \bar{Q_{X}^{(4)}}}])$ where X is the domain of I, T _I{./x} (ψ) : X → [0, 1] is an interval intuitionistic fuzzy-valued mapping such that T _I{./x} (ψ) (u) = T _I{u/x} (φ) for all u ∈ X; and I {u/x} is the interval-valued intuitionistic interpretation which differs from I only in the assignment of the individual variable x, i.e., y ^I{u/x} = y ^I for all y ≠ x and x ^I{u/x} = u;

If φ = ¬ ψ, then $\begin{matrix} T_{I} (φ) = \bar{T_{I} (ψ)} \\ = ([T_{I} (ψ)^{(3)}, T_{I} (ψ)^{(4)}], [T_{I} (ψ)^{(1)}, T_{I} (ψ)^{(2)}]); \end{matrix}$

If φ = φ ₁ ∧ φ ₂, then $\begin{matrix} T_{I} (φ) = T_{I} (φ_{1}) \land T_{I} (φ_{2}) \\ = ([T_{I} (φ_{1})^{(1)} \land T_{I} (φ_{2}^{(1)}, T_{I} (φ_{1})^{(2)} \land T_{I} (φ_{2}^{(2)}], \\ [T_{I} (φ_{1})^{(3)} \lor T_{I} (φ_{2}^{(3)}, T_{I} (φ_{1})^{(4)} \lor T_{I} (φ_{2}^{(4)}]) . \end{matrix}$

Proposition 3.9. Let Q be an interval-valued intuitionistic quantifiers and x an individual variable, and let φ∈Wff. Then for any interval-valued intuitionistic interpretation I, we have

T _I ((∀ x) φ) = ⋀ _u∈X T _I{u/x} (φ), T _I ((∃ x) φ)

= ⋁ _u∈X T _I{u/x} (φ).

Proof. For any interval-valued intuitionistic interprettation I, we have

$T_{I} ((\forall x) φ) = (\tilde{s}) \int_{X} T_{I {. / x}} (φ) \circ \forall_{X}$

= ⋁ _F⊆X [(⋀ _u∈F T _I{u/x} (φ)) ∧ ∀ _X (F)]

$= 0_{L} \lor [(⋀_{u \in X} T_{I {u / x}} (φ)) \land 1_{L}]$

= ([⋀ _u∈X T _I{u/x} (φ) ⁽¹⁾, ⋀ _u∈X T _I{u/x} (φ) ⁽²⁾] ,

[⋁ _u∈X T _I{u/x} (φ) ⁽³⁾, ⋁ _u∈X T _I{u/x} (φ) ⁽⁴⁾])

= ⋀ _u∈X T _I{u/x} (φ) .

Similarly, T _I ((∃ x) φ) = ⋁ _u∈X T _I{u/x} (φ).

Proposition 3.10. For any interval-valued intuitionistic quantifier Q and any φ∈Wff, if I is an interval-valued intuitionistic interpretation with the domain X = {u}, then T_I ((Qx) φ) = T_I (φ).

Proof. By the definition we have

T _I ((Qx) φ)

$= (\tilde{s}) \int_{X} T_{I {. / x}} (φ) \circ Q_{X}$

= ⋁ _F⊆X [(⋀ _u∈F T _I{u/x} (φ)) ∧ Q _X (F)]

= [(⋀ _u∈∅ T _I{u/x} (φ)) ∧ Q _X (F)] ∨

[T _I{u/x} (φ) ∧ Q _X (u)]

$= (1_{L} \land 0_{L}) \lor [T_{I {u / x}} (φ) \land Q_{X} (u)]$

$= 0_{L} \lor [T_{I {u / x}} (φ) \land 1_{L}]$

= T _I{u/x} (φ)

= T _I (φ) .

Theorem 3.11. For any interval-valued intuitionistic quantifier Q and any φ∈Wff, we have ¬ (Qx) φ = (Q^*x) ¬ φ, where Q^* is the dual of Q.

Proof. Firstly, we prove that

$(\tilde{s}) \int_{X} \tilde{f} \circ π^{*}$

$= ([⋁_{α \in [0, 1]} (α \land π^{* (1)} ({\tilde{f}}_{\geq α}^{(1)})), ⋁_{α \in [0, 1]} (α \land π^{* (2)} ({\tilde{f}}_{\geq α}^{(2)}))],$

$[⋀_{α \in [0, 1]} (α \lor π^{* (3)} ({\tilde{f}}_{\geq α}^{(3)})), ⋀_{α \in [0, 1]} (α \lor π^{* (4)} ({\tilde{f}}_{\geq α}^{(4)}))])$

$= ([⋁_{α \in [0, 1]} (α \land π^{* (3)} ({\tilde{f}}_{< α}^{(1)})), ⋁_{α \in [0, 1]} (α \land π^{* (4)} ({\tilde{f}}_{< α}^{(2)}))],$

$[⋀_{α \in [0, 1]} (α \lor π^{* (1)} ({\tilde{f}}_{\geq α}^{(3)})), ⋀_{α \in [0, 1]} (α \lor π^{* (2)} ({\tilde{f}}_{\geq α}^{(4)}))])$

$= (\bar{[⋀_{α \in [0, 1]} (α \lor π^{* (3)} ({\tilde{f}}_{< α}^{(1)})), ⋀_{α \in [0, 1]} (α \lor π^{* (4)} ({\tilde{f}}_{< α}^{(2)}))]},$

$\bar{[⋁_{α \in [0, 1]} (α \land π^{* (1)} ({\tilde{f}}_{\geq α}^{(3)})), ⋁_{α \in [0, 1]} (α \land π^{* (2)} ({\tilde{f}}_{\geq α}^{(4)}))]})$

$= (\bar{[⋁_{α \in [0, 1]} (α \land π^{* (1)} ({\tilde{f}}_{\geq α}^{(3)})), ⋁_{α \in [0, 1]} (α \land π^{* (2)} ({\tilde{f}}_{\geq α}^{(4)}))]},$

$\bar{[⋀_{α \in [0, 1]} (α \lor π^{* (3)} ({\tilde{f}}_{< α}^{(1)})), ⋀_{α \in [0, 1]} (α \lor π^{* (4)} ({\tilde{f}}_{> α}^{(2)}))]})$

$= \bar{(\tilde{s}) \int_{X} ([{\tilde{f}}^{(3)}, {\tilde{f}}^{(4)}], [{\tilde{f}}^{(1)}, {\tilde{f}}^{(2)}]) \circ π}$

$= \bar{(\tilde{s}) \int_{X} \bar{\tilde{f}} \circ π}$ . Then for each interval-valued intuitionistic interpretation I with domain X,

$T_{I} ((Q^{*} x) \neg φ) = (\tilde{s}) \int_{X} T_{I {. / x}} (\neg φ) \circ Q_{X}^{*}$

$= \bar{(\tilde{s}) \int_{X} \bar{T_{I {. / x}} (\neg φ)} \circ Q_{X}}$

$= \bar{(\tilde{s}) \int_{X} T_{I {. / x}} (φ) \circ Q_{X}}$

$= \bar{T_{I} ((Qx) φ)}$

= T _I (¬ (Qx) φ) .

Theorem 3.12. For any interval-valued intuitionistic quantifier Q and any φ, ψ∈Wff, if individual variable x is not free in ψ, then we have (Qx) φ ∧ ψ = (Qx) (φ ∧ ψ) , (Qx) φ ∨ ψ = (Qx) (φ ∨ ψ) .

Proof. The proof is similar to Theorem 2 in [5].

Theorem 3.13. For any interval-valued intuitionistic quantifiers Q ₁ and Q ₂, individual variable x and φ∈Wff, we have

((Q ₁∩ Q ₂) x) φ = (Q ₁ x) φ ∧ (Q ₂ x) φ ;

((Q ₁ ∪ Q ₂) x) φ = (Q ₁ x) φ ∨ (Q ₂ x) φ .

Proof. (i) For any interval-valued intuitionistic interpretation I with domain X, we have T _I (((Q ₁ ∩ Q ₂) x) φ) ≤ T _I ((Q ₁ x) φ ∧ (Q ₂ x) φ) .

Conversely, we get

T _I ((Q ₁ x) φ ∧ (Q ₂ x) φ)

$= ((\tilde{s}) \int_{X} T_{I {. / x}} (φ) \circ Q_{1 X}) ⋀$

$((\tilde{s}) \int_{X} T_{I {. / x}} (φ) \circ Q_{2 X})$

= (⋁ _α (α ∧ Q _1X (T _I{./x} (φ) _≥α))) ⋀

(⋁ _β (β ∧ Q _2X (T _I{./x} (φ) _≥β)))

$= ([(⋁_{α_{1}} (α_{1} \land Q_{1 X}^{(1)} (T_{I {. / x}} (φ)_{\geq α_{1}}^{(1)}))) ⋀$

$(⋁_{β_{1}} (β_{1} \land Q_{2 X}^{(1)} (T_{I {. / x}} (φ)_{\geq β_{1}}^{(1)}))),$

$(⋁_{α_{2}} (α_{2} \land Q_{1 X}^{(2)} (T_{I {. / x}} (φ)_{\geq α_{2}}^{(2)}))) ⋀$

$(⋁_{β_{2}} (β_{2} \land Q_{2 X}^{(2)} (T_{I {. / x}} (φ)_{\geq β_{2}}^{(2)})))],$

$[(⋀_{α_{3}} (α_{3} \lor Q_{1 X}^{(3)} (T_{I {. / x}} (φ)_{\leq α_{3}}^{(3)}))) ⋁$

$(⋀_{β_{3}} (β_{3} \lor Q_{2 X}^{(3)} (T_{I {. / x}} (φ)_{\leq β_{3}}^{(3)}))),$

$(⋀_{α_{4}} (α_{4} \lor Q_{1 X}^{(4)} (T_{I {. / x}} (φ)_{\leq α_{4}}^{(4)}))) ⋁$

$(⋀_{β_{4}} (β_{4} \lor Q_{2 X}^{(4)} (T_{I {. / x}} (φ)_{\leq β_{4}}^{(4)})))])$

$\leq ([(⋁_{α_{1}, β_{1}} (α_{1} \land β_{1} \land Q_{1 X}^{(1)} (T_{I {. / x}} (φ)_{\geq α_{1}}^{(1)}) \land$

$Q_{2 X}^{(1)} (T_{I {. / x}} (φ)_{\geq β_{1}}^{(1)})),$

$⋁_{α_{2}, β_{2}} (α_{2} \land β_{2} \land Q_{1 X}^{(2)} (T_{I {. / x}} (φ)_{\geq α_{2}}^{(2)}) \land$

$Q_{2 X}^{(2)} (T_{I {. / x}} (φ)_{\geq β_{2}}^{(2)}))],$

$[(⋀_{α_{3}, β_{3}} (α_{3} \lor β_{3} \lor Q_{1 X}^{(3)} (T_{I {. / x}} (φ)_{\leq α_{3}}^{(3)}) \lor$

$Q_{2 X}^{(3)} (T_{I {. / x}} (φ)_{\leq β_{3}}^{(3)})),$

$⋀_{α_{4}, β_{4}} (α_{4} \lor β_{4} \lor Q_{1 X}^{(4)} (T_{I {. / x}} (φ)_{\leq α_{4}}^{(4)}) \lor$

$Q_{2 X}^{(4)} (T_{I {. / x}} (φ)_{\leq β_{4}}^{(4)}))])$

$\leq ([(⋁_{α_{1}, β_{1}} (α_{1} \land β_{1} \land Q_{1 X}^{(1)} (T_{I {. / x}} (φ)_{\geq α_{1} \land β_{1}}^{(1)}) \land$

$Q_{2 X}^{(1)} (T_{I {. / x}} (φ)_{\geq α_{1} \land β_{1}}^{(1)})),$

$⋁_{α_{2}, β_{2}} (α_{2} \land β_{2} \land Q_{1 X}^{(2)} (T_{I {. / x}} (φ)_{\geq α_{2} \land β_{2}}^{(2)}) \land$

$Q_{2 X}^{(2)} (T_{I {. / x}} (φ)_{\geq α_{2} \land β_{2}}^{(2)}))],$

$[(⋀_{α_{3}, β_{3}} (α_{3} \lor β_{3} \lor Q_{1 X}^{(3)} (T_{I {. / x}} (φ)_{\leq α_{3} \lor β_{3}}^{(3)}) \lor$

$Q_{2 X}^{(3)} (T_{I {. / x}} (φ)_{\leq α_{3} \lor β_{3}}^{(3)})),$

$⋀_{α_{4}, β_{4}} (α_{4} \lor β_{4} \lor Q_{1 X}^{(4)} (T_{I {. / x}} (φ)_{\leq α_{4} \lor β_{4}}^{(4)}) \lor$

$Q_{2 X}^{(4)} (T_{I {. / x}} (φ)_{\leq α_{4} \lor β_{4}}^{(4)}))])$

$\leq ([(⋁_{γ_{1}} (γ_{1} \land Q_{1 X}^{(1)} (T_{I {. / x}} (φ)_{\geq γ_{1}}^{(1)}) \land$

$Q_{2 X}^{(1)} (T_{I {. / x}} (φ)_{\geq γ_{1}}^{(1)})),$

$⋁_{γ_{2}} (γ_{2} \land Q_{1 X}^{(2)} (T_{I {. / x}} (φ)_{\geq γ_{2}}^{(2)}) \land$

$Q_{2 X}^{(2)} (T_{I {. / x}} (φ)_{\geq γ_{2}}^{(2)}))],$

$[(⋀_{γ_{3}} (γ_{3} \lor Q_{1 X}^{(3)} (T_{I {. / x}} (φ)_{\leq γ_{3}}^{(3)}) \lor$

$Q_{2 X}^{(3)} (T_{I {. / x}} (φ)_{\leq γ_{3}}^{(3)})),$

$⋀_{γ_{4}} (γ_{4} \lor Q_{1 X}^{(4)} (T_{I {. / x}} (φ)_{\leq γ_{4}}^{(4)}) \lor$

$Q_{2 X}^{(4)} (T_{I {. / x}} (φ)_{\leq γ_{4}}^{(4)}))])$

$= ([(⋁_{γ_{1}} (γ_{1} \land (Q_{1} \cap Q_{2})_{X}^{(1)} (T_{I {. / x}} (φ)_{\geq γ_{1}}^{(1)})),$

$⋁_{γ_{2}} (γ_{2} \land (Q_{1} \cap Q_{2})_{X}^{(2)} (T_{I {. / x}} (φ)_{\geq γ_{2}}^{(2)}))],$

$[(⋀_{γ_{3}} (γ_{3} \lor (Q_{1} \cap Q_{2})_{X}^{(3)} (T_{I {. / x}} (φ)_{\leq γ_{3}}^{(3)})),$

$⋀_{γ_{4}} (γ_{4} \lor (Q_{1} \cap Q_{2})_{X}^{(4)} (T_{I {. / x}} (φ)_{\leq γ_{4}}^{(4)}))])$

$= (\tilde{s}) \int_{X} T_{I {. / x}} (φ) \circ (Q_{1} \cap Q_{2})_{X}$

= T _I (((Q ₁ ∩ Q ₂) x) φ) ,

where α, α _i, β _i, γ, γ _i ∈ [0, 1] for i = 1, 2, 3, 4.

(ii) For any interval-valued intuitionistic interpretation I with domain X, we have

T _I (((Q ₁ ∪ Q ₂) x) φ)

$= (\tilde{s}) \int_{X} T_{I {. / x}} (φ) \circ (Q_{1} \cup Q_{2})_{X}$

$= ([⋁_{α_{1}} (α_{1} \land (Q_{1} \cup Q_{2})_{X}^{(1)} (T_{I {. / x}} (φ)_{\geq α_{1}}^{(1)})),$

$⋁_{α_{2}} (α_{2} \land (Q_{1} \cup Q_{2})_{X}^{(2)} (T_{I {. / x}} (φ)_{\geq α_{2}}^{(2)}))],$

$[(⋀_{α_{3}} (α_{3} \lor (Q_{1} \cup Q_{2})_{X}^{(3)} (T_{I {. / x}} (φ)_{\leq α_{3}}^{(3)})),$

$⋀_{α_{4}} (α_{4} \lor (Q_{1} \cup Q_{2})_{X}^{(4)} (T_{I {. / x}} (φ)_{\leq α_{4}}^{(4)}))])$

$= ([(⋁_{α_{1}} (α_{1} \land (Q_{1 X}^{(1)} (T_{I {. / x}} (φ)_{\geq α_{1}}^{(1)}) \lor$

$Q_{2 X}^{(1)} (T_{I {. / x}} (φ)_{\geq α_{1}}^{(1)}))),$

$⋁_{α_{2}} (α_{2} \land (Q_{1 X}^{(2)} (T_{I {. / x}} (φ)_{\geq α_{2}}^{(2)}) \lor$

$Q_{2 X}^{(2)} (T_{I {. / x}} (φ)_{\geq α_{2}}^{(2)})))],$

$[(⋀_{α_{3}} (α_{3} \lor (Q_{1 X}^{(3)} (T_{I {. / x}} (φ)_{\leq α_{3}}^{(3)}) \land$

$Q_{2 X}^{(3)} (T_{I {. / x}} (φ)_{\leq α_{3}}^{(3)}))),$

$⋀_{α_{4}} (α_{4} \lor (Q_{1 X}^{(4)} (T_{I {. / x}} (φ)_{\leq α_{4}}^{(4)}) \land$

$Q_{2 X}^{(4)} (T_{I {. / x}} (φ)_{\leq α_{4}}^{(4)})))])$

$= ([(⋁_{α_{1}} (α_{1} \land Q_{1 X}^{(1)} (T_{I {. / x}} (φ)_{\geq α_{1}}^{(1)}))) ⋁$

$(⋁_{α_{1}} (α_{1} \land Q_{2 X}^{(1)} (T_{I {. / x}} (φ)_{\geq α_{1}}^{(1)}))),$

$(⋁_{α_{2}} (α_{2} \land Q_{1 X}^{(2)} (T_{I {. / x}} (φ)_{\geq α_{2}}^{(2)}))) ⋁$

$(⋁_{α_{2}} (α_{2} \land Q_{2 X}^{(2)} (T_{I {. / x}} (φ)_{\geq α_{2}}^{(2)})))],$

$[(⋀_{α_{3}} (α_{3} \lor Q_{1 X}^{(3)} (T_{I {. / x}} (φ)_{\leq α_{3}}^{(3)}))) ⋀$

$(⋀_{α_{3}} (α_{3} \lor Q_{2 X}^{(3)} (T_{I {. / x}} (φ)_{\leq α_{3}}^{(3)}))),$

$(⋀_{α_{4}} (α_{4} \lor Q_{1 X}^{(4)} (T_{I {. / x}} (φ)_{\leq α_{4}}^{(4)}))) ⋀$

$(⋀_{α_{4}} (α_{4} \lor Q_{2 X}^{(4)} (T_{I {. / x}} (φ)_{\leq α_{4}}^{(4)})))])$

= T _I ((Q ₁ x) φ) ∨ T _I ((Q ₂ x) φ)

= T _I ((Q ₁ x) φ ∨ (Q ₂ x) φ) ,

where α _i ∈ [0, 1] for i = 1, 2, 3, 4.

Combining the results of Theorems 3.11 and 3.12, we obtain a prenex normal form for logical formulas with interval-valued intuitionistic linguistic quantifiers.

Theorem 3.14. (Prenex Normal Form) For any φ∈Wff, there exists φ∈Wff such that φ ≡ (Q ₁ x ₁) . . . (Q _n x _n) φ,

where n ≥ 0, Q ₁, . . . , Q _n are interval-valued intuitionistic fuzzy quantifiers, φ∈Wff does not contain interval-valued intuitionistic fuzzy quantifier.

Remark 3. From Remarks 1 and 2, we know that above results are natural generalization of related results in [34] when the “interval-valued intuitionistic fuzzy measure (Sugeno integral, quantifier)” induce to “fuzzy measure (Sugeno integral, quantifier)”.

4 Application examples

Now, we consider the applications of interval-valued intuitionistic linguistic quantifiers.

4.1 Application in multi-criteria decision making

First, we recall some notions. In [20,30, 20,30], the notions of score function and accuracy function are proposed. In the recent paper [23,24,25 , 23,24,25], J. Wu and F. Chiclana studied new score and accuracy functions for ranking interval-valued intuitionistic fuzzy numbers.

A function Q : [0, 1] → [0, 1] such that Q (0) =0, Q (1) =1 and Q (x) ≥ Q (y) if x ≥ y is called a basic unit-monotonic (BUM) function. $\int_{0}^{1} Q (y) dy$ is called the attitudinal character of Q.

Definition 4.1. [24] Let α = ([a ⁽¹⁾, a ⁽²⁾] , [a ⁽³⁾, a ⁽⁴⁾]) be an interval-valued intuitionistic fuzzy number, then the interval-valued score function of α is defined by $S_{WC} (α) = [a^{(1)} - a^{(4)}, a^{(2)} - a^{(3)}] .$

The attitudinal expected score function associated to α is defined by $E (S_{WC} (α))_{λ} = (1 - λ) (a^{(1)} - a^{(4)}) + λ (a^{(2)} - a^{(3)}) .$ where λ is the attitudinal character of a BUM function Q.

Definition 4.2. [24] Let α = ([a ⁽¹⁾, a ⁽²⁾] , [a ⁽³⁾, a ⁽⁴⁾]) be an interval-valued intuitionistic fuzzy number, then the interval-valued accuracy function of α is defined by $A_{WC} (α) = [a^{(1)} + a^{(3)}, a^{(2)} + a^{(4)}] .$

The attitudinal expected score function associated to α is defined by $E (A_{WC} (α))_{λ} = (1 - λ) (a^{(1)} + a^{(3)}) + λ (a^{(2)} + a^{(4)}) .$ where λ is the attitudinal character of a BUM function Q.

Definition 4.3. [24] Let α = ([a ⁽¹⁾, a ⁽²⁾] , [a ⁽³⁾, a ⁽⁴⁾]) be an interval-valued intuitionistic fuzzy number, then the membership uncertainty index function of α is defined by T (α) = (a ⁽²⁾ - a ⁽¹⁾) - (a ⁽⁴⁾ - a ⁽³⁾) .

Definition 4.4 [24]. Let α and β be any two interval-valued intuitionistic fuzzy numbers. We say that α ≺ β if and only if one the following conditions is true:

E (S _WC (α)) _λ < E (S _WC (β)) _λ;

E (S _WC (α)) _λ = E (S _WC (β)) _λ and

E (A _WC (α)) _λ < E (A _WC (β)) _λ;

E (S _WC (α)) _λ = E (S _WC (β)) _λ,

E (A _WC (α)) _λ = E (A _WC (β)) _λ and

T (α) > T (β).

Example 4.3 We consider multi-criteria decision making problems (see Example 3 in [6]). Assume X = {x ₁, x ₂, . . . , x _n} is a collection of criteria to be satisfied. Let S = {s ₁, s ₂, . . . , s _n} be a set of possible solution alternatives. For any s _j ∈ S and for any criterion x _i, the two interval-valued coordinates of $x_{i} (s_{j}) \in L$ indicate the degrees of membership and non-membership (interval-valued) of s _j satisfies x _i, respectively. We desire to find out the most preferred alternative such that “Q of criteria are satisfied by it”, where Q is an interval-valued intuitionistic linguistic quantifier with domain X.

Now, consider an air-condition system selection problem. Assume S = {s ₁, s ₂, s ₃} and three criteria X = {x ₁, x ₂, x ₃}, the degrees of membership and non-membership of the alternative s _j satisfying the criterion x _i are those presented in Table 1. Take Q=“almost all”, for any E ⊆ X, $\begin{matrix} Q_{X} (E) & = & ([0.80 {(\frac{μ (E)}{μ (X)})}^{2}, 0.90 {(\frac{μ (E)}{μ (X)})}^{2}], \\ [0.05 {(\frac{μ (E)}{μ (X)})}^{2}, 0.10 {(\frac{μ (E)}{μ (X)})}^{2}]) . \end{matrix}$ where μ (E) is the cardinality of E.

Then by calculations using Theorem 2.11 (iv), $\begin{matrix} D (s_{1}) = (\tilde{s_{1}}) \int_{X} s_{1 X} \circ Q_{X} \\ = ([0.55, 0.65], [0.05, 0.10]), \\ D (s_{2}) = (\tilde{s_{2}}) \int_{X} s_{2 X} \circ Q_{X} \\ = ([0.40, 0.45], [0.10, 0.15]), \\ D (s_{3}) = (\tilde{s_{3}}) \int_{X} s_{3 X} \circ Q_{X} \\ = ([0.356, 0.45], [0.05, 0.10]) . \end{matrix}$

Assume the following BUM function Q (y) = y ^1/3, then the attitudinal character of Q is λ = 3/4. The attitudinal expected score function associated to above interval-valued intuitionistic fuzzy number areas follows E (S _WC (D (s ₁))) _λ = 0.5625,

E (S _WC (D (s ₂))) _λ = 0.325,

E (S _WC (D (s ₃))) _λ = 0.364 .

Thus D (s ₁) ≻ D (s ₃) ≻ D (s ₂), s ₁ is the optimum candidate.

If fuzzy quantifier Q=“almost all” is given as $\begin{matrix} Q_{X} (E) = ([0.9 {(\frac{μ (E)}{μ (X)})}^{2}, {(\frac{μ (E)}{μ (X)})}^{2}], \\ [0.9 {(\frac{μ (X - E)}{μ (X)})}^{2}, {(\frac{μ (X - E)}{μ (X)})}^{2}]) . \end{matrix}$

Then, we can get the same decision result.

4.2 Application in linguistic summarization of data

A linguistic data (base) summary is meant as a concise, human-consistent description of a (numerical) data set. This concept has been introduced by Yager [32], and then presented in a more implementable form and further developed by Kacprzyk and Yager [7], and Kacprzyk et al. [8]. Recently, some new methods of linguistic summarization are presented, for examples, linguistic summarization using IF-THEN rules is proposed (see [28]); Yager’s approach is generalized based on the cardinality of the interval-valued fuzzy sets (see [13]) and type-2 fuzzy sets (see [14]).

“To summarize a database linguistically” means to build a natural language sentence which describes amounts of elements that have the chosen properties. A linguistic summary of a dataset consists of

a summarizer S (e.g., young)

a quantity in agreement Q (e.g., most);

truth degree T, e.g., 0.7, as, e.g.,

“T (most of employees are young) = 0.7.”

The summarizer S is a linguistic expression that is semantically represented by a fuzzy set. The meaning of S, i.e., its corresponding fuzzy set is, in practice, subjective and may be either predefined or elicited from the user.

In general, a linguistic summary of a data set is in the form of Q objects are P [T] (1)

Q objects being R are P [T] (2)

where P and R are the labels associated with fuzzy sets, and Q is a linguistic pronouncement of quantity.

In [13], the degree of truth of a statement in the form (1) is computed by using interval-valued fuzzy quantifiers. By using the notions of interval-valued intuitionistic linguistic quantifiers and interval intuitionistic fuzzy-valued Sugeno integrals in this paper, we can give a new method of linguistic summarization. This method is similar to the example in [13], the detailed algorithm is omitted here, we only give a example of final form, for instance, we can get the final form of the summary as following:

almost all scores are high ([0.31, 0.37], [0.60, 0.62]).

Remark 4. In [4], the authors discuss flexible querying of relational databases based on Sugeno fuzzy integral. This method can be generalized by using interval intuitionistic fuzzy-valued Sugeno integral, the further results will be written in another paper. For fuzzy queries to relational databases, the literatures [9,17, 9,17] are interesting.

5 Summary

Linguistic quantification is a very important topic in the field of high level knowledge representation and reasoning. Ying established the theory of linguistic quantifiers modeling by fuzzy measures and Sugeno integral. This paper we generalized Ying’s model to interval-valued intuitionistic linguistic quantifiers based on interval intuitionistic fuzzy-valued Sugeno integrals. An interval-valued intuitionistic fuzzy quantifier is represented by a family of interval-valued intuitionistic fuzzy measures and the interval-valued intuitionistic truth value of a quantified proposition is calculated by using interval intuitionistic fuzzy-valued Sugeno integral. We proved some logical properties of interval-valued intuitionistic linguistic quantifiers in new model, and discussed related applications in uncertainty decision making, linguistic summarization of data and fuzzy queries of relational database.

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