Measuring contradiction regarding a negation on AIFS

Measuring contradiction regarding a negation on AIFS

S. Cubillo, E. Casti˜neira, C. Torres-Blanc

Facultad de Inform´atica

28660 Boadilla del Monte Universidad Polit´ecnica de Madrid

{scubillo, ecastineira, ctorres}@fi.upm.es

Abstract

In [5], an axiomatic model for con-tradiction measures on Atanassov Intuitionistic fuzzy sets was pre-sented; there, different kinds of those measures, depending on the continu-ity conditions required, were estab-lished. But in previous papers (see [4]), not only the contradiction in general, but also the contradiction with respect to a given strong intu-itionistic fuzzy negation were stud-ied. This is due to the fact that in some applications, in order to fix a suitable model, not any negation is valid, but it is necessary to use a particular one. Thus, the problem of the axiomatization of the differ-ent types of contradiction measures regarding a given strong negation re-mained open. This is the main aim of the present work.

Keywords: Atanassov Intuitionis-tic fuzzy sets,N-contradiction mea-sures, continuity from below and from above.

1 Preliminaries

1.1 An Atanassov intuitionistic fuzzy set (AIFS) is a set A = {(x, µA(x), νA(x)) : x ∈

X}, whereµA:X→[0,1],νA:X→[0,1] are

called the membership and non-membership

∗ _{This paper is partially supported by CICYT} (Spain) under Project TIN 2005-08943-C02-01.

functions, respectively, and such that, for all x ∈ X, µA(x) +νA(x) ≤ 1 (see [1]). Let us

denote the set of all intuitionistic fuzzy sets on X asIF(X).

An AIFS could also be considered as an L-fuzzy set as defined by Goguen in [10], where the latticeLis the setL={(α1, α2)∈[0,1]2 :

α1 + α2 ≤ 1}, with the partial order ≤L defined as follows: given α = (α1, α2), β

= (β1, β2)∈L,

α≤_Lβ⇐⇒ α1≤β1andα2≥β2.

(L,≤_L) is a complete lattice with smallest element 0L = (0,1), and greatest element 1L = (1,0).

So, an AIFS A is an L-fuzzy set whose L -membership function χA _{∈ L}X _{= {χ : X →} L} is defined for each x ∈ X as χA_{(x) =}

(µ_A(x), ν_A(x)). The order ≤_L induces, in a natural way, a partial order in LX_{, that we}

denote in the same way. In this way (LX_,≤

L) is a bounded and complete lattice.

Furthermore, let us recall that a decreasing function N :L→ L is an intuitionistic fuzzy negation (IFN) if N(0_L) = 1_L and N(1_L) = 0L hold. Moreover, N is a strong IFN if the equalityN(N(α)) =α holds for allα∈L.

Bustinceet al. introduced in [3] the intuition-istic fuzzy generators, which can be used to construct intuitionistic fuzzy negations, and Deschrijver et al. focused on this problem in [8] and [9], and proved that any strong IFN N is characterized by a strong negation N : [0,1] → [0,1] by means of the formula

N(α1, α2) = (N(1−α2),1−N(α1)), for all

negation associated to N.

1.2 The study of contradiction in the frame-work of intuitionistic fuzzy sets was initiated in [6]. Similarly to the fuzzy case, an AIFS A, or alternatively χA_{, is said to be}

contra-dictory with respect to some strong IFN N, or, to be short, N-contradictory, if χA_(x)≤

L (N ◦ χA_{)(x) for all x ∈ X. Also A, or}

χA_{, is said to be contradictory (without}

de-pending on any specific negation) if there ex-ists a strong negation N, such that A is N -contradictory.

Nevertheless, it is interesting to know not only if a set is contradictory, but also the extent to which this property holds; that is, it is neces-sary to measure somehow the degree of con-tradiction of any AIFS. In order to do this, in [4] some functions were proposed to measure both the degree of N-contradiction with re-spect to a strong negationN, and the degree of contradiction of an AIFS. And in [5], an axiomatic model to measure contradiction is given. In a similar way, this paper focuses on establishing an axiomatic model to measure

N-contradiction.

1.3. In the previous paper [4], Casti˜neira

et al. analyzed the regions of L in which contradictory sets with respect to a given negation are located, with the purpose of suggesting the way to measure how contra-dictory an AIFS is. In [6] it was proved that, given χA _{= (µ}

A, νA) ∈ LX, and N a

strong IFN associated with the strong nega-tion N, χA _{is N-contradictory if and only if}

N(µA(x)) +νA(x) ≥ 1, for all x ∈ X. Thus

a region free of contradiction is determined in L, as well as other region where contra-dictory sets remain. Being more specific, if χA_{(X) ={χ}A_{(x) :x∈X}is the range ofχ}A_,

the set A isN-contradictory if and only if

χA(X)⊂ {(α1, α2)∈L|N(α1) +α2 ≥1}

Moreover, let L_N ={(α1, α2) ∈ L:N(α1) +

α2 ≤ 1}, and the boundary curve N(α1) +

α2= 1 satisfies the following properties:

1) It determines an increasing function ofα1.

2) It contains the point (0,0).

3) Its intersection with the line α1+α2 = 1

is the point (αN,1−αN), beingαN the

equi-librium point of the negation N.

-contradictory IFS

N

N

aN

1 - a

boundary of -contradiction N

non

-contradictoryIFS N

region of -contradiction N

c

(X₎

c

(X₎

I

L

Figure 1: Regions of N-contradiction and non-N-contradiction

2 Measures of N-Contradiction

In [4], in order to measure the N -contradiction of AIFS, the following functions

CN

i : LX → [0,1], i = 1,2,3, were proposed.

If χ= (µ, ν)∈LX_{, then:}

C₁N(χ) = Max(0,Inf

x∈X(N(µ(x)) +ν(x)−1)) C₂N(χ) = Max(0,1−Sup

x∈X

(g(µ(x))+g(1−ν(x)))),

where g : [0,1] → [0,1] is an order automor-phism satisfyingN(x) =g−1_{(1−g(x)) for all}

x∈[0,1].

CN

3 (χ) = d(dχ(0(XL,)L,LNN)), wheredis the Euclidean

distance.

But it is necessary to determine what is un-derstood as a measure of N-contradiction. That is, which are the properties demanded to a function to accept it measures adequately the N-contradiction.

Before introducing the N-contradiction mea-sures, we need a previous definition.

Definition 2.1. Let χ ∈ LX_{; we say that χ}

is LN-normal ifχ(X)∩LN 6=∅, where χ(X) is the closure of χ(X) in the usual topology in R2_.

Furthermore, χ is said to be L-normal if χ(X)∩ {(α1, α2)∈L; α2 = 0} 6=∅.

The set of all LN-normal AIFS will be de-noted by LX_N. And the set of all L-normal AIFS,LX

Let us observe thatχ∈LX _{isL-normal if and}

only if it is LN-normal for all strong IFN N. That is, T

N LX

N =LX0 .

Now a first proposal is given.

Definition 2.2. Let X 6= ∅ be a universe of discourse and N a strong IFN; a func-tion C_N : LX _{→ [0,1] is a measure of N} -contradiction on IF(X), or equivalently on

LX_{, if the following is satisfied:}

(c.i) CN(χ0L) = 1, whereχ0L(x) = 0L for all x∈X.

(c.ii) If χ∈LX

N, thenCN(χ) = 0.

(c.iii) Anti-monotonicity: If χA_{, χ}B _{∈ L}X

verify χA_{(x) ≤}

L χB(x) for all x ∈ X, thenC_N(χA_{)≥ C}

N(χB).

Remark. If in the axiom (c.ii) we replaceLX

N withLX₀ , the definition is just that of contra-diction measure given in [5].

The set of all measures ofN-contradiction on

LX will be denoted by N CM(LX). Recall that the set of all contradiction measures is denoted by CM(LX_).

Remark. Obviously,N CM(LX_{)⊂ CM(L}X_).

In [4] it was proved that the functions CN

1 , CN

2 ,C3N defined above satisfy the axioms (c.i)

and (c.iii), moreover it is not difficult to show that they also satisfy axiom (c.ii); hence CN

1 , CN

2 ,C3Nare measures ofN-contradiction.

Furthermore, thoseN-contradiction measures seem to vary their values in a gradual way; nevertheless the previous definition does not guarantee any kind of continuity in the mea-sures, as the following example shows: The functionCN :LX →[0,1], given by

CN(χ) =

½

1, ifχ=χ0L

0, otherwise

is a measure ofN-contradiction, that changes sharply inχ0L.

So, if we want to modelize the continuity in theN-contradiction measures, we need to im-pose some additional conditions. The follow-ing two sections are devoted to this subject.

3 Completely Semi-continuous

N-Contradiction measures

In order to demand a measure changes smoothly, we propose a new definition.

Definition 3.1. Let X 6=∅ and N a strong IFN; anN-contradiction measureC_N :LX _→

[0,1] is to be said completely semi-continuous from below on LX _{if the following axiom is}

satisfied:

(c.iv) For all {χi}i∈I ⊂ LX, where I is an arbitrary set of indexes,

Inf

i∈ICN(χ

i_{) =C}

N

µ

Sup

i∈I χi

¶

holds, where Sup

i∈I χ

i _{∈ L}X _{is defined as} µ

Sup

i∈I χi

¶

(x) = Sup

i∈I

χi_{(x), for allx∈X.}

It is easy to prove that (c.iv) implies (c.iii).

The set of all completely semi-continuous from below N-contradiction measures onLX

will be denoted by N CM_csc(LX_).

Remark. N CMcsc(LX) ⊂ CMcsc(LX),

where CMcsc(LX) is the set of contradiction

measures satisfying axiom (c.iv).

Proposition 3.2. Let N be a strong IFN, N the strong fuzzy negation associated with

N and αN the equilibrium point of N. For

each p∈(0, α_N], letC_{N, p} :LX _{→[0,1] be the}

function defined for eachχ= (µ, ν)∈LX _by:

CN, p(χ) =   

 

0, if Sup

x∈Xµ(x)> p

Max

µ

0,xInf∈Xν(x)−1+N(p)

N(p)

¶

,else

Then CN, p∈ N CMcsc(LX).

Proof. Before confirming the axioms, let us notice that the function has a simple geomet-rical interpretation (see figure 2) since it can be written as

CN, p(χ) =       

     

0, if

 



Sup

x∈X

µ(x)> p or

Inf

x∈Xν(x)≤1−N(p)

Inf

x∈Xν(x)−1+N(p)

N(a _{1 2})+a =1

a

c

(

X

)

p

1 -

N

(

p

)

-1 +N(p)

n

xeX

Inf

Figure 2: MeasureCN, p∈ N CMcsc(LX).

Now, let us prove the conditions.

(c.i) CN, p(χ0L) = Inf

x∈Xν(x)−1+N(p)

N(p) = 1

(c.ii) Let χ = (µ, ν) ∈ LX_N, then if there ex-ists x ∈ X such that µ(x) > p or ν(x) < 1 −N(p) then CN, p(χ) = 0 by the

defini-tion; if on the contrary, there is not such an x, then there exists {xn}n∈N ⊂ X such that

lim

n→∞χ(xn) = (p,1−N(p)), thus CN, p(χ) =

lim

n→∞ν(xn)−1+N(p)

N(p) = 0.

(c.iv) Let {χi_}

i∈I be a family of AIFS.

a) If Sup

i∈I

χi _{= (Sup} i∈I

µ_i,Inf

i∈Iνi) is such

that Sup

x∈XSupi∈I µi(x) > p, by definition C_{N, p}(Sup

i∈I

χi_{) = 0 is satisfied, and}

further-more, there exist x ∈ X and j ∈ I satis-fying µj(x) > p. Then CN, p(χj) = 0 and

Inf

i∈ICN, p(χ

i_{) = 0 =C}

N, p(Sup i∈I

χi_).

b) If Sup

x∈X

Sup

i∈I

µi(x)≤p, thenCN, p(Sup i∈I

χi_{) =}

Max

µ

0,xInf∈XInfi∈Iνi(x)−1+N(p)

N(p)

¶

Furthermore, for allx∈Xand i∈I,µi(x)≤

p, and so,

Inf

i∈ICN, p(χ

i_{) = Inf} i∈IMax

µ

0,xInf∈Xνi(x)−1+N(p)

N(p)

¶

= Max



_0,Infi∈IxInf∈Xνi(x)−1 +N(p)

N(p)





= Max



_0,xInf∈XInfi∈Iνi(x)−1 +N(p)

N(p)



 ¤

From now on, many proofs will be omitted due to limits of space.

Remark. Would we change in the defini-tion of C_{N, p} the condition Sup

x∈X

µ(x) > p by

Sup

x∈Xµ(x)≥p?

If we want to preserve the continuity of the measure, the answer is not. In fact, if we would have

C(χ) =

  

 

0, if Sup

x∈X

µ(x)≥p

Max

µ

0,xInf∈Xν(x)−1+N(p)

N(p)

¶

, else

taking m, with 1−N(p) < m < 1, and the family of constant AIFS {χn}n∈N, defined by (see figure 3)

χn(x) =

³

p− p

n, m

´

for all x∈X,

it holds Sup

n∈N

χn_{(x) = (p, m) and C(Sup} n∈N

χn_{) =}

0.

Nevertheless, for all n ∈ N, C(χn_{) =} m−1+N(p)

N(p) >0, and thus

Inf

n∈NC(χ

n_{) =} m−1 +N(p)

N(p) 6=C

µ

Sup

n∈N χn

¶

N

(a

)+a

=1

a

1 -

N

(

p )

m

c

c

_c

.

.

. ..

Figure 3: Counterexample.

Remark. In the extremal case p =αN, the

measure will be given as (see figure 4)

C_N∧(χ) = Max



_0,xInf∈Xν(x)−1 +αN

αN



N

_(a

_)+a

=1

a

N 1 - a

a

{

c

Figure 4: MeasureC∧

N ∈ N CMcsc(LX).

Proposition 3.3. Let f : [0,1] → [0,1] be a continuous and strictly decreasing function such that f(1) = 0 and α +f(α) < 1 for all α ∈ (0,1). Let (p, f(p)) ∈ L satisfying f(p) +N(p) = 1. For all β ∈[f(p), f(0)) let us consider the region

Lβ = ©

(α1, β)|α1∈[0, f−1(β)] ª S ©

(f−1(β), α2)|α2 ∈[β,1−f−1(β)] ª

and L_f(0) = {(0, α2)| α2 ∈ [f(0),1]}. Then

the functionCl

N :LX →[0,1] defined for each χ= (µ, ν)∈LX _{as (see figure 5)}

C_Nl (χ) =

   

  

1, if Sup

x∈Xχ(x) ∈ Lf(0) β−f(p)

1−f(p), if Sup x∈X

χ(x)∈Lβfor someβ

0, otherwise

satisfies that Cl

N ∈ N CMcsc(LX).

p b

f(0)

f(p)

.

L

c

Sup a = _{2 1}f (a )

-1

f (b)

c

Figure 5: MeasureCl

N ∈ N CMcsc(LX).

In a similar way, it is possible to define mea-sures demanding the continuity from above.

Definition 3.4. Let X 6=∅ and N a strong IFN; anN-contradiction measureCN :LX → [0,1] is to be said completely semi-continuous from above on LX _{if the following axiom is}

satisfied:

(c.v) For all {χi_}

i∈I ⊂LX\LXN,

Sup

i∈I

C_N(χi_{) = C}

N

µ

Inf

i∈Iχ

i ¶

holds, where

Inf

i∈Iχ

i _{∈ L}X _{is defined as} µ

Inf

i∈Iχ

i ¶

(x) =

Inf

i∈Iχ

i_{(x) for all x∈X.}

Remark. Notice that it is necessary to con-sider the AIFS are notL_N-normal in the pre-vious axiom. Indeed, let X = {x1, x2} and

the AIFS defined as follows:

χ1(xi) = ½

0L, ifi= 1

(αN,1−αN), ifi= 2

χ2(xi) = ½

(αN,1−αN), ifi= 1

0_L, ifi= 2

Then Inf{χ1_{, χ}2_}(x

i) = 0L, for i = 1,2, and thus C_N(Inf{χ1_{, χ}2_{}) = 1, nevertheless} CN(χ1) =CN(χ2) = 0 as χ1, χ2∈LXN.

Once again, axiom (c.v) implies axiom (c.iii). The set of all completely semi-continuousN -contradiction measures from above onLX will be denoted by N CMcsc(LX_).

Remark. N CMcsc_(LX_{) ⊂ CM}csc_(LX_),

where CMcsc(LX_{) is the set of contradiction}

measures satisfying axiom (c.iv).

Example 3.5. Let C∨

N : LX → [0,1] be a function defined for each χ= (µ, ν)∈LX by (see figure 6):

C_N∨(χ) =

(

0, if χ∈LX_N

Sup

x∈X

ν(x), otherwise

Then C_N∨ ∈ N CMcsc(LX). Furthermore,

C∨

N ∈ N CM/ csc(LX).

Remark. The measure Cl

N is not a com-pletely semi-continuousN-contradiction mea-sure from above.

Indeed, let X be a universe of discourse with Card(X)≥2, and x₁, x₂ ∈X such thatx₁ 6= x2. Let us take fori= 1,2 the AIFS

χi(x) =

½

(0, f(p)), ifx=xi

N_{(a 1 2)+a} =1

aN

c

N 1 - a

{

xeX

Sup

Figure 6: MeasureC∨

N ∈ N CMcsc(LX).

Then

µ

Inf

i=1,2χ i

¶

(x) = 0_L for allx∈X. So,

Cl

N( Inf_i=1,2χi) =CNl (χ0L) = 1.

But,Cl

N(χ1) =CNl (χ2) = Sup

i=1,2C l

N(χi) = 0.

Proposition 3.6. Let f : [0,1] → [0,1] be a continuous and strictly decreasing function such that f(1) = 0 and α +f(α) < 1 for all α ∈ (0,1). Let (p, f(p)) ∈ L satisfying f(p) +N(p) = 1. For allβ ∈ [f(p), f(0)] let us consider the region

Mβ = {(f−1(β), α2)|α2∈[0, β]} S

{(α₁, β)|α₁ ∈[f−1(β),1−β]}

and M_β = {(α₁, β)| α₁ ∈ [0,1−β]} if β ∈

(f(0),1]. The function Cu

N : LX → [0,1] de-fined for each χ= (µ, ν)∈LX _{as (see fig. 7):}

C_Nu(χ) =

(

0, ifχ∈LX_N β−f(p)

1−f(p), ifχ /∈LXN & Inf_x∈Xχ(x)∈Mβ

satisfies Cu

N ∈ N CMcsc(LX). Furthermore,

C_Nu ∈ N CM/ csc(LX).

On the other hand, measuresCN

1 ,C2N andC3N

defined in [4] do not satisfy the conditions de-manded in this section, as we are going to show.

Proposition 3.7. IfX 6=∅andN is a strong negation, N-contradiction measures on LX C₁N,C₂N and C₃N, defined at the beginning of section 2, are neither completely semicontin-uous from below nor from above.

Proof. First, let us see that, for i = 1,2,3,

CN

i ∈ N CM/ csc(LX). Let us fix β such that

p

b f(0)

f(p)

.

a = _{2 1}f (a )

c

-1

f (b)

c

xe_X Inf

M

Figure 7: MeasureCu

N ∈ N CMcsc(LX).

0< β <1−αN, and letαsuch that N−1(1−

β)< α < αN. We consider the AIFS

χ1_{(x) = (0, β)}

χ2_{(x) = (α,1−α)} ¾

∀x∈X

Then

Ã

Sup

j=1,2

χj !

(x) = (α, β) for all x ∈ X,

and it is easy to prove that for i= 1,2,3,

0< Inf

j=1,2Ci(χ j_)6=C

i( Sup j=1,2

χj_{) = 0.}

1-a

N

L

I

.

.

.

aN N

1 - a

N_{(a 1 2)+a} =1

b

a

N- 1 _{( 1 -b)}

Figure 8: CN

1 ,C2N,C3N are not inN CMcsc(LX).

Second, let us see that, for i = 1,2,3, CN

i ∈/ N CMcsc(LX). Let us fixαsuch that 1−αN <

α < 1, and β with α < 1− β. Now, we consider the AIFS

χ1_{(x) = (0, α)}

χ2_{(x) = (β,1−β)} ¾

∀x∈X

Then

µ

Inf

j=1,2χ j

¶

(x) = (0,1−β) for allx, and it can be proved that for i= 1,2,3,

Sup

j=1,2

Ci(χj)6=Ci( Inf j=1,2χ

j_).

a

b

1 -b

aN N

1 - a

.

.

.

I

L

Figure 9: CN

1 ,C2N,C3N are not inNCMcsc

So, we need to weaken the conditions, in order to acceptCN

1 ,C2N and C3N asN-contradiction

measures with some kind of continuity, in such a way that the mathematical model be con-sistent with the intuition.

4 Semi-continuous

N-Contradiction measures

Let us remember that a set S ⊂ LX _{is a}

semilattice from below if for all χA_{, χ}B _{∈ S,}

Sup{χA_{, χ}B_{} ∈ S holds; and similarly, a set}

S ⊂LX _{is a semilattice from above if for all}

χA_{, χ}B _{∈ S, Inf{χ}A_{, χ}B_{} ∈S holds (see, for}

example, [2]).

Definition 4.1. Let X 6=∅ and N a strong IFN, anN-contradiction measureCN :LX → [0,1] is to be said semicontinuous from below

if the following axiom is satisfied:

(c.vi) For all semi-lattice from below

{χi_}

i∈I⊂LX, where I is an arbitrary set, the following is satisfied

Inf

i∈ICN(χ

i_{) =C}

N

µ

Sup

i∈I χi

¶

Notice that axiom (c.vi) implies axiom (c.iii).

The set of all semi-continuous from belowN -contradiction measures onLX _{will be denoted}

by N CMsc(LX).

Remark. Obviously, N CMcsc(LX) ⊂ N CMsc(LX).

Proposition 4.2. Let X 6= ∅ and N and strong IFN. Given a fixed p ∈ (0,+∞), for all β ∈ [0,1] let us consider the following

re-gion

L_β =

(

(α₁, α₂)∈L| α1∈[0, β],

α2 = (α1+_βp₊)(1_p−β) )

,

that is,L_β is a segment on the line joining the points (−p,0) and (β,1−β).

Given the function CL

N :LX → [0,1] defined for each χ= (µ, ν)∈LX _{by (see figure 10):}

C_NL(χ) =

(

0, ifχ∈LX

N

1−β, ifχ /∈LX_N & Sup

x∈Xχ(x)∈Lβ

we have CL

N ∈ N CMsc(LX)\ N CMcsc(LX).

b

(-p,0)

I

L

Lb

a_N

1 -b

c

.

1 - a_N

Figure 10: MeasureCL

N ∈ N CMsc(LX).

Similarly, we have

Definition 4.3. Let X 6= ∅, an N -contradiction measure CN : LX → [0,1] is to be said semicontinuous from above if the following axiom is satisfied:

(c.vii) For all semilattice from above

{χi_}

i∈I ⊂ LX \LX_N, where I is an arbitrary set, the following holds

Sup

i∈I

CN(χi) =CN

µ

Inf

i∈Iχ

i ¶

Again, (c.vii) implies (c.iii).

The set of all semi-continuous from aboveN -contradiction measures onLX _{will be denoted}

by N CMsc(LX).

Let CU

N :LX → [0,1] be the function defined for each χ= (µ, ν)∈LX _{by (see figure 11):}

C_NU(χ) =

(

0, ifχ∈LX

N

1−β, ifχ /∈LX_N & Inf

x∈Xχ(x)∈Lβ

Then CU

N ∈ N CMscN(LX)\ N CMcscN (LX).

b

(-p,0)

I

L

L_b

a_N

1 -b

c

1 - a_N

Figure 11: MeasureCU

N ∈ N CMscN(LX).

Now, we have the following result.

Proposition 4.5. For i= 1,2,3, each mea-sure C_iN defined at the beginning of section 2 satisfies that CN

i ∈ N CMsc(LX), but, in

general,CN

i ∈ N CMsc(LX) do not hold.

Finally, the functions presented through this paper show the following result.

Proposition 4.6. For any strong IFNN, the following inequalities hold:

N CMcsc(LX)(N CMsc(LX)(N CM(LX) N CMcsc(LX)(N CMsc(LX)(N CM(LX)

Conclusions

Contradictory sets can result inconvenient in certain applications, for instance, in the processes of fuzzy inference. Until now, a mathematic model had been defined to mea-sure in which degree an AIFS is contradictory. However, demanding that an object have a small contradictory degree can be very restric-tive and it may result more interesting to mea-sure that degree regarding a given negation, if that negation is the one used in a specific application. That is why, in this work, we have presented a mathematic model to mea-sure the N-contradiction of an AIFS. More-over, we have obtained families of measures that satisfy different kinds of continuity.

References

[1] K.T. Atanassov (1999). Intuitionistic fuzzy sets. (Physica-Verlag, Heidelberg, New York).

[2] T.S. Blyth (2005). Lattices and Ordered Algebraic Structures, Springer-Verlag, London.

[3] H. Bustince, J. Kacprzyk, V. Mohedano (2000). Intuitionistic fuzzy generators -Application to intuitionistic fuzzy com-plementation. InFuzzy Sets and Systems, 114, pages 485-504.

[4] E. Casti˜neira, S. Cubillo, C. Tor-res (2006). Searching Degrees of Self-Contradiction in Atanassov’s Fuzzy Sets. In Mathware and Soft-Computing, vol-ume XIII, n.3, pages 136-156.

[5] E. Casti˜neira, S. Cubillo. Measures of Self-Contradiction in Atanassov’s Fuzzy Sets: An Axiomatic Model. Sent to In-ternational Journal Intelligent Systems

[6] S. Cubillo, E. Casti˜neira (2004). Contradiction in Intuitionistic Fuzzy Sets. In Proceedings of the conference IPMU’2004, pages 2180-2186, Perugia, Italy, 2004.

[7] S. Cubillo, E. Casti˜neira (2005). Measur-ing contradiction in fuzzy logic. In Inter-national Journal of General Systems, 34 (1), pages 39-59, February 2005.

[8] G. Deschrijver, C. Cornelis, E. Kerre (2002). Intuitionistic fuzzy connectives revisited. In Proceedings of conference IPMU’2002, pages 1839-1844. Annecy, France, 2002.

[9] G. Deschrijver, C. Cornelis, E. Kerre. (2004). On the representation of intu-itionistic fuzzy t-norms and t-conorms. InIEEE Transactions on Fuzzy Systems, volume 12, (1), pages 45-61.

[10] J. A. Goguen (1967). L-Fuzzy Sets. In