Example 2.4 Let us consider the set X = {xn}nIN {yn}nIN and, [0,1]X such that (x )=n/(n+1), n (y )=1/(n+2) and (x )=1/(n+2), (y )=n/(n+1). Then (x )+(x )=(n2+3n+1)/(n2+3n+2)<1 and n n n n n (y )+(y )=(n2+3n+1)/(n2+3n+2)<1 and it follows that, are N n n s-contradictory between them. Nevertheless and are not self-contradictory (Sup(x)=1 and Sup(x)=1).

Information Models N g-self-contradiction and Self-contradiction Degrees Clearly, self-contradiction of a fuzzy set could be viewed as contradiction of the set with itself. Taking this into account, the degrees of contradiction defined in above papers, provide us the respective degrees of selfcontradiction, as in this section is shown.

In [5], some functions were defined as a model to determine different degrees of N g-contradiction between two fuzzy sets.

Definition 3.1 Given, [0,1]X and N a strong FN, we define the following contradiction measure functions:

g Ng i) C1 (, ) = Max0, Inf (N ( (x)) - (x)) g xX Ng ii) C2 (, ) = Max0, Inf (N ((x)) - (x)) g xX Ng iii) C3 (, ) = Max 0,1- Sup(g((x)) + g( (x))) xX d(X, RN ) Ng g iv) C4 (, ) = d((0,0), RN ), where d is the Euclidean distance, X = {((x),(x)) : x X} and g RN ={( y1, y2 ) [0,1]2 : N ( y1) < y2} is the region free of contradiction. Therefore, g g d(X, RN )= Inf {d(((x), (x)), ( y1, y2 )) : x X, ( y1, y2 ) RN } and g g d((0,0), RN )= Inf {d((0,0),(y1, y2 )) : (y1, y2 ) RN }.

g g Another new function could serve as definition of contradiction degree:

d(X, RN ) Ng Ng g 1- v) C5 (, ) = N = N (1- C4 (, )) g g d((0,0), RN ) g Ns For the standard negation N (y)=1-y the equality C5 (, ) = CiNs (, ) is verified, for all i=1,2,3,4.

s Ng Ng 2 Ng And for N with g(y)=y2 is C5 (, ) = 1-(1- C4 (, )) = C3 (, ).

g Considering N g-self-contradiction as a particular case of N g-contradiction between two fuzzy sets with =, the N g-contradiction degrees given in 3.1 are turned into the following N g-self-contradiction degrees:

Ng Ng Ng Ng i) Cs1 () = C1 (, ) = Max0, Inf (N ((x)) - (x)) = C2 (, ) = Cs2 () g xX N Ng ii) Cs3g () = C3 (, ) = Max0,1- 2 Sup(g((x))) xX N This measure of N g-self-contradiction, Cs3g (), was also defined in [3].

d Sup (x), Sup (x), RN g d(X, RN ) Ng Ng g xX xX iii) Cs4 () = C4 (, ) = d((0,0), RN )= d((0,0), RN ) g g d Sup (x), Sup (x), RN g d(X, RN ) Ng Ng xX xX g 1- iv) Cs5 () = C5 (, ) = N g g d((0,0), RN ) = N 1- d((0,0), RN ) g g Fourth International Conference I.TECH 2006 1- y r r Proposition 3.2 Let N be a Yager strong negation, N ( y) = (1- y ), with 0g g 1+ y a Sugeno strong negation, then for all [0,1]X it is:

0 if Sup (x) ng xX d(X, RN )= g ) in other case dSup (x), Sup (x),(ng, ng xX xX and d((0,0), RNg )= d((0,0),(ng, ng ))= 2ng = 2ng Consequently, Sup((x)) Sup((x)) Sup((x)) N Ng xX xX xX Cs4g () = Max and Cs5 () = N Max0,1- = N g 1- Min 0,1ng ng g 1, ng Ng this last measure of N g-self-contradiction, Cs5 (), was introduced in [1].

However, a similar result is not true for all strong fuzzy negation, as the following example shows.

Example 3.3 Let N be with g(y)=y3; in this case d((0,0), RN )= 1 and d((0,0), (ng, ng ))= 26 > 1 ( ng = ( ) ).

g g 1 d((, ), RNg ) 1 N 10 Given [0,1]X such that X =, it is Cs4g () = = 0.9 and however d((0,0), RNg ) 10 1 Sup(x) N N 10 Max0,1- = 1- = 0.874. Moreover, Cs5g () = N (1- Cs4g ())= N (0.1) N.

g g g ng ng ng Until now, we have managed contradiction depending on a fixed strong negation. We continue studying contradiction without depending on any fixed negation.

The following degrees of contradiction, between two fuzzy sets, were given in [5].

Definition 3.4 Given, [0,1]X, we have the following contradiction measure functions:

i) C1(, ) = Min(d(X, L1), d(X, L2)), denoting L1 the line y =1 and L2 the line y =1.

1 ii) C2 (, ) = 0 if there exists {x } X such that lim{(xn )}=1 or lim{ (xn )}=1 and, in other case n n N n n Sup((x) + (x)) d1(X, (1,1)) xX C2 (, ) = 1- =, being d the reticular distance.

2 d1((0,0), (1,1)) iii) C3 (, ) = 0 if there exists {x } X such that lim{(xn )}=1 or lim{ (xn )}=1, and, in other case n n N n n d(X, (1,1)) C3 (, ) =.

d((0,0),(1,1)) Newly, considering self-contradiction as a particular case of contradiction between two fuzzy sets with =, the contradiction degrees given in 3.4 are turned into the following self-contradiction degrees:

i) Cs1() = C1(, ) = Inf (1- (x)) = 1- Sup((x)) = C2 (, ) = Cs2 (), the measure of self-contradiction xX xX 1- Sup((x)) was also introduced in [1].

xX d Sup (x), Sup (x), (1,1) 2 1- Sup (x) d(X, (1,1)) xX xX xX ii) Cs3 () = C3 (, ) = = = = Cs1() 2 2 Information Models Contradiction Degrees and Connectives In this section, the problem of consistency with connectives, will be managed. In fact, if we have noncontradictory premises, and these ones are relaxed (by an OR connective, that is, by means of a t-conorm), then the new information must also be non-contradictory. And, in a similar way, if we have contradictory premises, and we add new information ( by an AND connective, of a t-norm), the information must also be contradictory.

The following results handle this subject.

Proposition 4.1 Given [0,1]X, if is not N g–self-contradictory, for a strong fuzzy negation N, then S(,) is g not N g–self-contradictory, for all S t-conorm and for all [0,1]X.

In particular, if, [0,1]X are not N g-contradictory then or is not N g-self-contradictory and subsequently S(,) is not N g-self-contradictory, for all S t-conorm.

Proposition 4.2 Given [0,1]X, if is not self-contradictory (Sup{(x): x X}=1), then S(,) is not selfcontradictory for all S t-conorm and for all [0,1]X (Sup{S((x),(x)): x X}=1).

In particular, if, [0,1]X are not contradictory then or is not self-contradictory and subsequently S(,) is not self-contradictory, for all S t-conorm.

Then, it is obtained that the disjunction with non-contradictory information provides non-self-contradictory information.

In addition, the definitions of measures of contradiction also must be consistent with the idea that a disjunction with non-contradictory information remains non-contradictory. Indeed, we have the following result:

Ng Proposition 4.3 Given Ci with i=1,2,3,4,5 (or Ci with i=1,2,3), and, [0,1]X, if CiN g (, ) = 0 (or N Ci (, ) = 0 ), then for any t-conorm S it holds that Csi g (S(, )) = 0 (or Csi (S(, )) = 0 ).

In general, for all weak measure of self-contradiction (that is, C :[0,1]X [0,1] such that C( ) = 1, C() = 0 if normal and C anti-monotonic, as defined in [3]) it is verified that: if C()=0 then C(S(,))=0 for all S t-conorm X X and [0,1]X. Furthermore, for all weak measure of contradiction (that is, C :[0,1] [0,1] [0,1] such that C(, ) = 1, C(, ) = 0 if normal and C symmetric and anti-monotonic [3]) it is verified that: if C(,)=then C(S(,),S(,))=0 for all S t-conorm.

Proposition 4.4 Given [0,1]X, if is N g–self-contradictory, for some strong fuzzy negation N, then T(,) is g N g–self-contradictory, for all t-norm T and for all [0,1]X.

Moreover, if, [0,1]X are N g-contradictory then T(,) is N g-self-contradictory, for all t-norm T.

Proposition 4.5 Given [0,1]X, if is self-contradictory (Sup{(x): x X}<1), then T(,) is self-contradictory for all t-norm T and for all [0,1]X (Sup{T((x),(x)): x X}<1).

Moreover, if, [0,1]X are contradictory then they are N g-contradictory, for some strong fuzzy negation N, g and consequently T(,) is N g-self-contradictory, and therefore T(,) self-contradictory, for all t-norm T.

Then, it is obtained that the conjunction with contradictory information provides self-contradictory results.

Similarly, definitions of measures of contradiction also must be consistent with the idea that a conjunction with contradictory information must remain contradictory. Indeed, we have the following result:

Ng Ng Proposition 4.6 Given Ci with i=1,2,3,4,5 ( Ci with i=1,2,3), and,[0,1]X, if Ci (, ) > 0 (or Ng Ci (, ) > 0 ), then for any t-norm T it holds that Csi (T (, )) > 0 (or Csi (T (, )) > 0 ).

Fourth International Conference I.TECH 2006 -In particular, if T is a t-norm of the Lukasiewicz’s family, that is, T = g W (g g), with W(x,y)=Max(0,x+y-1), Ng Ng where g is an order automorphism in the unit interval, it holds that if Ci (, ) > 0 then Csi (T (, )) = 1, or equivalently, T(,)=.

In general, for all weak measure of self-contradiction it is verified that: if C()>0 then C(T(,))>0 for all t-norm T and [0,1]X. In a similar way, for all weak measure of contradiction it is verified that: if C(,)>0 then C(T(,),T(,))>0 for all t-norm T.

Contradiction in Inference For inference purposes in both classical and fuzzy logic, neither the information itself should be contradictory, nor should any of the items of available information contradict each other. In order to avoid these troubles in fuzzy logic, it is necessary to study self-contradiction and contradiction in the fuzzy inference systems.

The Compositional Rule of Inference ([4]) is based on the Zadeh’s Logical Transform:

TJ ()(y) = SupT ((x), J (x, y)) xX Where J : X X [0,1] is a given fuzzy relation, T a t-norm and [0,1]X any fuzzy set. We aim to study the relationship between the contradiction in the input and the contradiction in the output TJ (). Also, we want to research the relationship between the degrees of contradiction of the input and the degrees of contradiction of the output TJ ().

Proposition 5.1 Given [0,1]X, if is N g–self-contradictory (or self-contradictory), then TJ () is N g–selfcontradictory (or self-contradictory), for all t-norm T and all fuzzy relation J.

Reciprocals are not true, as the following example shows.

Example 5.2 Let us consider the set X = [0,1], [0,1]X such that (x)=1-x, J(x,y)=Min(x,y) and T(x,y)=Min(x,y) 1 for all x, y [0,1]. Therefore, TJ ()(y) = Min, y and thus Sup TJ ()(y) =. Then, TJ () is N s-self2 y[0,1] contradictory and self-contradictory but is neither N s-self-contradictory nor self-contradictory ( Sup (x) = 1).

x[0,1] Moreover, if is N g-self-contradictory (or self-contradictory) then, from proposition, 5.1 and 2.1 (or 2.3), it is obtained that and TJ () are N g-contradictory (or contradictory) between them, for all t-norm T.

Proposition 5.3 Given [0,1]X and a reflexive fuzzy relation J, (that is, J(x,x)=1 xX), is N g–selfcontradictory (or self-contradictory) if and only if TJ () is N g–self-contradictory (or self-contradictory), for all tnorm T.

In addition, if J is a reflexive fuzzy relation, then is N g–self-contradictory (or self-contradictory) if and only if and TJ () are N g–contradictory (or contradictory) between them, for all t-norm T.

Now, let us study if there is some relationship between the contradiction measures of the input and those of the inference output TJ ().

Proposition 5.4 Given a reflexive fuzzy relation J and [0,1]X such that C( )=0 then C(TJ () )=0, for all C weak contradiction measure.

If J is not reflexive the last proposition is not true, in general, as the following example shows.

Information Models Example 5.5 Let us consider X,, T and J as in the example 5.2; J(x,x)=Min(x,x)=x. Then, J is not reflexive.

Moreover, (x)=1-x is a normal fuzzy set, so C()=0 for all C weak contradiction measure (in particular for C ), si however Csi (TJ ())= 1- Sup TJ ()(y) = y[0,1] Also, if J is a reflexive fuzzy relation it is TJ () and therefore C() C(TJ ()), for all weak contradiction measure C.

Finally, let us see that for the N g-self-contradiction and self-contradiction degrees considered in this paper, the equality between the contradiction degree of the input and the contradiction degree of the output TJ () is verified; being of interest, for it, to consider a previous proposition.

Proposition 5.6 Given [0,1]X, for all J fuzzy relation and all t-norm T, the inequality Sup TJ ()(x) Sup (x) xX xX holds.

Consequently, if J is reflexive, it is Sup TJ ()(x) = Sup (x).

xX xX Ng Ng Corollary 5.7 Given [0,1]X, if J is a reflexive fuzzy relation it is Csi ()= Csi (TJ ()) and Ng Csi ()= Csi (TJ ()) for all i and for all t-norm T, being Csi () and Csi () the N g-self-contradiction and selfcontradiction degrees given in definition 3.1 and 3.4.

Conclusion This paper deepens on the study of contradictoriness in fuzzy sets. New self-contradiction measures have been obtained by means of contradiction measures between two fuzzy sets when the two sets are the same.

Furthermore, some results about the propagation of contradictoriness throughout connectives (t-norms and tconorms) have been attained. As it was expected, these results are coherent with the human intuition.

Finally, the compositional rule of inference, commonly used in reasoning processes, is studied from the point of view of the contradiction. Results prove non-contradictoriness of input, assure the same property in the output.

Bibliography [1] E. Castieira, S. Cubillo and S. Bellido. Degrees of Contradiction in Fuzzy Sets Theory. Proceedings IPMU'02, 171-176.

Annecy (France), 2002.

[2] E. Castieira, S. Cubillo. and S. Bellido. Contradiccin entre dos conjuntos. Actas ESTYLF'02, 379-383. Len (Spain), 2002, (in Spanish).

[3] S. Cubillo and E. Castieira. Measuring contradiction in fuzzy logic. International Journal of General Systems, Vol. 34, N1, 39-59, 2005.

[4] H. T. Nguyen and E. A. Walker. A first course in fuzzy logic. CRC Press, 1997.

[5] C. Torres, E. Castieira S Cubillo and V. Zarzosa. A geometrical interpretation to define contradiction degrees between two fuzzy sets. International Journal “Information Theories and Applications”. 2005.

[6] E. Trillas. Sobre funciones de negacin en la teora de conjuntos difusos. Stochastica III/1, 47-60, 1979 (in Spanish).

Reprinted (English version) (1998) in Avances of Fuzzy Logic. Eds. S. Barro et altr, 31-43.

[7] E. Trillas, C. Alsina and J. Jacas. On Contradiction in Fuzzy Logic. Soft Computing, 3(4), 197-199, 1999.

[8] E. Trillas and S. Cubillo. On Non-Contradictory Input/Output Couples in Zadeh's CRI. Proceedings NAFIPS, 28-32. New York, 1999.

[9] L. A. Zadeh. Fuzzy Sets. Inf. Control, volume 20, 301-312, 1965.

Fourth International Conference I.TECH 2006 Authors' Information Carmen Torres – Dept. Applied Mathematic. Computer Science School of University Politcnica of Madrid.

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