1 + x2 1/4=P (A) P (A) = 2 area(R)=1/f f R - 3 /p 0 3 /p A = (-,- 3/p] U [+ 3/p, + ) Figure 2: Density function of the Cauchy distribution.

4. A Survey of the Coherence in Some Notable Distributions Cases In this section, we deal with the coherence between some notable distributions and the possibility measures generated by the density functions of the above distributions.

4.1. Coherence and Normal Distribution Bearing in mind how important the normal distribution is, this section is given over to studying the coherence between the probability and possibility generated by its density function.

Fourth International Conference I.TECH 2006 As discussed in section 3.1, it holds that the density functions of the distributions with R, are N(,1/ 2), also possibility distributions; furthermore, they are the only ones within the normal family, as it should hold that -(x- )1 2 sup e = = xR 2 then necessarily has to be = 1 2.

Theorem 4.1. Let f be the density function of the normal distribution if and Pf are, respectively, N(,1/ 2), f the possibility and probability measures generated by f, then Pf (A) (A) for all A B.

f Proof: It can be proven, without loss of generality, for f (x) = e-x which corresponds to since any N(0,1/ 2), of the others is a translation of this one, and the relationship between probability and possibility will be the same.

Firstly, we will check that Pf ((-,-a] [a,+)) ((-,-a] [a,+)) for all a 0. Indeed, if a 1, it is f 2 e-x xe-x for any x a, then + 2 + e-a Pf ((-,-a] [a,+))= 2 e-x dx 2 xe-x dx = < f (a) = ((-,-a] [a,+)).

f aa 2 + If a [0,1), the function G(a) = f (a) - Pf ((-,-a] [a,+))= e-a - 2 e-x dx is non-negative. Indeed, a 2 + 2 a d from G'(a) = -2ae- a - 2 e-x dx - e-x dx = 2e-a (-a + 1) it follows that G is increasing 0 da in [0,1/ )and decreasing in(1/,1], moreover as G(0) = 0 and + 2 + then G(a) 0 for all a [0,1].

G(1) = e- - 2 e-x dx e- - 2 e-xdx = e- 1- > 0, 1 Finally, let us see that Pf (A) (A) for any A B. If 0 is an accumulation point of A, then f Pf (A) 1 = f (0) = (A). If 0 is not an accumulation point of A, then there exists a > 0 such that f A (-,-a] [a,+) and a or - a is either an element of A or an accumulation point of A. Therefore, Pf (A) Pf ((-,-a] [a,+)) ((-,-a] [a,+))= f (a) = (A). f f 2 P (A) = area(2R)#e-pa = P (A) f f e-pa = P (A) f R -a a A=(-,-a] U [a,+ ) Figure 3: Density function of the normal distribution.

4.2. Coherence and Other Distributions Even though important distributions, like the Cauchy distribution, do not generate coherent probabilities and possibilities as they are considered here, we can find other common distributions, apart from the important case of the normal distribution, which also generate coherent probabilities and possibilities. Let us take a look at some of these.

Information Models 1. Uniform distribution, with density function f (x) = 1 if | x - a | 1/ 2 and f (x) = 0 if | x - a |> 1/ 2.

Trivially, Pf (A) (A) is satisfied for any A B, since f (x)dx = L (A [a -1/ 2,a +1/ 2]).

f A 2. Simpson's distribution, with density function f (x) = 1- | x - a | if | x - a |1 and f (x) = 0 if | x - a |>1.

Let A B, if a is an accumulation point of A, then (A) = 1 P (A). If a is not an accumulation point of f f A, there exists (0,1)such that A (-, a - ][a +,+)and (A) = f (a + ) = f (a - ) = 1 -.

f Therefore, Pf (A) Pf ((-, a - ][a +,+)) = (1 - )2 < 1 - = (A).

f P f(A) P (A)=2area(R) < P f(A) f R a-1 0 a-e a a+e a +Figure 4: Density function of Simpson’s distribution.

3. Exponential distribution, with density function f (x) = e- x if x 0, and f (x) = 0 if x < 0. For each a R:

+ a 0, • If it is Pf ([a,+))= e-xdx = e-a = ([a,+)).

f a + • If a < 0, it is Pf ([a,+))= e-xdx = 1 = ([a,+)).

f a For each A B, there exists a R such that A [a,+) and a A or a is an accumulation point of A;

thus, Pf (A) Pf ([a,+))= ([a,+))= (A).

f f P (A)= area(2R) < e-2a = P (A) f fm m P (A)=area(R)=P (A) f e-2a f P (A) f R R -a a a A=[a,+ ) A=(-,-a] U [a,+ ) (a) (b) Figure 5: (a) Density function of the exponential distribution, and (b) density function f (x) = e-2|x| 4. Finally, going back to the example in section 3.1, let f (x) = e-2|x| be the density function associated with the possibility distribution (x) = e-|x|.The probability and possibility measures generated by f are also coherent.

Indeed, for all a 0, + Pf ((-,a] [a,+))= 2 e-2xdx = e-2a = f (a) = ((-,a] [a,+)), f a from which we can deduce, just as we did for the normal law, that for all A B, Pf A) (A).

f Fourth International Conference I.TECH 2006 Conclusions and Further Works In this paper, we have discussed the topic of the coherence between probability and possibility measures in the continuous case, that is, when these measures are defined on -algebras in the set R of real numbers. For this purpose, we have firstly found functions that are density functions and possibility distributions at the same time and, then we have studied the coherence between probability and possibility measures generated by the same density function. Moreover, the case of some significant distributions has been analysed.

The problem of finding the closest probability to a given possibility is an interesting open problem, technically more complex than in the finite case, in which it was successfully accomplished in [2].

Acknowledgements This paper is supported by CICYT (Spain) under Project TIN 2005-08943-C02-01.

Bibliography [1] G. Birkhoff. Lattice Theory. American Mathematical Society, Providence, 1973.

[2] E. Castieira, S. Cubillo and E. Trillas. On Possibility and Probability Measures in finite Boolean algebras. SoftComputing, 7 (2), 89-96, 2002.

[3] H. Cramer. Mtodos matemticos de estadstica. Aguilar, Madrid. (in Spanish), 1970.

[4] M. Delgado and S. Moral. On the concept of Possibility-Probability Consistency. Fuzzy Sets and Systems, 21, 311-318, 1987.

[5] D. Dubois and H. Prade. Thorie des possibilits. Applications la reprsentation des connaissances en informatique.

Masson, Paris. (in French), 1988.

[6] D. Dubois, H. T. Nguyen and H. Prade, Possibility Theory, Probability and Fuzzy Sets: Misunderstandings, Bridges and Gaps" in Fundamentals of Fuzzy Sets, D.Dubois and H. Prade Eds. Kluwer Academic Publishers, 2000.

[7] D. Dubois, H. Prade and S. Sandri. On possibility/probability transformations. Proceedings of 4th IFSA Conference, (Brussels), 50-53, 1991.

[8] J. A. Drakopoulus. Probabilities, possibilities and fuzzy sets. Fuzzy Sets and Systems, 75, 1-15, 1995.

[9] P. R. Halmos. Measure Theory. Springer-Verlag, New York, 1988.

[10] M. Love. Probability theory I. Springer-Verlag, New York, 1977.

[11] E. Trillas. Sobre funciones de negacin en la teora de conjuntos difusos". Stochastica, 1 (3), 47-60 (in Spanish), 1979.

Reprinted (English version) in Avances in Fuzzy Logic, (eds. S. Barro et altri) Universidad de Santiago de Compostela, 31-43, 1998.

[12] L.A. Zadeh. Fuzzy sets as a basis for a Theory of Possibility. Fuzzy Sets and Systems, 1, 3-28, 1978.

Authors' Information Elena Castieira – Dept. Applied Mathematic. Computer Science School of Technical University of Madrid.

Campus Montegancedo. 28660 Boadilla del Monte (Madrid). Spain; e-mail: ecastineira@fi.upm.es Susana Cubillo – Dept. Applied Mathematic. Computer Science School of Technical University of Madrid.

Campus Montegancedo. 28660 Boadilla del Monte (Madrid). Spain; e-mail: scubillo@fi.upm.es Enric Trillas – Dept. Artificial Intelligence. Computer Science School of Technical University of Madrid. Campus Montegancedo. 28660 Boadilla del Monte (Madrid). Spain; e-mail: etrillas@fi.upm.es Information Models RELATIONSHIP BETWEEN SELFCONTRADICTION AND CONTRADICTION IN FUZZY LOGIC* Carmen Torres, Susana Cubillo, Elena Castineira Abstract: This paper focuses on the study of self-contradiction as a particular case of contradiction between two fuzzy sets, and so, some self-contradiction degrees are defined from the contradiction degrees between two fuzzy sets. Furthermore, the definitions of measures of contradiction must be consistent with the idea that a disjunction with non-contradictory information remains non-contradictory, and a conjunction with contradictory information must remain contradictory; in this sense, some results are attained. Finally, contradiction in the compositional rule of inference is studied.

Keywords: fuzzy sets, t-norm, t-conorm, strong fuzzy negations, contradiction, measures of contradiction, fuzzy relation, compositional rule of inference.

Introduction and Preliminary Definitions The study about contradiction was initiated by Trillas et al. in [7] and [8]. They introduced the concepts of both self-contradictory fuzzy set and contradiction between two fuzzy sets. Moreover, the need to study not only contradiction but also the degree of such contradiction is pointed out in [1] and [2], suggesting some measures for this purpose. In [5] new ways to measure the contradiction degree are obtained dealing with the problem from a geometrical point of view.

This paper begins, as a previous step, with a study on the relation between self-contradiction and contradiction between two fuzzy sets. Then, taking into account that the self-contradiction of a fuzzy set could be understand as the contradiction with itself, remembering some contradiction degrees defined in [5], the corresponding selfcontradiction degrees for a fuzzy set, will be proposed, firstly, depending on a given strong negation, and later, without depending on any fixed negation.

In the following section, the problem of consistency with connectives will be managed. In fact, it is necessary to obtain non-contradictory knowledge, when the premises of non-contradictory information are relaxed. And, in a similar way, the information obtained adding contradictory premises, must also be contradictory.

Finally, last section will be devoted to study how contradictoriness is transmitted in the reasoning throughout the Compositional Rule of Inference.

Previously, we will remember some definitions and properties necessary throughout this article.

Definition 1.1 ([9]) A fuzzy set (FS) P, in the universe X, is a set given as P={(x, (x)): x X} such that, for all x X, (x) [0,1], and where the function [0,1]X is called membership function. We denote F(X) the set of all fuzzy sets on X.

Definition 1.2 PF(X) with membership function [0,1]X is said to be a normal fuzzy set if Sup{(x) : xX}=1.

Definition 1.3 A fuzzy negation (FN) is a non-increasing function N: [0,1] [0,1] with N(0)=1 and N(1)=0.

Moreover, N is a strong fuzzy negation if the equality N(N(y))=y holds for all y [0,1].

* This work is supported by cicyt (Spain) under project tin 2005-08943-c02-001.

Fourth International Conference I.TECH 2006 N is a strong negation if and only if, there is an order automorphism g in the unit interval (that is, g:[0,1] [0,1] is an increasing continuous function with g(0)=0 and g(1)=1) such that N(y)=g-1(1-g(y)) for all y [0,1] (see [5]);

from now on, let us denote N =g-1(1-g). Furthermore, the only fixed point of N is n =g-1(1/2).

g g g Definition 1.4 ([4]) A function T: [0,1] x [0,1] [0,1] is said to be a t-norm if it is a commutative, associative and non-decreasing in both variables function verifying T(y,1)=y for all y [0,1].

Definition 1.5 ([4]) A function S: [0,1] X [0,1] [0,1] is said to be a t-conorm if it is a commutative, associative and non-decreasing in both variables function verifying S(y,0)=y for all y [0,1].

Definition 1.6 ([7]) Given, [0,1]X and a strong negation N, then and are N g g-contradictory if and only if (x) N ((x )), for all x X. This inequality is equivalent to g((x))+g((x)) 1, for all x X.

g Definition 1.7 ([7]) Given [0,1]X and a strong negation N, is said to be N g g-self-contradictory if and only if (x) N ((x)), for all x X. This inequality is equivalent to g((x)) 1/2, for all x X.

g Therefore, the definition of N g-self-contradictory fuzzy set is a particular case from that of N g-contradictory fuzzy sets, where the two sets are the same.

Definition 1.8, [0,1]X are contradictory if they are N g-contradictory regarding some strong FN N. And is g self-contradictory if it is N g-self-contradictory for some strong FN N. This condition is equivalent to the fact that g is not a normal fuzzy set (Sup{(x) : xX}<1). Again, the definition of self-contradiction is a particular case from that of contradiction.

Self-contradiction and Contradiction between Two FS The goal of this section is study if there exists some direct relation between the self-contradiction of two fuzzy sets and the contradiction between them. In fact, we have the following properties.

Proposition 2.1 Given, [0,1]X, if and are N g–self-contradictory, for some strong fuzzy negation N, g then, are N g–contradictory.

The following example shows that reciprocal is not true.

Example 2.2 Let us consider the set X={x,y} and, [0,1]X such that (x)=3/4, (y)=0 and (x)=0, (y)=3/4;

and the standard negation N =1-id. Then (x)+(x)=3/4 and (y)+(y)=3/4 and so, are N s s-contradictory between them. Nevertheless and are not N s-self-contradictory ((x)>1/2 and (y)>1/2).

Proposition 2.3 Given, [0,1]X, if and are self-contradictory, then, are contradictory.

Proof: As, [0,1]X are self-contradictory there exist order automorphisms g and g’ on [0,1], such that g((x))1/2 and g’((x))1/2 for all xX. Let us take the following function on [0,1], g’’=Min{g,g’}. This function is continuous because g and g’ are continuous; g’’(0)=0, g’’(1)=1. Let us see that g’’ is increasing: let y,y [0,1] be 1 such that y

Newly, reciprocal is not true as the following example shows.

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