Elena Castiñeira

Computing with Antonyms

This work tries to follow some agreements linguistic seem to have on the semantical concept of antonym, and to model by means of a membership function an antonym aP of a predicate P, whose use is known by a given μP

On conjectures in orthocomplemented lattices

Abstract A mathematical model for conjectures in orthocomplemented lattices is presented. After defining when a conjecture is a consequence or a hypothesis, some operators of conjectures, consequences and hypotheses are introduced and some properties they show are studied. This is the case, for example, of being monotonic or non-monotonic operators. As orthocomplemented lattices contain orthomodular lattices and Boolean algebras, they offer a sufficiently broad framework to obtain some general results that can be restricted to such particular, but important, lattices. This is, for example, the case of the structures theorem for hypotheses. Some results are illustrated by examples of mathematical or linguistic character, and an appendix on orthocomplemented lattices is included.

On the aggregation of some classes of fuzzy relations

This paper deals with the aggregation of fuzzy relations under a closure constraint, that is, it studies how to combine a collection of fuzzy relations that present some common Properties in order to obtain a single one with the same properties. This aggregation problem is addressed for two important classes of closely related fuzzy relations: generalized distances and indistinguishability operators.

Measuring contradiction in fuzzy logic

Several methods have been proposed within fuzzy logic for inferring new knowledge from the original premises. However, there must be some guarantee that the results contradict neither each other nor the initial information. In 1999, Trillas etu2009u2009al. introduced the concepts of both contradictory set and contradiction between two sets. Moreover, we established the need to study not only contradiction but also the degree of such contradiction in E. Castiñeira etu2009u2009al., “Degrees of contradiction in fuzzy sets theory”, Proceedings IPMU02, 2002a, pp. 171–176, Annecy (France), E. Castiñeira etu2009u2009al., “Contradicción entre dos conjuntos”, Actas ESTYLF02, 2002b, pp. 379–383, León (Spain) (in Spanish) establishing some measures for this purpose. Nevertheless, contradiction could have been measured in some other way. Elena Castiñeira Holgado was born in Asturias (Spain). She received her Bachelor Degree in Mathematics from the Complutense University of Madrid in 1985, and her Ph.D. in Computer Sciences from the Technical University of Madrid in 1994, with research on fractals. She has been working on non-classical logics, and, especially, fuzzy logic since 1997. The main goal of this paper is to propose an axiomatic definition of measure of contradiction both for a set and between two sets. A requirement for such modelling is that some minimum conditions be met. We also examine how well some measures proposed throughout the fuzzy logic literature fit this definition. Finally, we obtain some results on the relationship between measures of contradiction and the measures of ambiguity introduced by Fishburn for classical logic and by Yager for imprecision.

Measuring incompatibility between Atanassov's intuitionistic fuzzy sets

The aim of this paper is to establish an axiomatic definition of incompatibility measure in the framework of Atanassovs intuitionistic fuzzy sets and use geometrical methods to build some families of such incompatibility measures. First, we construct several functions to measure incompatibility for an intuitionistic t-norm that can be represented by an adequate t-norm and t-conorm. Additionally, we establish some relations between some particular cases of these functions. Similarly, we then obtain incompatibility measures for a family of non-representable intuitionistic t-norms.

SELF-CONTRADICTION AND CONTRADICTION BETWEEN TWO ATANASSOV'S INTUITIONISTIC FUZZY SETS

The paper focuses on the study of the contradiction between two Atanassovs intuitionistic fuzzy sets. First, taking into account some characterizations obtained in previous papers, some functions are defined in order to measure the degrees of contradiction. Besides the principal properties of these measures are pointed out. Finally, some results relating self-contradiction and contradiction between two Atanassovs intuitionistic fuzzy sets are achieved.

Measuring contradiction on a-IFS defined in finite universes

The work outlined here aims to build and examine contradiction measures on Atanassov intuitionistic fuzzy sets (A-IFS) that are defined in this particular case in finite universes. The axiomatic definition of contradiction measure in the A-IFS framework was given in [7]. A number of axioms formalizing the concept of continuity for the above measures were also given. In this paper, Section 1, which briefly discusses the preliminaries required to develop the work, is followed by a section analysing how the restriction of the universe of discourse to a finite set influences the continuity axioms. The following three sections look at three types of specific contradiction measures. In Section 3, continuous t-norms and fuzzy negations are used to construct a large family of measures. These measures satisfy different types of continuity, which are examined at length. Then, Sections and take up other families introduced in , and , proving that their behaviour with respect to continuity is better than it was in the earlier articles because the universes considered here are finite.

An axiomatic model for measuring contradiction and N-contradiction between two AIFSs

The importance of dealing with contradictory information or of deriving contradictory consequences in inference processes justifies undertaking a theoretical study on the subject of contradiction. In [S. Cubillo, E. Castineira, Contradiction in intuitionistic fuzzy sets, in: Proceedings of the Conference IPMU2004, Perugia, Italy, 2004, pp. 2180-2186] we defined contradictory and N-contradictory Atanassov intuitionistic sets, where we established that two sets A and B are N-contradictory, with respect to a given intuitionistic negation N, if A implies N(B), and are contradictory if they are N-contradictory for some negation N. The purpose of this article is to thoroughly examine the model for measuring contradiction between two Atanassov intuitionistic fuzzy sets irrespective of a fixed negation, proposed in [C. Torres-Blanc, E.E. Castineira, S. Cubillo, Measuring contradiction between two AIFS, in: Proceedings of the Eighth International FLINS Conference, Madrid, Spain, 2008, pp. 253-258], and also to introduce a mathematical model to measure N-contradiction between sets, where N is an intuitionistic negation. First, we justify and determine the minimum axioms that a function must satisfy to be able to be used as a measure of contradiction or a measure of N-contradiction. Also, we introduce some early examples of valid functions that conform to the model. Then, we establish the conditions for these measures to be continuous from below or continuous from above. Finally, we build families of contradiction and N-contradiction measures, establishing how they are relate to each other, and we look at how they behave with respect to continuity.

On possibility and probability measures in finite Boolean algebras

Abstractu2002In this paper, we study what possibility and necessity measures are like in a finite Boolean algebra, establishing a classification of possibility measures in this type of algebras. The relation between probability and possibility measures is studied. Some conditions for obtaining probabilities that are coherent with a possibility are given. Lastly, Euclidean distance is used for finding probabilities that are close to a given possibility. Also the closest probability is identified and turns out to be coherent.

Multi-argument fuzzy measures on lattices of fuzzy sets

In this paper, we axiomatically introduce fuzzy multi-measures on bounded lattices. In particular, we make a distinction between four different types of fuzzy set multi-measures on a universe X, considering both the usual or inverse real number ordering of this lattice and increasing or decreasing monotonicity with respect to the number of arguments. We provide results from which we can derive families of measures that hold for the applicable conditions in each case.