Enric Trillas

A Survey on Fuzzy Implication Functions

One of the key operations in fuzzy logic and approximate reasoning is the fuzzy implication, which is usually performed by a binary operator, called an implication function or, simply, an implication. Many fuzzy rule based systems do their inference processes through these operators that also take charge of the propagation of uncertainty in fuzzy reasonings. Moreover, they have proved to be useful also in other fields like composition of fuzzy relations, fuzzy relational equations, fuzzy mathematical morphology, and image processing. This paper aims to present an overview on fuzzy implication functions that usually are constructed from t-norms and t-conorms but also from other kinds of aggregation operators. The four most usual ways to define these implications are recalled and their characteristic properties stated, not only in the case of [0,1] but also in the discrete case.

ON IMPLICATION AND INDISTINGUISHABILITY IN THE SETTING OF FUZZY LOGIC

Many implication connectives used in the Fuzzy Set Theory are obtained from DeMorgan triples by using the formalism of either the Boolean Logic, the Intuitionistic Logic or the Quantum Logic. This paper aims to give a comprehensive survey of the characteristics and main properties of such implications by paying special attention to the two properties which are considered as the most essential to an implication connective, i.e. a) by means of the implication an order relation is obtained, and b) the basic rule of inference, the Modus Ponens, is satisfied. The properties of the material equivalence obtained from each of these implications are also checked.

On the law [p/spl and/q/spl rarr/r]=[(p/spl rarr/r)V(q/spl rarr/r)] in fuzzy logic

This paper deals with the logical equivalence of the classical propositional calculus [p/spl and/q/spl rarr/r]=[(p/spl rarr/r)V(q/spl rarr/r)]. This equality seems to play a central role in a recent discussion around a paper of Combs and Andrews (1998). After reconsidering the equivalence in lattices, its validity in the standard theories of fuzzy sets endowed with an implication operator is studied.

On the use of words and fuzzy sets

This paper only tries to stimulate some reflections, by showing a possible way towards rethinking fuzzy sets from their roots. A rethinking that does not change the view that fuzzy sets are mathematical entities giving extension to predicates. In such a way that, if the predicates are precise, the corresponding new entities are nothing else than classical sets. What, perhaps, is a key idea is that the use of a predicate organizes, in some way, the universe of discourse. When this organization is a preorder, the extension or L-set, appears once a degree, numerical or not, but consistent with the organization, can be defined. The ultimate goal of those above mentioned reflections, provided they would be realized in the future, is to extend the current theories of fuzzy sets to wider areas of both language and reasoning. Our objective is to reach a better knowledge of the links between language and its representations for the progress of computing with words and perceptions.

On the representation of fuzzy rules

In fuzzy logic, connectives have a meaning that, can frequently be known through the use of these connectives in a given context. This implies that there is not a universal-class for each type of connective, and because of that several continuous t-norms, continuous t-conorms and strong negations, are employed to represent, respectively, the and, the or, and the not. The same happens with the case of the connective If/then for which there is a multiplicity of models called T-conditionals or implications. To reinforce that there is not a universal-class for this connective, four very simple classical laws translated into fuzzy logic are studied.

On antonym and negate in fuzzy logic

The abilities to speak well and to conceptualize seem to be closely linked. It has been maintained that the human brain has a preference for binary oppositions or polarities. The notions of antonym and negate are examples of polarity between the pairs of predicates P−no P, P−ant P. Other characteristics as mutual exclusivity, complementation, or reciprocity are applied, in some cases, to them. However, if negation is a general phenomenon in natural languages, the use of antonyms is more usual for graduate predicates. For this reason, antonyms were considered very early in fuzzy set theory. 1–4 In this work, some relations between antonyms and negates are analyzed in the frame of fuzzy set theory, showing both similarities and dissimilarities between these two concepts. The last goal is to get automatic methods to build concepts with an adequate and easy interpretation. This paper is an experimental‐theoretic intent on the way of establishing a mathematical model of antonymy in agreement both with some linguistic facts and with some uses in fuzzy control. ©1999 John Wiley & Sons, Inc.

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

When QM-operators are implication functions and conditional fuzzy relations

Some fuzzy reasoning systems base inference processes on fuzzy implication functions. Although there has been a great deal of work done on characterizing R‐ and S‐implications, little is known about QM‐implications in spite of their long history since they came to fuzzy logic by analogy with the quantum mechanic logic. This paper tackles the study of some characteristics of this type of operator. It focuses on the QM‐implication operator both as an implication function and also as a T‐conditional function, giving useful tools to characterize them. © 2000 John Wiley & Sons, Inc.

On contradiction in fuzzy logic

Abstract We clarify which space of functions in [0, 1]E would be reasonable in fuzzy logic in order to avoid self-contradiction.

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.