Searching for the roots of non-contradiction and excluded-middle

This paper tries to obtain frameworks in which one can prove as theorems, and with few assumptions, the laws of non-contradiction (NC) and excluded-middle (EM), for a large class of very general systems including orthocomplemented and De Morgan lattices, as well as the theories of L -fuzzy sets. Its goal is nothing else than to begin with a search for the algebraic roots of these laws. For such a goal, and similarly to ancient Aristotelian logic, the laws are referred to self-contradiction and not to falsity as it is done in modern logics. It is additionally shown that there are some structures where it is possible to find special operations that separate those operations that verify NC from those verifying EM.     

An essay on the linguistic roots of fuzzy sets

This paper mainly tries to show that the membership function of a fuzzy set labeled P does show some intrinsic property related with how P is actually managed in the universe of discourse. Its final goal is to analyze an answer to the question, which intrinsic but simple property allows a function to represent a fuzzy set labeled P? The presented property exhibits that the membership function just ‘measures’ in some scale the extent up to which x is P in language, for all x in the universe of discourse.Such study is done in a form allowing to consider how to represent the ‘collective’ originated by a predicate reflecting a collective noun. As particular cases of what is presented, and when the degrees can be some kinds of numerical subsets, the Zadeh’s fuzzy sets, the interval-valued, the intuitionistic, and the type-2 fuzzy sets, appear as particular cases and to some extent are discussed. A ‘unification’ of all different kinds of fuzzy sets based on a linguistic origin is achieved.     

Photocatalyzed Degradation of Phenol, 2,4-Dichlorophenol,Phenoxyacetic Acid and 2,4-Dichlorophenoxyacetic Acid over SupportedTiO2 in a Flow System

The photodegradation of phenol, 2,4-dichlorophenol, phenoxyacetic acid and 2,4-dichlorophenoxyacetic acid using TiO₂ as photocatalyst is investigated. The photodegradations of these compounds have been conducted in a continuous mode by means of a flow system, in which TiO₂ remains fixed onto glass pearls. The use of this system gives high yields of degradation for the chemicals tested, except for 2,4-dichlorophenol for which a slow dechlorination process is observed. The rate of photodegradation depends on the pH of the solution, the point of zero charge of TiO₂ and the pK_a of the chemicals being the key parameters. The main aromatic intermediates detected have been hydroquinone,paraquinone and hydrohydroquinone during phenol degradation; phenol and hydroquinone during phenoxyacetic acid degradation; chlorohydroquinone and chlorophenol during 2,4-dichlorophenol degradation; and dichlorophenol during 2,4-dichlorophenoxyacetic acid degradation. Finally, some long term irradiations with phenol as model compound have been performed, showing high degrees of photodegradation. It has been observed that only a periodic evacuation of the effluent out of the reactor is needed to sustain high percentages of photodegradation.     

Some algebraic clues towards a syntactic view on the Principles of Non-Contradiction and Excluded-Middle

This short paper just considers the possibility of a new view for posing and proving Aristotle’s Principles of Non-Contradiction and Excluded-Middle. It is done by means of their refutability, or deducibility, respectively, under Tarski’s Consequence Operators.     

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 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.     

On the difference of fuzzy sets

We formulate and solve a collection of functional equations arising in the framework of fuzzy logic when modeling the concept of a difference operation between couples of fuzzy sets. © 2008 Wiley Periodicals, Inc.     

Characterizing non-monotonic fuzzy relations

 This paper is on &-T-monotonic Fuzzy Relations and on the particular case of &-Monotonic Fuzzy Relations (in the crisp case &-Weakly-monotonic and &-Strongly-monotonic Relations, respectively). It goes a little further of the previous work the authors did on this kind of Fuzzy and Crisp Relations by proving a Characterization Theorem of &-T-monotonicity for Reflexive Fuzzy Relations that contains all the former results. This theorem is not only particularized to the crisp case but it is shown that then only crisp Logical States are needed. Of course, also Non-&-T-monotonic Fuzzy Relations and Non-Weak-&-monotonic crisp Relations became characterized. Received: 20 May 1997/Accepted: 18 June 1997     

On the law [p∧q→r]=[(p→r)V(q→r)] in fuzzy logic

This paper deals with the logical equivalence of the classical propositional calculus [p∧q→r]=[(p→r)V(q→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