The paradoxical success of fuzzy logic

International Journal of Intelligent Systems, 1991

One of the main features of Fuzzy Logic is its capability to deal with the concept of compatibility between two propositions, in such a way that the inference process modeled through the Compositional Rule of Inference is independent from the particular possibility distributions involved. It is in this context that the compatibility functions can be considered as fuzzy truth values, labels or qualifications, playing the same role as the values true and false play in the Classical Logic, where the meaning of propositions is nothing but its truth value. In this article we consider a restricted family of labels having the following desirable properties: (a) easy parametric representation, (b) easy semantic interpretation, (c) to allow a gradation in the family according to the modifications performed by each label, and (d) to be closed under inference processes (FRfunctions), and also under some suitable and meaningful operations between them.

Mathematical Principles of Fuzzy Logic, 1999

ArXiv, 2021

A presentation is provided of the basic notions and operations of a) the propositional calculus of a variant of fuzzy logic-canonical fuzzy logic, CFL-and in a more succinct and introductory way, of b) the theory of fuzzy sets according to that same logic. The propositional calculus of bivalent classical logic and classical set theory can be considered as particular cases of the corresponding theories of CFL if the numerical value of a specific parameter w is restricted to only two possibilities, 0 and 1.

Towards Hybrid and Adaptive Computing, 2010

1999

Electronic Notes in Theoretical Computer Science, 2006

There are several ways to extend the classical logical connectives for fuzzy truth degrees, in such a way that their behavior for the values 0 and 1 work exactly as in the classical one. For each extension of logical connectives the formulas which are always true (the tautologies) changes. In this paper we will provide a fuzzy interpretation for the usual connectives (conjunction, disjunction, negation, implication and bi-implication) such that the set of tautologies is exactly the set of classical tautologies. Thus, when we see logics as set of formulas, then the propositional (classical) logic has a fuzzy model.

Fuzzy Sets and Systems, 1998

We discuss some issues on fuzzy logic developed by Pavelka [Z. Math. Logic 25 (1979) 45-52, 119-134, 447-464] vs. the many-valued Lukasiewicz's logic. The focus is on the proper choice of fuzzy implication operations, a question which has been addressed many times in the fuzzy research literature. Pavelka had shown in 1979 that the only natural way of formalizing fuzzy logic for truth values in the unit interval [0, 1] is by using Lukasiewicz's implication operator a --+ b = min { 1, 1 -a + b} or some isomorphic form of it. Many other papers around the same time had attempted to formulate alternative definitions for a --* b by giving intuitive justifications. There continues to be some confusion, however, even today about the right notion of fuzzy logic. Much of this confusion lies in the use of improper "and" ("or") and the "not" operations and a misunderstanding of some of the key differences between "proofs" or inferencing in fuzzy logic and those in Lukasiewicz's logic. We point out the need for defining the strong conjunction operator "®" in connection with fuzzy Modus-ponens rule and why we do not need the fuzzy Syllogism inference rule. We formulate two requirements of the fuzzy implication operator, which are satisfied by Lukasiewicz's a --* b, but which fail for many of the alternative definitions for "--* ".

The representation of human-originated information and the formalization of commonsense reasoning has motivated different schools of research in Artificial or Computational Intelligence in the second half of the 20th century. This new trend has also put formal logic, originally developed in connection with the foundations of mathematics, in a completely new perspective, as a tool for processing information on computers. Logic has traditionally put emphasis on symbolic processing at the syntactical level and binary truth-values at the semantical level. The idea of fuzzy sets introduced in the early sixties and the development of fuzzy logic later on [Zadeh, 1975a] has brought forward a new formal framework for capturing graded imprecision in information representation and reasoning devices. Indeed, fuzzy sets membership grades can be interpreted in various ways which play a role in human reasoning, such as levels of intensity, similarity degrees, levels of uncertainty, and degrees of preference.