Interval-valued algebras and fuzzy logics

In this chapter, we present a propositional calculus for several interval-valued fuzzy logics, i.e., logics having intervals as truth values. More precisely, the truth values are preferably subintervals of the unit interval. The idea behind it is that such an interval can model imprecise information. To compute the truth values of ‘p implies q’ and ‘p and q’, given the truth values of p and q, we use operations from residuated lattices. This truth-functional approach is similar to the methods developed for the well-studied fuzzy logics. Although the interpretation of the intervals as truth values expressing some kind of imprecision is a bit problematic, the purely mathematical study of the properties of interval-valued fuzzy logics and their algebraic semantics can be done without any problem. This study is the focus of this chapter

REVIEW OF MODELING PREFERENCES FOR DECISION MODELS

A group decision problem is set in environments where there is a common issue to solve, a set of possible options to choose, and a set of individuals who are experts and express their opinions about the set of possible alternatives with the intention to reach a collective decision as the unique solution of the problem in question. The modeling of the preferences of the decision-maker is an essential stage in the construction of models used in the theory of decision, operations research, economics, etc. On decision problems experts use models of representation of preferences that are close to their disciplines or fields of work. The structures of information most commonly used for the representation of the preferences of experts are vectors of utility, orders of preference and preference relations. In decision problems, the expression of preferences domain is the domain of information used by the experts to express their preferences, the main are numerical, linguistic, and intervalar stressing the multi-granular linguistic. This paper is a review of these concepts. Its purpose is to provide a guide of bibliographic references for these concepts, which are briefly discussed in this document

The *-composition -A Novel Generating Method of Fuzzy Implications: An Algebraic Study

Fuzzy implications are one of the two most important fuzzy logic connectives, the other being t-norms. They are a generalisation of the classical implication from two-valued logic to the multivalued setting. A binary operation I on [0; 1] is called a fuzzy implication if (i) I is decreasing in the first variable, (ii) I is increasing in the second variable, (iii) I(0; 0) = I(1; 1) = 1 and I(1; 0) = 0. The set of all fuzzy implications defined on [0; 1] is denoted by I. Fuzzy implications have many applications in fields like fuzzy control, approximate reasoning, decision making, multivalued logic, fuzzy image processing, etc. Their applicational value necessitates new ways of generating fuzzy implications that are fit for a specific task. The generating methods of fuzzy implications can be broadly categorised as in the following: (M1): From binary functions on [0; 1], typically other fuzzy logic connectives, viz., (S;N)-, R-, QL- implications, (M2): From unary functions on [0,1], typically monotonic functions, for instance, Yager’s f-, g- implications, or from fuzzy negations, (M3): From existing fuzzy implications