We explore the rating system used by credit agencies with a focus on problems that justify the use of fuzzy set theory. We prove that a fuzzy market is viable if and only if an equivalent martingale measure exists, from which we construct the forward probability measure and under which the discounted price of a default-free bond is a martingale. We model the evolution of credit migration of a d...

Indeterminacy makes the main distinction between fuzzy/intuitionistic fuzzy (and other extensions of fuzzy) set/logic vs. neutrosophic set/logic, and classical probability probability. Also, statistics plithogenic statistics, algebraic structures structures, crisp numbers numbers. We present a broad definition indeterminacy, various types indeterminacies, many practical applications.

In fuzzy Decision Support Systems, methods are strongly required for eliciting knowledge in the form of interpretable fuzzy sets from numerical data. In medical settings, statistical data are often available, or can be obtained from rough data, typically in the form of probability distributions. Moreover, since physicians are used to think and work according to a statistical interpretation of m...

A technique to perform design calculations on imprecise representations of parameters using the calculus of fuzzy sets has been previously developed [25]. An analogous approach to representing and manipulating uncertainty in choosing among alternatives (design imprecision) using probability calculus is presented and compared with the fuzzy calculus technique. Examples using both approaches are ...

In testing statistical hypotheses, as in other statistical problems, we may be confronted with fuzzy concepts. This paper deals with the problem of testing hypotheses, when the hypotheses are fuzzy and the data are crisp. We first give new definitions for notion of mass (density) probability function with fuzzy parameter, probability of type I and type II errors and then state and prove the seq...

We introduce a fuzzy metric on the set of probability measures on a fuzzy metric space. The construction is an analogue, in the realm of fuzzy metric spaces, of the Prokhorov metric on the set of probability measures on compact metric spaces. © 2011 Elsevier B.V. All rights reserved.

In this paper, we point out that the definition of fuzzy attribute implication actually implies a minimal degree and thus may be impractical in some applications. To solve this, we introduce probability information to fuzzy inclusion degree, and discuss some properties of this definition. Illustrations verifies the correctness of probability fuzzy attribute implication.

The main subject of this paper is the embedding of fuzzy set theory—and related concepts—in a coherent conditional probability scenario. This allows to deal with perception-based information—in the sense of Zadeh—and with a rigorous treatment of the concept of likelihood, dealing also with its role in statistical inference. A coherent conditional probability is looked on as a general non-additi...

The satisfiability problem for the logic FP(Ł) (fuzzy probability logic over Łukasiewicz logic) is shown to be NP-complete; satisfiability in FP(ŁΠ) (the same over the logic joining Łukasiewicz and product logic) is shown to be in PSPACE.

We compare two approaches to probability on systems of fuzzy sets—measures on tribes as introduced by Butnariu and Klement [3] and measures on MV-algebras with product studied by Riečan and Mundici [21]. We show that these two approaches overlap significantly and they confirm the original proposal of probability measure on fuzzy sets by Zadeh [24].