FUZZY LOGIC AND PROBABILITY FUNCTIONS

Fuzzy Sets and Systems, 2006

This paper presents a backward analysis on the interpretation, modelling and impact of the concept of fuzzy random variable. After some preliminaries, the situations modelled by means of fuzzy random variables as well as the main approaches to model them are explained. We also summarize briefly some of the probabilistic studies concerning this concept as well as some statistical applications.

International Journal of Theoretical Physics, 2002

Our main aim from this work is to see which theorems in classical probability theory are still valid in fuzzy probability theory. Following Gudder's approach to fuzzy probability theory (see [8, 10]), the basic concepts of the theory, that is of fuzzy probability measures and fuzzy random variables (observables), are presented. We show that fuzzy random variables extend the usual ones. Moreover, we prove that for any separable metrizable space, the crisp observables coincide with random variables. Then we prove the existence of a joint observable for any collection of observables, and we prove the weak law of large numbers and the central limit theorem in the fuzzy context. We construct a new definition of almost everywhere convergence. After proving that Gudder's definition implies ours and present an example indicates that the converse is not true, we prove the strong law of large numbers according to this definition.

2013

In spite of there is relationship between fuzzy logic principles and a probability principles in calculate values for computations. There are some distinct from values different probability distributions a values of a membership f unctions .This led to finding a variation in these computations. Some continuous distributions and continuous membership functions has several command properties .In this paper we choose an exponential distribution and find a corresponding calculation in f uzzy logic through membership functions.

Cybernetics and Systems Analysis, 2020

An approach to finding the probabilistic characteristics of fuzzy events is proposed. The probabilities are described by triangular fuzzy numbers that can satisfy certain conditions. Examples of calculating fuzzy probabilities of fuzzy events are considered.

International Symposium on Multiple-Valued Logic, 1976

This paper is concerned with the foundations of fuzzy reasoning and its relationships with other logics of uncertainty. The definitions of fuzzy logics are first examined and the role of fuzzification discussed. It is shown that fuzzification of PC gives a known multivalued logic but with inappropriate semantics of implication and various alternative forms of implication are discussed. In the main section the discussion is broadened to other logics of uncertainty and it is argued that there are close links, both formal and semantic, between fuzzy logic and probability logics. A basic multivalued logic is developed in terms of a truth function over a lattice of propositions that encompasses a wide range of logics of uncertainty. Various degrees of truth functionality are then defined and used to derive specific logics including probability logic and Lukasiewicz infinitely valued logic. Quantification and modal operators over the basic logic are introduced. Finally, a semantics for the basic logic is introduced in terms of a population (of events, or people, or neurons) and the semantic significance of the constraints giving rise to different logics is discussed.

Journal of Mathematical Analysis and Applications, 1982

A general framework for a theory is presented that encompasses both statistical uncertainty. which falls within the province of probability theory, and nonstatistical uncertamty. which relates to the concept of a fuzzy set and possibility theory [L. A. Zadeh, J. FUZZJ Sers I (1978). 3-281. The concept of a fuzzy integral ts used to define the expected value of a random vartable. Properties of the fuzzy expectation are stated and a mean-value theorem for the fuzzy integral is proved. Comparisons between the fuzzy and the Lebesgue integral are presented. After a new concept of dependence IS formulated, various convergence concepts are defined and their relationshtps are studied by using a Chebyshevlike inequality for the fuzzy Integral. The possibility of using this theory m Bayestan estimation with fuzzy prior mformation IS explored.

Journal of Mathematical Analysis and Applications, 1997

The aim of this paper is to show that fuzzy logic is a suitable tool to manage several types of probability-like functionals. Namely, we show that the superadditive functions, the necessities, the upper and lower probabilities, and the envelopes can be considered theories of suitable fuzzy logics. Some general results about the compactness in fuzzy logic are also obtained. ᮊ

1999

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Computational Statistics & Data Analysis, 2006

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-additive "uncertainty" measure m(•) = P (E|•) of the conditioning events. This gives rise to a clear, precise and rigorous mathematical frame, which allows to define fuzzy subsets and to introduce in a very natural way the counterparts of the basic continuous T-norms and the corresponding dual T-conorms, bound to the former by coherence. Also the ensuing connections of this approach to possibility theory and to information measures are recalled.

Facta universitatis - series: Electronics and Energetics, 2005

This paper gives a review of some classical and new applications of fuzzy logic. Some of the fundamentals of fuzzy logic that support these applications will be explained in order to make the paper sound.