Granular Computing An Introduction
This book is about Granular Computing (GC) - an emerging conceptual and of information processing. As the name suggests, GC concerns computing paradigm processing of complex information entities - information granules. In essence, information granules ari
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FUZZY SETS As discussed in Chapter 2, sets or intervals are generic models of information granules. They dwell on a fundamental notion of dichotimization that bluntly states that any given element belongs to a certain concept or becomes excluded from it. The dichotomy is the underlying philosophical doctrine that goes back to Aristotle. While permeating logic and mathematics and being fully endorsed there, its validity and omnipresence has been challenged in different ways. This challenge arose through the pioneering developments of many valued logic led by Jan Lukasiewicz and Emil Post and a non-Aristotelian approach to philosophy of science promoted by Alfred Korzybski. While these were deeply rooted in the realm of philosophy and logic, the notion of fuzzy sets introduced by Lotfi Zadeh in 1965 has become highly appealing at the applied end of the spectrum of the challenges to the two-valued logic. As a consequence, they have found a vast array of applications and seriously questioned the very essence of the principle of dichotomy. In this chapter, we introduce the idea of fuzzy sets, discuss their underlying fundamentals and elaborate on the essential processing schemes of such information granules.
3. 1 THE CONCEPT AND FORMALISM Fuzzy sets offer a possibility to formally express concepts of continuous boundaries. These concepts are everywhere and they are the core of our perception processes. When expressing ideas, describing concepts and communicating them to others, we always, we use terms to which the yes-no quantification does barely apply. Natural language is an evident environment of such communication. When forming and using concepts and communicating ideas and even simple observations, we use linguistic terms. The core issue there is that of building these underlying concepts (no matter in which area of human endeavors we are talking about). For instance, low inflation rate, high pressure, small approximation error, medium income are just few illustrative examples. Objects occurring in an image do not exhibit sharp boundaries. It becomes apparent that while these concepts are useful in the context of a certain problem as well as convenient in any communication realized in natural language, their set based formal model will lead us to a serious representation A. Bargiela et al., Granular Computing © Springer Science+Business Media New York 2003
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3 Fuzzy Sets
drawback. We sense that determining a binary (yes-no, included-excluded) boundary between the elements that satisfy the term of low inflation rate is neither realistic nor formally sound. Our perception suggests that there could be (and are) some elements whose membership to the concept could be only partial. This problem was succinctly raised in the past when struggling with the better and more comprehensive understanding the nature of such perception problems. In Duhem (1906) we fmd the following observation
... it is impossible to describe a practical fact without attenuating by the use of the word" approximately" or "nearly": on the othe
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