This chapter presents the mathematical formulation of the fuzzy logic essentials and sets and serves as a useful background for entering the mathematical expression of the knowledge representation in the fuzzy world. Particular examples and application spaces are explored for an integrated presentation of the facets of fuzziness, both from a theoretical and practical contemplation. This knowledge could assist the comprehension of the mathematical typology and terminology of the fuzzy concepts, leading to the fuzzy inference process that it is examined in the next chapter.
TopIntroduction
At the early stage of the fuzzy logic (FL) history it became clear that there is an intimate relationship between FL and many-valued logic. In parallel, soon it became obvious that FL, quite naturally, could be placed within the language of many-valued logic (Gottwald, 1979). FL (and fuzzy set theory) is composed of an organized body of mathematical tools, particularly well-suited for handling incomplete information, uncertainty, the unsharpness of classes of objects or situations, or the gradualness of preference profiles, in a flexible way. It offers a unifying framework for modeling various types of information, ranging from precise numerical, interval-valued data, to symbolic and linguistic knowledge, with a stress on semantics rather than syntax (Dubois & Prade, 2000).
FL is all about the relative importance of precision, that is, the importance to be exactly right when a rough answer will do. Two relevant famous quotations, from A. Einstein and L. Zadeh (“Law of Incompatibility”), respectively, stress this perspective:
So far, as the law of mathematics refer to reality, they are not certain. And so far, as they are certain, they do not refer to reality. (Einstein, 1954, p. 83)
As complexity rises, precise statement lose meaning and meaningful statements lose precision. (L. Zadeh in McNeill & Freiberger, 1993, p. 43)
In fact, FL does a good job of treading off between significance and precision, something that humans have been managing for a very long time. FL is conceptually easy to understand; it is natural and flexible; it is tolerant of imprecise data; can model nonlinear function of arbitrary complexity; can built on top of the experience of experts, letting, at the same time, to rely on experience of people who already understand the system. FL can be blended with conventional control techniques, augmenting and simplifying the implementation of conventional control methods. FL is based on natural language, which is used by ordinary people on a daily basis and which is convenient and efficient. Apparently, the basis for FL is the basis for human communication; hence, FL is easy to use.
Overall, FL plays an important role in the information science and engineering, as it bridges the gap between human and computers, through a major contribution to data and knowledge representation. Due to the nature of the FL, the latter could be extended to the field of social sciences and humanities, placing the field of FL as an environment that could provide complete and uniquely optimal means for solving problems and managing the uncertainty usually met in the analyzed data associated with problems examined in these areas (e.g., modeling human interaction in the blended-learning settings via the analysis of online learning data).
This chapter is meant to provide an overview of the main components of FL in their mathematical representation, setting the corresponding typology for embracing fuzziness (the degree of precision) in the modeling processes of real-world systems. Indicative examples of FL applications are further employed, to facilitate the comprehension of the introduced mathematical concepts and ideas.