Toward a theory of fuzzy information granulation and its centrality in human reasoning and fuzzy logic

There are three basic concepts that underlie human cognition: granulation, organization and causation. Informally, granulation involves decomposition of whole into parts; organization involves integration of parts into whole; and causation involves association of causes with effects. Granulation of an object A leads to a collection of granules of A, with a granule being a clump of points (objects) drawn together by indistinguishability, similarity, proximity or functionality. For example, the granules of a human head are the forehead, nose, cheeks, ears, eyes, etc. In general, granulation is hierarchical in nature. A familiar example is the granulation of time into years, months, days, hours, minutes, etc. Modes of information granulation (IG) in which the granules are crisp (c-granular) play important roles in a wide variety of methods, approaches and techniques. Crisp IG, however, does not reflect the fact that in almost all of human reasoning and concept formation the granules are fuzzy (f-granular). The granules of a human head, for example, are fuzzy in the sense that the boundaries between cheeks, nose, forehead, ears, etc. are not sharply defined. Furthermore, the attributes of fuzzy granules, e.g., length of nose, are fuzzy, as are their values: long, short, very long, etc. The fuzziness of granules, their attributes and their values is characteristic of ways in which humans granulate and manipulate information. The theory of fuzzy information granulation (TFIG) is inspired by the ways in which humans granulate information and reason with it. However, the foundations of TFIG and its methodology are mathematical in nature. The point of departure in TFIG is the concept of a generalized constraint. A granule is characterized by a generalized constraint which defines it. The principal types of granules are: possibilistic, veristic and probabilistic. The principal modes of generalization in TFIG are fuzzification (f-generalization); granulation (g-generalization); and fuzzy granulation (f.g-generalization), which is a combination of fuzzification and granulation. F.g-generalization underlies the basic concepts of linguistic variable, fuzzy if-then rule and fuzzy graph. These concepts have long played a major role in the applications of fuzzy logic and differentiate fuzzy logic from other methodologies for dealing with imprecision and uncertainty. What is important to recognize is that no methodology other than fuzzy logic provides a machinery for fuzzy information granulation. *Tel: + 1-510-642-4959; fax: + 1-510-642-1712; e-mail: [email protected]. ~Research supported in part by NASA Grant NCC 2-275, ONR Grant N00014-96-1-0556, LLNL Grant 442427-26449, ARO Grant DAAH 04-961-0341, and the BISC Program of UC Berkeley. :To Didier Dubois and Henri Prade, who have contributed in so many major ways to the development of fuzzy logic and its applications. 0165-0114/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved PII S0165-01 14 (97 )00077-8 112 L.A. Zadeh / Fuzzy Sets and Systems 90 (199D 111-127 TFIG builds on the existing machinery for fuzzy information granulation in fuzzy logic but takes it to a significantly higher level of generality, consolidates its foundations and suggests new directions. In coming years, TFIG is likely to play an important role in the evolution of fuzzy logic and, in conjunction with computing with words (CW), may well have a wide-ranging impact on its applications. The impact of TFIG is likely to be felt most strongly in those fields in which there is a wide gap between theory and reality. © 1997 Elsevier Science B.V.