Connecting a tenable mathematical theory to models of fuzzy phenomena

Abstract

It is evident that fuzzy logic should be studied from various scientific points of departure, however, fuzzy logic appears different depending on this viewpoint: from the standpoint of a philosopher or applied computer scientist fuzzy logic is a contrast to binary logic and crispness, while a mathematician examines fuzzy logic from a pure mathematical angle: what are the mathematical principles and algebraic structures behind fuzzy logic? Thus, for a mathematician there is nothing really fuzzy in fuzzy logic, indeed, it is an exact logic of inexact concepts and phenomena. An analogy can be found in probability theory: it is not relevant to ask what the probability is that Central Limit Theorem holds; this is a matter of exact proof, not a probability. Intuitionist mathematics is a branch of mathematical research where a theorem is accepted only if it can be proved on the basis of intuitionist logic: to prove, for example, that α holds it is not enough to show that ¬ α leads to contradiction. In this sense intuitionist logic is a challenger for Boolean logic. In contrast, in mathematical fuzzy logic that has been developed as a formal system e.g. in [5], [11] and [17], the meta logic is Boolean logic. To our knowledge there is no approach to fuzzy logic where the situation would be different. The point of view in this reviewing paper is that of mathematicians'. © 2009 Springer-Verlag Berlin Heidelberg.