xvi PREFACE TO THE SECOND EDITION
with the latter, the axioms of a probability theory referred to as the excluded middle will
hereinafter only be referred to as axioms – never as laws. The operations due to De Morgan
also will not be referred to as a law, but as a principle
... since this principle does apply to
some (not all) uncertainty theories (e.g., probability and fuzzy). The excluded middle axiom
(and its dual, the axiom of contradiction) are not laws; Newton produced laws,Kepler
produced laws, Darcy, Boyle, Ohm, Kirchhoff, Bernoulli, and many others too numerous to
list here all developed laws. Laws are mathematical expressions describing the immutable
realizations of nature. It is perhaps a cunning, but now exposed, illusion first coined by
probabilists in the last two centuries to give their established theory more legitimacy by
labeling their axioms as laws. Definitions, theorems, and axioms collectively can describe
a certain axiomatic foundation describing a particular kind of theory, and nothing more; in
this case the excluded middle and other axioms (see Appendix A) can be used to describe
a probability theory. Hence, if a fuzzy set theory does not happen to be constrained by an
excluded middle axiom, it is not a violation of some immutable law of nature like Newton’s
laws; fuzzy set theory simply does not happen to have an axiom of the excluded middle – it
does not need, nor is constrained by, such an axiom. In fact, as early as 1905 the famous
mathematician L. E. J. Brouwer defined this excluded middle axiom as a principle in his
writings; he showed that the principle of the excluded middle was inappropriate in some
logics, including his own which he termed intuitionism. Brouwer observed that Aristotelian
logic is only a part of mathematics, the special kind of mathematical thought obtained if
one restricts oneself to relations of the whole and part. Brouwer had to specify in which
sense the principles of logic could be considered ‘‘laws’’ because within his intuitionistic
framework thought did not follow any rules, and, hence, ‘‘law’’ could no longer mean
‘‘rule’’ (see the detailed discussion on this in the summary of Chapter 5). In this regard, I
shall take on the cause advocated by Brouwer almost a century ago.
In addition, the term coherence does not connote a law. It may have been a clever term
used by the probabilists to describe another of their axioms (in this case a permutation of
the additivity axiom) but such cleverness is now an exposed prestidigitation of the English
language. Such arguments of the past like ‘‘no uncertainty theory that is non-coherent
can ever be considered a serious theory for describing uncertainty’’ now carry literally no
weight when one considers that the term coherence is a label and not an adjective describing
the value of an axiomatic structure. I suppose that fuzzy advocates could relabel their
axiom of strong-truth functionality to the ‘‘law of practicability’’ and then claim that any
other axiomatic structure that does not use such an axiom is inadequate, to wit ‘‘a theory
that violates the practicability axiom is a violation of the law of utility,’’ but we shall not
resort to this hyperbole. With this edition, we will speak without the need for linguistic
slight-of-hand. The combination of a fuzzy set theory and a probability theory is a very
powerful modeling paradigm. This book is dedicated to users who are more interested in
solving problems than in dealing with debates using misleading jargon.
To end my discussion on misleading definitional terms in the literature, I have made
two subtle changes in the material in Chapter 15. First, following prof. Klir’s lead of a
couple years ago, we no longer refer to ‘‘fuzzy measure theory’’ but instead describe it now
as ‘‘monotone measure theory’’. The former phrase still causes confusion when referring
to fuzzy set theory; hopefully this will end that confusion. And, in Chapter 15 in describing
the monotone measure, m, I have changed the phrase describing this measure from a ‘‘basic
probability assignment (bpa)’’ to a ‘‘basic evidence assignment (bea)’’. Here we attempt to
avoid confusion with any of the terms typically used in probability theory.
5星 · 超过95%的资源 需积分: 33 150 浏览量
更新于2024-08-01
3
收藏 5.41MB PDF 举报 
我的内容管理
展开
