Re: FUZZY LOGIC: Fuzzy Identity Law

S. F. Thomas (
Fri, 26 Jan 1996 15:06:25 +0100

Ian Goddard ( wrote:

: The Core Principle of Logic
: (c) 1996 Ian Williams Goddard
: The Three Laws of Thought are the foundational laws of Aristotelian
: logic. However, while recent advances in logic known as fuzzy logic
: have overthrown the second and third laws, the first law, the Law of
: Identity -- the primary axiom of Aristotelian logic -- remains
: unchallenged. This short essay overthrows the first law as it
: simultaneously establishes the true core axiom of logic: the Law of
: Fuzzy Identity.
: The first law of Aristotelian logic establishes a crisp -- that is,
: a non-fuzzy -- structure of identity. Here is this first law,
: followed by the Law of Fuzzy Identity:
: THE LAW OF (crisp) IDENTITY: If statement P is true, then P is true.
: If a thing, A, is A, then it is A. A = A.
: THE LAW OF FUZZY IDENTITY: If not-A is 80% of W, then A is 20% of W.
: A thing, A, is A relative to not-A. A = (A, not-A).
: Aristotelian logic (AL) divides a universe of fuzzy gray-scale
: continuums into crisp black and white. Having swept the all
: pervasive reality of fuzzyness under the rug, AL then tries to rip
: black and white apart, proclaiming that black is black -- A is A --
: free from any relation to white, and vice versa. Identity is
: therefore assumed to be absolute, not relative, crisp, not fuzzy.
: This unchallenged fallacy is here, now, and forever abolished.
(( cuts ))
: In review: the identity of A is a derivative of the relational
: contrast between A and not-A. Therefore, the identity set of A
: contains both A and not-A. This is proven by the fact that the
: elimination of not-A from the identity set of A results in the
: simultaneous elimination of A. Consequently, the long standing
: Aristotelian Law of (crisp) Identity, which fails to perceive the
: relational basis of identity, is prove to be false.

It is obvious that in some sense fuzzy contains crisp, or that
crisp is a special case of fuzzy. From this elementary insight,
it is easy to conclude that fuzzy is or should be logically prior
to crisp, and many thinkers have attempted to establish a fuzzy
logic that is without reference to a starting Boolean or crisp
logic. This endeavor is not doomed to fail, but it is to my mind
quixotic, and unnecessary. It is sufficient to realize that what
is fuzzy in one domain may be rendered crisp in another domain:
the accomplishment of fuzzy set theory and fuzzy logic is
precisely to render fuzzy terms, relations, etc. in an object
language into crisp terms, relations, etc. in a crisp meta-
language. The fuzzy theory, perhaps paradoxically, is not fuzzy
at all, rather a branch of ordinary bivalent mathematics. The
Boolean logic of the mathematical metalanguage is used to
*bootstrap* our way into the representation of that which is fuzzy
in ordinary language. In the process, it is possible to show
(Thomas, 1995. Fuzziness and Probability) that a crisp subclass may
drop out of the otherwise fuzzy terms and relations which populate
the object-language, and that the rules appropriate to this crisp
subclass mirror the bivalent rules familiar from the bivalent
metalanguage in which the whole development is couched.
(Nor is it necessary to reject the Aristotelian dicta, even
amongst the fuzzy terms and relations which populate the object
language.) This reassures us of the essential integrity of an
approach to elaborating fuzzy, which proceeds from the classical
(including Aristotelian) assumptions of crisp bivalent mathematics.
(It also, btw, assures us that there is no inherent contradiction
in using crisp, binary computers to elaborate our fuzzy models.)
Having accomplished that, it seems unnecessary to attempt a logical
development of fuzzy which eschews the canons of ordinary bivalent
mathematics. Goddard's piece not only attempts to eschew such
canons, but to assert their falsity. To my mind, this is a huge
mistake, being ultimately circular. What constitutes "proof" in
the metalanguage, if the rules of logic in the metalanguage
themselves depend on the laws sought to be asserted/denied in the
development itself?

(( cuts ))

: It is therefore 100% logical to state that: different than = same
: as. Indeed, Zen mind is enlightenment mind is logical mind.

Perhaps I am missing something.

: -----------------------------------------------------------------------
: IAN GODDARD <> FREEDOM: to have it, give it.
: -----------------------------------------------------------------------
: SEE THE PROOF:'s-law.htm
: -----------------------------------------------------------------------

S. F. Thomas