FUZZY LOGIC: Fuzzy Identity Law

Ian Goddard (igoddard@mail.erols.com)
Fri, 26 Jan 1996 14:11:36 +0100


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.


The first law of Aristotelian logic, the Law of Identity, states
that A = A. The statement "A = A" proclaims that identity is 100%
self-referential, that A is derived entirely from an internal
relation of A to itself. But this is false, for everything, every A,
is 100% relative, and 100% relative = 0% self-referential, and
therefore the Law of (crisp) Identity must always be 100% false
(which shall be proven).

RELATIVE: something [ A ] dependent upon EXTERNAL [ not-A ]
conditions for its specific nature [ identity ]. (Random House,
Webster’s Dictionary.)

As we can see, the definition of relative observes that the relative
identity of A = (A, not-A).

Because A is relative, A is a derivative of the relational contrast
between A and not-A. Therefore, the identity set of A -- the set
that contains the necessary components of A -- must contain both A
AND not-A, thus: A = (A, not-A). Consequently, the identity of A is
NOT crisp but fuzzy, not internally self-contained but spread out
over the continuum of the space and time in which both A and not-A


IF A = (A, not-A), and thus Aristotle's first law is false,
THEN the elimination of not-A from the identity set of A, (A,
not-A), must result in the simultaneous elimination of A. Here then
is this necessary proof.

The set of a whole universe, W, contains two members: solid mass, A,
and empty space, not-A. Each member fills a percentage of W.

W = whole universe, A = solid mass, not-A = empty space

W = (A, not-A) = W

If we subtracted all not-A from W, then W is 100% A.

IF W = (A, not-A) - not-A
THEN W = (100% A)

Or, if we subtracted all A from W, then W would be 100% not-A.

IF W = (A, not-A) - A
THEN W = (100% not-A)

In each case, (1) W = (100% solid) and (2) W = (100% empty), W,
being 100% uniform, contains zero contrasting components, zero
parts, and thus contains zero members. And as 100% undifferentiated
W itself stands in contrast to nothing, W does not even contain
itself. In each case, containing zero members, W is a null set:

W = (100% solid) = 0
W = (100% empty) = 0

ERGO, the proof required is satisfied:

W = (A, not-A) - not-A = (100% A) = 0
W = (A, not-A) - A = (100% not-A) = 0

This proves that the elimination of not-A from the identity set of A
results in a null set and thus in the elimination of A. This proves
that, (1) the Law of Fuzzy Identity is 100% true, (2) the
Aristotelian Law of (crisp) Identity, which states that A = 100% A
and 0% not-A, is 100% false, and (3) the proposition that a thing,
A, is always 100% relative is 100% true. As relational-fuzzy
identity is absolute, it is properly defined as a law: the Law of
Fuzzy Identity.

What this proof also reveals is that -- as the existence of the
whole is dependent upon the contrast of its parts, A and not-A, in
the very same fashion that the existence of the parts, A and not-A,
are dependent upon their contrast with one another -- (a) the whole
contains the parts 100%, (b) the parts contain each other 100%, and
thus (c) each part contains the whole 100%. In other words, the
identity set of A contains both its internal and its external area
and thus contains the whole.

W = (A, not-A)
A = (A, not-A)
not-A = (A, not-A)

Therefore, the whole and its parts are a unified identity-family:

W = A = not-A = W

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.


The identity of A is always unique and yet always identical to
not-A. For example, the following two statements, the first defining
A’s unique identity and the second defining not-A’s unique identity,
are both unique and identical as each contains the other 100%:

(A = 20% of 100% W) = (not-A = 80% of 100% W)


These two statements clearly demonstrate how the identities of A and
not-A are simultaneously unique and yet identical. This apparent
paradox is the necessary structure of identity, for, as we have
observed, the unique identity of A is not self-contained but is
derived from the relation of A to not-A. A defines the unique
identity of not-A just as not-A defines that of A, and thus not-A is
contained within the identity set of A, such that:

A = (20% A, 80% not-A) = 100% W

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

IAN GODDARD <igoddard@cap.gwu.edu> FREEDOM: to have it, give it.
SEE THE PROOF: www.erols.com/igoddard/god's-law.htm