>Hello,
>
>In reading about fuzzy logic, I noticed that it is still possible to have
>absolute true and absolute false values. If some idea is absolutely true,
>take for example "This is water," then from my understanding, the 'This'
>falls into the set of things that are water with a truth value of 1.0, but
>it does not fall into the set of things that are not water. In every other
>instance along the continuum, however, the 'This' would fall into both sets.
>Correct me if I'm wrong.
>
>This doesn't make much sense to me. It seems more practical to include the
>'This' with a 0 truth value in the set of not water things as well. This
>reflects the idea that for something to exist, the idea of its opposite must
>also exist even if there is no real expression of that opposite. Basically,
>if something is water, there has to be a conception of something that is
>absolutely not water, otherwise, it is meaningless to be just water.
>
>At an extreme end, if all there was was water--I would say a universe of
>water, but then there would be a universe--then there could be no idea of
>water because there would be nothing to relate it to.
>
>Am I off the deep end with this?
>
No!
But it helps to explore the concept of "fuzzy entropy"
"Kosko illustrates these various degrees of ambiguity by geometrically
plotting various degrees of set membership inside a unit hypercube.
(See fig. below)
This sets-as-points approach holds that a fuzzy set is a point in a
unit-hypercube and a nonfuzzy set is a corner of the hypercube. Normal
engineering practice often visualizes binary logical values as the
corners of a hypercube, but only fuzzy theory uses the inside of the
cube. Fuzzy logic is a natural filling-in of traditional set theory.
Any engineer will recognize the three-dimensional representation of
all possible combinations three Boolean values: {0,0,0}, {0,0,1},
{0,1,0}, {0,1,1}, {1,0,0}, {1,0,1}, {1,1,0}, {1,1,1}, which correspond
to the corners of the unit hypercube. But fuzzy logic also allows any
other fractional values inside the hypercube, such as {.5,.7,.3}
corresponding to degrees of set membership.
Fuzzy logic holds that any point inside the unit hypercube is a fuzzy
set with Russell's paradox located at the point of maximum ambiguity
in the center of the hypercube .
Fuzzy Entropy
Degrees of fuzziness are referred to as entropy by Kosko. Fuzzy
entropy measures the ambiguity of a situation, information and entropy
are inversely related-if you have a maximum-entropy solution, then you
have a minimum-information solution, and visa versa, according to
Kosko. But minimum-information does not mean that too little
information is being used. On the contrary, the principle of maximum
entropy ensures that only the relevant information is being used.
This idea of maximizing entropy, according to Kosko, is present
throughout the sciences, although it is called by different names.
>From the quantum level up to astrophysics or anywhere in-between for
pattern recognition, you want to use all and only the available
information, Kosko claims. This emergent model proposes that
scientists and engineers estimate the uncertainty structure of a given
environment and maximize the entropy relative to the known
information, similar to the Lagrange technique in mathematics. The
principle of maximum entropy states that any other technique has to be
biased, because it has less entropy and thus uses more information
than is really available.
Fuzzy theory provides a measure of this entropy factor. It measures
ambiguity with operations of union, intersection and complement: U, |,
and -."
From: http://www.teleport.com/~cognizer/eet/Almanac/TUTOR/KOSKO.HTM
Kind regards,
Stephen Paul King