Engineering - Molecular Theory of the Living Cell - page 134

assigned degrees of truthfulness or certainty measured as a ratio of two
numbers, that is, D 1 /D 2, where D 1 is the distance between the fuzzy answer
(located at coordinate F) in the N-dimensional hypercube and its nearest vertex
located at C and D 2 is the distance between F and the vertex, C C , that is
irreconcilably opposite to C. (C C is called the complement of C.) The bit values
of crisp C C are obtained by subtracting the corresponding bit values of C from 1.
For example, the complement of C(1, 0, 1) is C C (1-1, 1-0, 1-1), or C C (0, 1, 0).
The distance, D AB , between the two points, A (a 1 ,a 2 ,a 3 ,

,a k ) and B (b 1 ,b 2 ,b 3 ,
... ,b k ), can be calculated using the Pythagorean theorem:

...

h i 1 = 2

2 2 2 2

D AB ¼ð a 1 b 1 Þ þð a 2 b 2 Þ þð a 3 b 3 Þ þ :::; þð a k b k Þ (5.23)

Applying Eq. 5.23 to points C and F, and C C and F in Fig. 5.8 , the ratio of D 1 over
D 2 can be calculated, which Kosko referred to as fuzzy entropy (Kosko 1993,
pp. 126-135), one of many fuzzy entropies defined in the literature. For conve-
nience, we will refer this ratio as the Kosko entropy , denoted by S K , in recogni-
tion of Kosko's contribution to the science of fuzzy logic. S K now joins the list of
other well-known entropies in physics and mathematics - the Clausius (which
may be denoted as S C ), Boltzmann (as S B ), Shannon (as S S ), Tsalis entropies (as
S T ), etc. The Kosko entropy of a fuzzy answer is then given by:

D FC c

S K ¼ D CF = (5.24)

where D CF is the distance between crisp point C and fuzzy point F and D FC c is
the distance between crisp point C C and fuzzy point F. Formally, Eq. 5.24
constrains the numerical values of S K to the range between 0 and 1:

1 S K 0 (5.25)

However, both the Principle of Ineffability, Statement 5.22, and the Einstein's
Uncertainty Thesis, Statement 5.38 (see below), strongly indicate that S K
cannot be equal to 1 or to 0, leading to Inequality 2.26 :

1 > S K > 0 (5.26)

According to Inequality 5.26 , the maximum value of S K is less than 1 and its
minimum value is greater than 0. If we designate the minimum uncertainty that
no human knowledge can avoid with u (from uncertainty) in analogy to the
Planck constant h below which no action (i.e., the energy integrated over time)
can exist, Inequality 5.26 can be rewritten as:

1 > S K u (5.27)

where u is a positive number whose numerical values probably depend on the
measurement system involved.