Some time back I realized that competition in populations behaved very
much like the classical pH and pOH equation in chemistry for titration
curves:
pH+pOH =14
http://www.ausetute.com.au/titrcurv.html
The Neanderthal - modern human population densities by area behave in
this manner.
Suppose that the implicit curves of fuzzy logic behaved the same way:
Log[x]+Log[y]=constant
It was the diagram on page 136 of Fuzzy Thinking by Bart Kosko that made
me think this approach might work :
http://www.amazon.com/Fuzzy-Thinking-New-Science-Logic/dp/078688021X/ref=sr_11_1/104-0029617-0633535?ie=UTF8&qid=1174578060&sr=11-1
This morning I did curves in Mathematica for implicit Fuzzy logics that
seem to work:
( these plots give analogs of the iterative plots and work better in
Mathematica than the iteratives do)
Kosko-Grim type:
Clear[x, y, a, b]
x'=1 - Abs[x - y]
y'=1 - Abs[x + y - 1]
f[x_, y_] = (1 - Abs[x - y])*(1 - Abs[x + y - 1])
ContourPlot[ f[x, y], {x, -0.5,
1.5}, {y, -0.5, 1.5}, PlotPoints -> {300, 300},
ImageSize -> 600,
ColorFunction -> (Hue[2#] &)]
Zadeh type:
x'=x*y
y'=x+y-x*y
Clear[x, y, a, b]
f[x_, y_] = x*y*(x + y - x*y)
ContourPlot[ f[x, y], {x, -2, 2}, {y, -2, 2}, PlotPoints -> {300, 300},
ImageSize -> 600,
ColorFunction -> (Hue[2#] &)]
My half dual iteration type:
( this type of fuzzy logic like the zero centered version was developed
for getting better Julias , pictures as output)
x'=0.5 - Abs[-x + y - 0.5]
y'=0.5 - Abs[x + y - 0.5]
Clear[x, y, a, b]
f[x_, y_] = (0.5 - Abs[-x + y - 0.5])*(0.5 - Abs[x + y - 0.5])
ContourPlot[ f[x, y], {x, -0.5, 0.5}, {y, 0, 1.}, PlotPoints -> {300,
300},
ImageSize -> 600,
ColorFunction -> (Hue[2#] &)]
The code search gives:
http://www.google.com/codesearch?q=fuzzy+logic+lang%3Amathematica&hl=en&btnG=Search+Code
The engineering versions ( Zadeh, Kosco) of fuzzy logic is a little
different than the mathematics and
the philosophical versions ( Grim).
The engineering version is more about making "categories" than about the
philosophical idea of truth and lies.
Mathematical versions are more about making an algebra that is more
general than Boolean.
Another area that is close is the Bayesian probability approach (
dependent logic).
Another related area from topology is the Hausdorff measure/ space
approach.
Another related area is the sigmoids of population theory.
The applications of this more abstract fuzzy logic are to things like
election theory and moral/ ethics theory.
It has been said that future wars will be fought over the right of
people to believe and behave in "gray"
areas. Not Republican or Democrat or religiously moral of reprobate
but in between.
A for instance is that in the United States people are allowed their own
religious beliefs, but in cleric controlled Islamic countries Christians
are often persecuted and imprisoned for nothing more than going to church.
So the terrorist war is meant to "correct" the Christian countries for
allowing atheists, Jews and agnostics to live
in peace; for allowing lifestyles that include both alcohol and other
immoral behavior by their
ideas.
So in a very real way the current war is a fuzzy logic war.
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