[om-list] Analytic Fuzzy Set Operators

[Mark Butler]

As the wind howled outside my window on this sleepless night, I had an idea
for fuzzy set operators that works quite well:

0 = false
1 = true
t(A) = a
t(B) = b
t(not A)    =  1 - a
t(A and B)  =  a * b
t(A or B)   =  a + b - (a * b)

This set of definitions follows DeMorgan's laws perfectly.  They do not follow
the LEM exactly, but the worst case error is 1/4 rather than 1/2 with the
min()/max() set of operators.

The big advantage is that the functions are analytic, i.e. you can easily
integrate and differentiate them, even when they are nested several levels
deep from complex logical expressions.  In addition, they are much easier to
reduce than expresssions containing min() and max() operators.

By the way, in my last message, the word should have been "eigenfunctions",
literally "proper functions" of an operator in the same way that eigenvectors
are "proper vectors" of a matrix.

- Mark