The Complex Domain of Probability

Given two independent events X,Y with probabilities x,y (relative 
to some sample space S), let x' and y' denote the probabilities of 
{X union Y} and {X intersection Y} respectively.  These values are 
given by the simple formulas

         x' = x + y - xy              y' = xy

This is based on the usual definition of the probability of an event 
X relative to a sample space S as

                         content[X intersection S]
             Pr{X|S}  =  -------------------------
                                content[S]

so the probability of the empty set is 0, and the probability of the 
entire sample space is 1.  Clearly all probabilities (according to
this definition) are real numbers on the interval 0,1.  We could, 
however, choose a different parameterization.  For example, we could 
assign the empty set a "probability" of -1 and the complete sample 
space a "probability" of +1.  

To understand why we might want to make such an unusual definition, 
consider the typical association of the probabilities 0 and 1 with 
the logical operators Falsity and Truth.  Letting F(x) and T(x) 
denote the Falsity and Truth, respectively, of an event x, we have 
the identities 

   T(T(x)) = F(F(x)) = T(x)        T(F(x)) = F(T(x)) = F(x)

Obviously these operators act just like +1 and -1 under ordinary
multiplication, where we have 

  (+1)(+1) = (-1)(-1) = (+1)       (+1)(-1) = (-1)(+1) = (-1)

Thus, by mapping the operators Truth and Falsity to the numerical
values +1 and -1, both of the operations are represented by ordinary
multiplication.  In contrast, with the usual associations False = 0
and True = 1, the simplest algebraic analogs are 

                t(x) = x        f(x) = 1 - x

Of course, this f(x) still has period 2, and we could generalize 
to other "existential operators" by assigning a linear fractional 
transformation to each one, such as g(x) = 1 - 1/x (which has 
period 3), or h(x) = (1+x)/(1-x) (which has period 4).  However,
these Mobius transformations are really just rotations of the
Riemann sphere, and the simplest expression of such rotations 
is just pure multiplication of (in general) complex numbers.  
In this sense, it could be argued that the associations 
{False = -1, True = +1} are more natural.

Another interesting feature of the {-1,+1} mapping is the fact that
it gives formally symmetrical probability equations for the union 
and intersection of independent events.  The "probability" u of X 
in the {-1,+1} framework is related to the corresponding old-style 
probability x by u = 2x-1.  In terms of the example cited earlier,
if we set u=2x-1, v=2y-1, u'=2x'-1, and v'=2v'-1 we find that the
equations for the union and intersection are

                     u + v      1 - uv
               u' =  -----  +   ------
                       2           2
                                                           (1)

                     u + v      1 - uv
               v' =  -----  -   ------
                       2           2

This formalization highlights the fact that if m is the mid-point 
between two probabilities u,v, then m is also the midpoint of the
union and intersection probabilities u' and v'.  (Of course, this
is equally true with the usual 0,1 mapping, but the symmetry between
these two operations is less apparent.  On the other hand, the
logical duality of AND and OR under negation is certainly clear 
in DeMorgan's Rules.)

Now that we've assigned the Truth and Falsity operators to +1 and 
-1 respectively, it's natural to ask what sort of existential 
operator might be assigned to i = sqrt(-1).  To make it fit into 
the same scheme, this mysterious operator - let's call it the 
"Width" W(x) - would have to satisfy the relation 

                      W(W(x)) = F(x)

In other words, the Width of the Width of x must equal the Falsity 
of x.  Similarly we could define the Height H(x) to correspond 
with -i.  In this scheme of things the "probabilities" could be
complex numbers.  It's tempting to think that we could restrict our
probabilities to the unit disk in the complex plane, but it turns
out that this set of points is not closed under the operations for
the union and intersection.  This leads to the question of what 
regions of the complex plane ARE closed under the independent 
probability operations.  Obviously the real interval (-1,1) is one 
such region, as so is the entire complex plane, but are there any 
others?  

Interestingly, it turns out that the disk of radius 2 centered on 
the real number 1 is closed under unions, and the disk of radius 
2 centered on -1 is closed under intersections.  But what about
a region closed under BOTH operations?  For this purpose the
definition of "closure" may need to be adjusted slightly.  

Recall that the midpoint of two events is also the midpoint of 
their union and intersection.  Therefore, given any two points 
in the complex plane, repeated application of equations (1) yields
successive pairs of points that essentially rotate about their
fixed midpoint and change their radius.  So let the word "closure" 
signify that repeated applications of (1) to a pair of points 
centered on a point c results in a sequence of pairs that remain 
bounded in their distance from c.  The set of all these c points 
could be called the natural domain of complex probability.

Notice that the above notion of closure referred to ANY two
initial points centered around c, but this isn't quite right.  
The quality of boundedness IS fairly insensitive to the choice
of points (for points in the neighborhood of c), but to be
more precise I should say that c is in the "closure region"
iff iteration of equations (1) on the initial pair u=v=c leads
to a bounded sequence.

Letting z denote the "radius" of the points u and v relative 
to their mid-point c, the repeated application of equations (1)
for a given fixed center c gives the iteration

                         c^2 - z^2 - 1
                z  -->   -------------
                               2

So the natural domain of complex probability may be considered
the set of complex points c such that this iteration beginning
with z=0 remains bounded.  Since this is just a simple quadratic
mapping, it's not surprising that the resulting set of points is
a modified version of the familiar Mandelbrot set.  However, like
everything else associated with the {-1,+1} scheme, the "True" 
and "False" regions are symmetrical, as shown in the figure
below.

 Close-up of Region A     Close-up of Region B