From: Osher Doctorow, Ph.D. [log in to unmask], Sat. March 3, 4:05PM

I have been generalizing logic-based probability (LBP) to geometric
proximity functions p1(x, y) = 1 + y - x for 0 < = y < = x < = 1 and pn(x1,
x2, ..., xn; y) = 1 + y - Sn/n where Sn = x1 + x2 + ... + xn for 0 < = y < =
xi < = 1 for i = 1, 2, ..., n.   We can also replace y by Tn = y1 + y2 + ...
+ yn and in various conditions replace the endpoints of the interval, 0 and
1, by other endpoints.  See geometry.research,
[log in to unmask], etc.  LBP and p1, pn are extremely
closely related to reliability because if P(A) is the probability of A, then
P(X-->Y) = 1 + F(x, y) - FX(x)  where F(x, y) is the joint cumulative
distribution function (cdf) of X and Y and FX(x) is the (marginal) cdf of
random variable X.   Since FX(x) = P(X < = x), it is just 1 - reliability at
x for x = t and X continuous.

The importance of all this is that proximity is intuitively "nearness" but
on a relatively tractable scale of usually 0 to 1 rather than
distance-function or metric scales of 0 to infinity.  It is technically a
one-sided partial inverse to Euclidean distance since y < = x, etc.  It is
maximum on [0, 1] when the distance is minimum and vice versa, subject to
the conditions.  It obeys the triangle inequality, etc.  However, the
proximity function is also intuitively influence.  Recall from mathematical
statistics that A and B are independent iff one or both do not influence the
other (the usual product equation is an equivalent formulation).  They are
dependent iff one or both do influence the other.  While dependence is not
the same as influence, it is a special case, and so things that can be
formulated using proximity tend to involve highly influential variables.
That reliability is so directly capable of being formulated via proximity
means that reliability is highly influential (and also involves highly
probable influence).  In my opinion, this puts reliability near the center
of both mathematics and engineering/physics.

Osher Doctorow
Ventura College, Doctorow Consultants, etc.