Subject: categories: An invariance property of 1 - x + y

From: "Osher Doctorow" <[email-masked]>
Date: Mon, 14 Aug 2000 23:49:54 -0700

From: Osher Doctorow, Ph.D. [email-masked], Mon. Aug. 14, 2000, 11:25PM

Dear Colleagues:

The quantity 1 - x + y which I introduced earlier to [email-masked] , and
which is the non-trivial case of Lukaciewicz implication as well as the
logic-based probability (LBP) probability of the set/event analogue of the
logical conditional Pr(A-->B) = 1 - Pr(A) + Pr(AB) for x = Pr(A), y =
Pr(AB), has an interesting invariance property.  I will use the LBP
formulation to show this.  Consider two sets A and B, possibly intersecting.
The universe is then divided into A U B (the union of A, B) and its
complement which is (A U B)' = A' B'  where intersection is indicated by
adjacent letters such as A' B' (which is the intersection of A' and B').
However, A U B is the union of AB and the part of B that does not intersect
A (which is BA') and the part of A which does not intersect B (which is
AB').   Thus the universe is the union of the disjoint (mutually exclusive
and exhaustive) sets AB', AB, A'B, and A'B'.

Lemma.  Pr(A-->B) = 1 - Pr(A) + Pr(AB)  =  Pr(B) + Pr(A' B' )

Proof.  The first equality has been established several weeks ago in my
contribution.  The second equality is true iff the equality with Pr(A) added
to both sides and Pr(AB) subtracted from both sides is true, but that
equation is the same as 1 = Pr(A) + Pr(B) - Pr(AB) + Pr(A' B')  = Pr(A U B)
+ Pr(A U B)' .   The latter is true by the definition of complement.  Q.E.D.

What is the invariance property?   From the lemma, let us write Pr(A' B' )
as Pr(A U B)'  = 1 - Pr(A U B).  Then Pr(A-->B) = 1 - Pr(A) + Pr(AB) = 1 -
Pr(A U B) + Pr(B).   Therefore, for x and y as above, the conjugate of 1 - x
+ y, which is 1 - y + x, equals 1 plus the excess of Pr(A) over Pr(AB) which
equals 1 plus the excess of Pr(A U B) over Pr(B).   Since 1 - x + y and its
conjugate involve either the excess of x to y or its negative, and since
Pr(A-->B) is the probable influence of A on B, in words the invariance
property of 1 - x + y says that the probable influence of A on B involves
the excess of Pr(A) over Pr(AB) or equivalently the excess of Pr(A U B) over
Pr(B).  Now, AB does involve an extra symbol B concatenated (technically,
intersected here) with symbol A.  Also, A U B involves an extra symbol A
"unioned" or "united" with B. Thus, the "excess of symbols" parallels the
excess of probabilities for the pairs.  In a sense, the "set excess"
parallels the probability/measure excess (I could write this symbolically,
but I will not for now) across union and intersection - that is the
invariance of 1 - x + y.   In mathematical physics, this would be expressed
by saying that set union and set intersection can be exchanged or are
invariant under the symmetry - although the symmetry holds under very
specific conditions.   This terminology comes from symmetries in which
particles and antiparticles are exchanged, protons/neutrons/electrons are
exchanged, bosons and leptons are exchanged, left-handed and right-handed
particles are exchanged, etc.  These types of symmetries and others are
quite fundamental in mathematical physics.

Osher Doctorow