Dear Colleagues: The quantity 1 - x + y which I introduced earlier to categories@mta.ca , 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