  In this note, we present a conjecture on intersections of set families, and a
rephrasing of the conjecture in terms of principal downsets of Boolean
lattices. The conjecture informally states that, whenever we can express the
measure of a union of sets in terms of the measure of some of their
intersections using the inclusion-exclusion formula, then we can express the
union as a set from these same intersections via the set operations of disjoint
union and subset complement. We also present a partial result towards
establishing the conjecture.
