There is plenty of evidence already of the increasing disjoint between players and referees, and also of a certain arrogance among the current generation of rather puffed-up and self-regarding professional referees.
One is that the growing disjoint between the American economy and the world economy (and Europe especially) means that divergences between core inflation and headline inflation (used in TIPS) are growing.
Hence x and y are not disjoint.
(This would also follow from Supplementation, as emphasized e.g. in Olson 2006, since the assumption that x and y have the same proper parts entails that no part of y is disjoint from x, at least so long as parthood is reflexive; but there is no need to invoke (P.4) here).
Mereologically, an atom (or "simple") is an entity with no proper parts, regardless of whether it is point-like or has spatial (and/or temporal) extension: By definition of 'PP', all atoms are pairwise disjoint and can only overlap things of which they are part.
For example, with ξ construed as overlap, (P.12ξ,1) is satisfied by the model of Figure 7, left: here z is a minimal upper bound of x and y, yet z hardly qualifies as a sum "made up" of x and y, since its parts include also a third, disjoint item w.
Why think that, in this scenario, the blocks really are disjoint while they are passing through one another?
And since no two simples can overlap, this would mean that it's possible for disjoint things (the simples) to have identical (hence overlapping) exact locations.
If one is an endurantist, one will find it natural to describe this situation as involving multi-location in space or spacetime: Suzy is exactly located at rA, a region with the size and shape of an adult; Baby Suzy is exactly located at rB, a disjoint but simultaneous region with the size and shape of a baby; and Suzy=Baby Suzy.
Then f is an isomorphism from M to a structure whose universe is the disjoint union of a set A and its power set P(A) and which assigns A to I, assigns P(A) to S, and assigns the membership relation ∈ to E. So roughly speaking, σ defines the power-set operation "to within isomorphism".
Axiom III amounts to finite additivity because Reichenbach's logic does not have infinite disjunctions: it is a finite restriction of Kolmogorov's third axiom, which postulates additivity of probabilities for countable, even infinite, disjoint sets.
