This post in an addendum to this post.
The equations that I used to define causation may need to be revised slightly to handle two kinds of cases, both of which used in the main body of the text, but which I did not explicitly define.
One case is where you are looking for the cause of a set of properties (“particles”). In this case we define z and DC as follows:
Under this definition the single particle case is covered when z is a set containing only one element.
A more complicated situation arises when the condition you wish to determine the cause of is not determined by any one state of the world. For example one of the applications I provided was determining the cause of a man’s death. However the state we are looking for, a dead man, is not determined uniquely by any one configuration. In this case we have to abandon first order logic and define z (or Z) as a predicate. We could then define Z to be true whenever the man is dead, and Z would then hold no matter how he died. In this case our definitions are like so:
The simpler cases where we are only testing for the cause of the state of a single “particle” or when we are testing for the cause of a set of particles can be recovered by defining Z as:
Single particle case: where p is the particle state we are interested in.
Set of particles case: where p1, p2, … are the particle states we are interested in.
Neither of these revisions however makes a difference to how C is defined, so they can be safely ignored when considering the cause of macroscopic objects, since we are only working with an approximation in those cases anyways.