Now for the “uniqueness” part. As before, we partition E into three subsets. I are the “inert” entities, which do not causally-depend on anything and have no causal dependencies themselves. (These could be abstract entities like numbers, propositions and so on). Formally I = {x in E: there is no y with x C’ y or y C’ x}. U are the “uncaused causes”—formally U = {x in E: there is no y with y C’ x, but there is z with x C’ z}. O are all the “other”, caused entities, so that formally O = {x in E: there is some y with y C’ x}. We will also let W be the “wholly uncaused causes”—formally W = {u in U: there is no x, y with u P y and x C’ y}.
B1. If S is any subset of W such that for any x, y in S we have x P y or y P x, (call such an S a “chain of parts”), then there is some entity z of which all members of S are parts.
B2. Suppose that y ⇐ x and z ⇐ x. Then there is some entity w ⇐ x such that: w ⇐ y or y P w; w ⇐ z or z P w.
Informally, the idea is that y and z can’t independently cause x without any further causal explanation. So there must be some common cause, however each of them may be part of that common cause.
Definition: Say that entities x and y are “causally-connected” if and only if x=y or there are finitely-many entities x=x1,..,xn=y with each xi C’ xi+1 or xi+1 C’ xi for i=1..n-1.
B3. Any two entities x, y in O are causally-connected.
Informally, O doesn’t “come apart” into disconnected components, such as a bunch of isolated universes. Premises B1-B3 turn out to be necessary for Theorem 6 to hold, as well as sufficient (see below). So they can’t be made any weaker!
Theorem 4: For any x in O, there is a unique entity f(x) in W such that: f(x) ⇐ x, and any y in U with y ⇐ x satisfies y P f(x).
Proof: For any x in O, define a subset W(x) = {y in W: y ⇐ x}; this is non-empty by Theorem 3. Consider any non-empty chain of parts S that is a subset of W(x). Since W(x) is a subset of W, then by B1, there is some z in E of which all members of S are parts, and such a z must also be in W. (If z is part of some dependent x then so is some s in S part of x, which is impossible because s is in W). Also since y ⇐ x for some member y of S, where y cannot be equal to x, and y P z, we have y C’ x2 ⇐ x for some x2, so z ⇐ x, as in the proof of Lemma 1, and z is also a member of W(x). Even if S is empty, we can take some z in W(x), and then trivially all members of S are parts of z. By application of Zorn’s Lemma to W(x), there is an element f(x) in W(x) which is maximal with respect to P i.e. there is no other y in W(x) with f(x) P y. Now, by B2, for any other y ⇐ x in U there must be some z ⇐ x with f(x) P z and y P z; this implies z is in W, and given f(x) is maximal in W(x) we have z = f(x) and so y P f(x). This makes f(x) the unique maximal element of W(x).
Theorem 5: For any x, y in O, f(x) = f(y) if and only if x and y are causally-connected.
Proof: It is clear that if f(x) = f(y) then x is causally-connected to y (just build a path backwards from x to f(x) and then forward again to y). Conversely, consider any two x, y in O:
a) If x C’ y, then f(x) is in U and satisfies f(x) ⇐ y so we have f(x) P f(y). Since x is not f(x), we have f(x) C’ x2 ⇐ x for some x2, and hence f(y) ⇐ x which means f(y) P f(x), and so f(x) = f(y).
b) If z is in U with z C’ x and z C’ y, then z P f(x) so f(x) ⇐ y and f(x) P f(y); similarly, f(y) P f(x) so that f(x) = f(y).
The result now follows by induction on the length of the causal path connecting x to y.
Theorem 6: If O is non-empty, then there is a single entity g in W such that: f(x) = g for every x in O, and y P g for every y in U.
Proof: Assuming O is non-empty, take any element y in O, and set g = f(y); then the result that f(x) = g for any x in O follows from Theorem 5 and B3; further, for any y in U, there is some x in O with y C’ x, so by Theorem 4, y P f(x). If there are no elements of O (meaning there are none in U either) then the Theorem is trivial.
Note that B1, B2 and B3 are entailed by the statement of Theorem 6. For B1, we can just take g as the relevant z. For B2, we can take g as the relevant w. B3 follows using using the first part of Theorem 5 (just track from x back to g, then forward to y again). The premises have been weakened a bit, but are still basically unjustifiable (we can imagine them being false without absurdity) and the above re-work shows they are needed for the uniqueness conclusion (Theorem 6).
One interesting fact is that Theorem 6 now has a loophole that wasn’t there before. If C is not equal to C’, then it is possible that there is some x C g, provided any such x is part of g (i.e. g contains any cause for itself within itself). This even allows Theorem 6 to be consistent with a premise like this:
B4: For any entity x, there is some y with y C x.
Informally, “every entity has a cause”, a premise which is usually considered a fatal inconsistency in a first cause argument! With a bit of renaming, the set O could be said to consist of “ordinary” entities (ones which have causes that are not parts of themselves) and the remaining non-inert entities are “extraordinary” entities (ones which contain all their own causes). W are “wholly extraordinary entities”, ones which are not part of any ordinary entity. Theorem 6 implies that every ordinary entity is causally dependent on a single wholly extraordinary entity, one which contains every other extraordinary entity.
Again, there aren’t any particularly theistic conclusions here, since g could very well be a maximum entity if there is one—say g is the whole universe or multiverse. In that case, g contains every ordinary entity as well as containing all the extraordinary entities.
Now for the “uniqueness” part. As before, we partition E into three subsets. I are the “inert” entities, which do not causally-depend on anything and have no causal dependencies themselves. (These could be abstract entities like numbers, propositions and so on). Formally I = {x in E: there is no y with x C’ y or y C’ x}. U are the “uncaused causes”—formally U = {x in E: there is no y with y C’ x, but there is z with x C’ z}. O are all the “other”, caused entities, so that formally O = {x in E: there is some y with y C’ x}. We will also let W be the “wholly uncaused causes”—formally W = {u in U: there is no x, y with u P y and x C’ y}.
B1. If S is any subset of W such that for any x, y in S we have x P y or y P x, (call such an S a “chain of parts”), then there is some entity z of which all members of S are parts.
B2. Suppose that y ⇐ x and z ⇐ x. Then there is some entity w ⇐ x such that: w ⇐ y or y P w; w ⇐ z or z P w.
Informally, the idea is that y and z can’t independently cause x without any further causal explanation. So there must be some common cause, however each of them may be part of that common cause.
Definition: Say that entities x and y are “causally-connected” if and only if x=y or there are finitely-many entities x=x1,..,xn=y with each xi C’ xi+1 or xi+1 C’ xi for i=1..n-1.
B3. Any two entities x, y in O are causally-connected.
Informally, O doesn’t “come apart” into disconnected components, such as a bunch of isolated universes. Premises B1-B3 turn out to be necessary for Theorem 6 to hold, as well as sufficient (see below). So they can’t be made any weaker!
Theorem 4: For any x in O, there is a unique entity f(x) in W such that: f(x) ⇐ x, and any y in U with y ⇐ x satisfies y P f(x).
Proof: For any x in O, define a subset W(x) = {y in W: y ⇐ x}; this is non-empty by Theorem 3. Consider any non-empty chain of parts S that is a subset of W(x). Since W(x) is a subset of W, then by B1, there is some z in E of which all members of S are parts, and such a z must also be in W. (If z is part of some dependent x then so is some s in S part of x, which is impossible because s is in W). Also since y ⇐ x for some member y of S, where y cannot be equal to x, and y P z, we have y C’ x2 ⇐ x for some x2, so z ⇐ x, as in the proof of Lemma 1, and z is also a member of W(x). Even if S is empty, we can take some z in W(x), and then trivially all members of S are parts of z. By application of Zorn’s Lemma to W(x), there is an element f(x) in W(x) which is maximal with respect to P i.e. there is no other y in W(x) with f(x) P y. Now, by B2, for any other y ⇐ x in U there must be some z ⇐ x with f(x) P z and y P z; this implies z is in W, and given f(x) is maximal in W(x) we have z = f(x) and so y P f(x). This makes f(x) the unique maximal element of W(x).
Theorem 5: For any x, y in O, f(x) = f(y) if and only if x and y are causally-connected.
Proof: It is clear that if f(x) = f(y) then x is causally-connected to y (just build a path backwards from x to f(x) and then forward again to y). Conversely, consider any two x, y in O:
a) If x C’ y, then f(x) is in U and satisfies f(x) ⇐ y so we have f(x) P f(y). Since x is not f(x), we have f(x) C’ x2 ⇐ x for some x2, and hence f(y) ⇐ x which means f(y) P f(x), and so f(x) = f(y).
b) If z is in U with z C’ x and z C’ y, then z P f(x) so f(x) ⇐ y and f(x) P f(y); similarly, f(y) P f(x) so that f(x) = f(y).
The result now follows by induction on the length of the causal path connecting x to y.
Theorem 6: If O is non-empty, then there is a single entity g in W such that: f(x) = g for every x in O, and y P g for every y in U.
Proof: Assuming O is non-empty, take any element y in O, and set g = f(y); then the result that f(x) = g for any x in O follows from Theorem 5 and B3; further, for any y in U, there is some x in O with y C’ x, so by Theorem 4, y P f(x). If there are no elements of O (meaning there are none in U either) then the Theorem is trivial.
Note that B1, B2 and B3 are entailed by the statement of Theorem 6. For B1, we can just take g as the relevant z. For B2, we can take g as the relevant w. B3 follows using using the first part of Theorem 5 (just track from x back to g, then forward to y again). The premises have been weakened a bit, but are still basically unjustifiable (we can imagine them being false without absurdity) and the above re-work shows they are needed for the uniqueness conclusion (Theorem 6).
One interesting fact is that Theorem 6 now has a loophole that wasn’t there before. If C is not equal to C’, then it is possible that there is some x C g, provided any such x is part of g (i.e. g contains any cause for itself within itself). This even allows Theorem 6 to be consistent with a premise like this:
B4: For any entity x, there is some y with y C x.
Informally, “every entity has a cause”, a premise which is usually considered a fatal inconsistency in a first cause argument! With a bit of renaming, the set O could be said to consist of “ordinary” entities (ones which have causes that are not parts of themselves) and the remaining non-inert entities are “extraordinary” entities (ones which contain all their own causes). W are “wholly extraordinary entities”, ones which are not part of any ordinary entity. Theorem 6 implies that every ordinary entity is causally dependent on a single wholly extraordinary entity, one which contains every other extraordinary entity.
Again, there aren’t any particularly theistic conclusions here, since g could very well be a maximum entity if there is one—say g is the whole universe or multiverse. In that case, g contains every ordinary entity as well as containing all the extraordinary entities.