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I am unsure what to make of it myself; whether classifying it into a first cause argument or not.
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It is.
Here is an excerpt that sums it up:
http://www.onecountry.org/e102/e10214xs.htm Quote:
Let V be the collection (universe) of all existing entities. Since V is composite it cannot be self-caused (see above) and so must have a cause G (different fromV itself). Thus, G ® V, G ¹ V Moreover, every existing phenomenon A is either an entity, and thus a component of V, or else a system all of whose components are in V -- in which case A is a subsystem of V. Thus, G is either a component or a subsystem of V. But, in either case, G ® G by the potency principle. Thus, G is self-caused and hence noncomposite (no composite can be self-caused as shown above). Finally, since G ® V and every phenomenon A is a part of V then by the potency principle, G is a universal cause (the cause of every existing phenomenon, including itself).
Finally, we show that G is the only uncaused phenomenon, for suppose there is another such phenomenon G'. Then G ® G' (since G is a universal cause). But since G' is self-caused it cannot be other-caused by the principle of sufficient reason. Thus, G = G' and the uniqueness of G is established.
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If G->V and V=^infinity, then G=V
G is composite, and not self-caused.
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It follows immediately from these principles that no composite phenomenon can be self-caused, for suppose A ® A where A is composite. Then, by the potency principle A ® E, where E is any component of A. But this contradicts the limitation principle.
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Consider A -> B, where A != B. Then any system containing A is a sufficient cause of B. For instance, consider the system denoted by {A,C}, which consists of A and some other different phenomenon C; then {A,C} -> B.
So, from Hatcher's proof, we know that G -> G. Since G is a universal cause, G is a sufficient cause for the system {G,x}, where x is some other phenomenon different from G. Thus G -> {G,x}.
From the principle I showed above, {G,x} -> {G,x}. {G,x} is a composite phenomenon, and yet {G,x} is a sufficient cause for {G,x}. This violates the limitation principle.