Proof of God's existence

Reproduced below is an interesting - and fairly simple - logical proof of God's existence. The argument was originally developed by a Muslim philosopher, Avicenna in the 10th century AD, and has been reformulated cogently by William Hatcher. Should be interesting to chew on for all you logicians out there...

We begin with a basic proposition.

P0. Something exists (there is not nothing).

This seems pretty true, I'm sure you'd all agree... :)

We now define reality as the totality of actual existence = everything there is (or was or will be). A phenomenon is some nonempty portion of reality. Let us designate reality as the set V.

Next, we consider a binary relationship → called causality which may hold between any two phenomena A and B. If the relationship A→B does indeed hold, then we say that A causes B. Also, causality need not be direct - A→B is an indirect causality if A→C→B where C is distinct from A and B.

Now we define three kinds of phenomena based on the above notion of causality.

D0. A phenomenon B is without a cause if, for no A, does A→B hold.
D1. B is caused (other-caused) if for some A≠B, A→B holds
D2. B is uncaused (self-caused) if B→B and never A→B for A≠B.

We now introduce a second binary relationship ∈ which may hold between two phenomena A and B. If A∈B holds, we say "A is a component of B". Given B, if A∈B for at least one A, then we say that B is composite. Otherwise, B is simple (noncomposite). Composites are phenomena which have parts. All known physical phenomena are composites except, possibly, the elementary particles of quantum mechanics. A composite phenomenon can also be called a system.

Based on these definitions, Hatcher goes on to state three propositions he calls empirically grounded.

P1. The principle of sufficient reason: All phenomena are either self-caused (i.e. A→A) or other-caused (B→A; B is not equal to A) but not both. Put another way, this principle says that the question "why?" is always meaningful. Everything happens for a reason. It also means that situation D0 can never occur - a phenomenon cannot be without cause.

P2. The potency principle: If A→B then for all C∈B, A→C. In other words if A is the cause of B then A is the cause of every part of B.

P3. The principle of limitation: For all A, where B∈A, A→B does not hold. This says a system (which Hatcher represents as a set) cannot be the cause of its own components. A car, for example, (the system) cannot be the cause of its own steering wheel (a part), because the car does not even logically exist until the steering wheel exists. Thus the car's existence cannot precede the steering wheel's existence.

P3 also implies that a composite or system (that has several elements) cannot be self-caused - because if it were self-caused, then such a composite system would be the cause of its own elements, which contradicts P3. Therefore all self-caused phenomena are noncomposite.

Notice that nothing excludes that a component may be the cause of a whole of which it is a part. Also, P1, P2 and P3 are all empirically grounded. P1 says that if we ask "why B?" the answer "there is no reason that’s just way it is" is not acceptable. P1 is thus the fundament and basis of (scientific) rationality. It is the essential logical precondition for all of science. P2 is virtually a definition of the notion of complete cause and P3 is, essentially, a special case of the second law of thermodynamics, which negates the possibility of purely "holistic" causality, i.e., the transfer of order from a whole to a proper part, without any input of organizing energy from outside the system.

Finally, one more definition :)

D3. By God, symbolized G, we mean a unique, self-caused (uncaused), noncomposite, universal cause, if such a phenomenon exists (which is the thing to be proved).

Now every phenomenon B is a part of V (all of reality). This allows us to state P0 more compactly:

P0. V is composite

Now that we have this foundation, here's the meat of the proof of the existence of G.

By P1, V is either self-caused or other-caused. Suppose V→V. By P0, V is composite. Thus, E∈V for some E. But then, by P2, V→E∈V, which contradicts P3. Hence, by P1, G→V for some phenomenon G≠V. Like every phenomenon, G is a part of V.

Thus by P2, G→G. G is therefore self-caused. But this means that G is noncomposite (else P3 would be contradicted). G is also universal because every phenomenon B is a part of V by definition. Thus, by P2 and G→V, it follows that, for every phenomenon B, G→B.

Finally, G is the unique uncaused phenomenon, for suppose that, for some phenomenon G1, G1→G1. Now we have already established that G is universal. Thus G→G1. By P1, G1 cannot be both self-caused and other-caused. But G is a cause of G1. Thus, G is not "other", i.e., G=G1 as claimed.

We have thus proved the existence of a unique uncaused, noncomposite, universal cause, which we can call God. Note that being the universal cause does not imply G is the direct cause of all phenomena - but directly or indirectly, the chain of causation leads back to G eventually. I've omitted a couple of other definitions/assumptions, but the crux of the argument is given above. If one accepts P0, P1, P2 and P3, then it logically leads to the existence of G as defined above. Rejecting the existence of G therefore requires one to reject one of P0, P1, P2 or P3 on some reasonable grounds.

Which one would you pick? :)