Additional Information

Is the G Proof based on the famous ontological argument?

No. The G Proof has no relationship with, and is superior to, the ontological argument. The ontological argument originated in the Eleventh Century, and was later developed by several great mathematicians: Rene Descartes, Gottfried Leibniz and ultimately Kurt Godel. We think Descartes, Leibniz and Godel would have preferred The G Proof, had they known about it.

Where did The G Proof come from?

The essential idea of the proof was developed by William S. Hatcher, who died in 2005. Mark Emerson was astonished to learn about Dr. Hatcher's work quite unexpectedly while browsing on Wikipedia in October 2011. Several changes have been made to Hatcher's axioms and we present the proof in the style of Kalish of Montague. Moreover, the quality of the presentation has been hugely improved over Dr. Hatcher's presentation. Additional information about this appears on our Technical Papers page.

Existential Generalization (EG) — by Mark Emerson

In Videos 4, 5 and 8, I say several times that we can pick ANY bound variable with EG. This is not precisely true.

For example, if we have Fx and Kx, then we know something (x) both F's and K's, so we can, by EG, write exists y (Fy and Ky). Or we can write exists x (Fx and Kx), and indeed we could pick ANY variable for the new bound variable.

However, suppose we have Fx and Ky, which means something F's and something K's, but does not necessarily mean there is anything that does both. To apply EG, we must decide whether we are generalizing on x or on y. If we generalize on x, then we can write, by EG, exists z (Fz and Ky). Or we can write exists x (Fx and Ky). But we CANNOT use y for the new bound variable, because that would additionally capture the y in Ky, producing exists y (Fy and Ky), which is invalid because it says there is something that both F's and K's, when we did not know that. Thus, with EG, we cannot pick a bound variable, other than the variable on which we are generalizing, that is free in the expression on which we are generalizing.

The rule, per Kalish and Montague, amounts to this. For EG to work, it must be possible to use UI to go back the other direction.

Hence, in the first example above, we can apply UI to exists y (Fy and Ky) to get Fx and Kx. But in the second example, if we apply UI to exists y (Fy and Ky), we can get Fx and Kx or we can get Fy and Ky, but we CANNOT get Fx and Ky, so that application of EG is invalid.

I omitted this detail from the video to save time.

How Is The G Proof Different from Related Arguments? — by Jonathan Emerson

G Theory is a formal mathematical theory which fully expresses and proves variations of the Cosmological Argument. These proofs collectively are called The G Proof.

Generally, a Cosmological Argument first assumes some version of the Principle of Sufficient Reason (PSR), which declares that causation or reason is requisite for existence: simply put, PSR says that everything must have a sufficient reason for its existence. The PSR is then used in conjunction with a few more assumptions to conclude that God exists as the causal agent or explanation for the universe and everything within it. In this context, God is understood to be omnipotent: not necessarily "omnipotent" in the traditional sense of an overseer who steps in and out of the world, influencing various events at will; but at least omnipotent in the naturalistic sense of initiating and/or perpetuating an immutable chain of events, i.e. all events.

Here are the most important objections to Cosmological Arguments:

Objection 1. There are aspects of the universe which transcend explanation — they just are.

Objection 2. In particular, the universe as a whole transcends explanation — it just is.

Objection 3. The PSR leads to an infinite regress of causes. (e.g. What caused God? What caused the cause of God? etc.)

G Theory recognizes Objection 1 by defining an Absolute phenomenon as anything which just is. This makes it possible to avoid infinite regression by falling back on an Absolute phenomenon (such as God, if God is Absolute). In fact, if the existence of Absolute phenomena is altogether denied, then infinite regression is required. Now supposing an infinite regress does in fact exist, one must further ask: is this infinite regress Absolute, or is there an explanation for it? This question is thoroughly analyzed in G Theory Version 2. Modeling Georg Cantor's foundational mathematics of transfinite numbers, Version 2 demonstrates that, even if a given infinite regress is not Absolute, there necessarily exists another infinite regress which is Absolute and which causes the first one.

This means that, for any given phenomenon, either the phenomenon is Absolute or it is caused by an Absolute phenomenon (even if an infinite regress is involved). Now, clearly the vast majority of phenomena experienced by most people are not Absolute: often when we analyze a situation, and we have a decent intuition about it, we can come up with a reasonable explanation. Even when we fail to come up with a true explanation, it is not too far-fetched to believe that some true explanation exists. Thus a significant portion of the universe is not Absolute, moreover this entire portion is fully explained by the Absolute portion. This Absolute portion is just the totality of all Absolute phenomena. We formally define God as this Absolute totality, and we see that this totality explains the rest of the universe.

Now all three objections are resolved as follows:

Objection 1 does not refute the argument; on the contrary it supplies the definition of God.

Objection 2 fails because any given phenomenon in the universe either is Absolute and hence part of God, or else is caused by God. Clearly God is not the entire universe.

Objection 3 only strengthens the argument because G Theory embraces infinite regression, rather than outright rejecting it as many versions of the Cosmological Argument do.

Another possible objection questions our formal definition of God: how does any of this prove that God is personal or benevolent or even conscious? The answer is simple: it neither proves nor attempts to prove any of these things. Why, then, should we call it God? Some prefer a different name such as the Absolute or Absolute Truth. For Cantor it was the Absolute Infinite, and for Aristotle it was the Unmoved Mover. But among religious and spiritual people from cultures all over the world, God is by far the most popular name in English to describe that which has generated — with or without the intent to do so — everything in the universe, and therefore this is the most appropriate name suiting our definition.

One more objection concerns the doctrine of free will. Ironically many people who accept some form of the Cosmological Argument also believe in free will, which seems to undermine God's omnipotence: if agents of free will can cause events independent of the natural course of action determined by God, then God is insufficient to cause these events. To resolve this objection, we simply ask: is free will Absolute? If so, then free will is just a part of God; and if not, then free will is ultimately caused by God, and our experience of it is akin to a character in a story who appears to be making choices, when it was in fact the author who made all of these choices. Thus free will is not independent of God.

G Theory does not commit the God of the gaps fallacy. There are countless phenomena which science is still unable to explain, for example existence itself: why is there something rather than nothing? When pondering any of these mysterious phenomena, we might conclude one of three possibilities:

1. the phenomenon transcends explanation — it just is (i.e. it is Absolute);

2. there exists a naturalistic explanation, but science does not know it yet; or

3. the explanation requires something supernatural, e.g. God did it.

A God of the gaps fallacy just asserts option 3. G Theory concisely and explicitly takes options 1 and 2 into account without ever resorting to option 3. This is easy to see because an Absolute phenomenon, by definition, precludes both options 2 and 3. When considering an Absolute phenomenon, there is no gap — no explanation for it is missing because no such explanation exists.

What about the Big Bang? Is the Big Bang the Absolute phenomenon that generated the universe? This raises an extraordinarily difficult question: How can we determine whether a phenomenon is Absolute? Is it even possible to do so? If we assume that the Big Bang or any other mysterious phenomenon is Absolute, then we are guilty of an alternate form of the God of the gaps fallacy: we have merely crossed out God and written, No explanation, it just is. Our inability to provide any explanation for something is never grounds for declaring it Absolute.

However, consider another possibility: if some Absolute phenomena turn out to be self-sufficient in a meaningful way, then we may eventually develop ways to understand them. For example mathematics is self-sufficient: it exists independent of the Big Bang and all the material energy in the universe. Indeed, mathematics appears to be Absolute, providing its own explanation — it just works. It is unchanging and perfect (though our conceptions and understanding of it are changing and imperfect). And mathematics plays a key role in providing scientific explanations. Therefore, through mathematics (including G Theory), and perhaps through additional means yet to be discovered, it may be possible to understand the parts and properties of God, such as self-sufficiency, and why something exists rather than nothing.

A Natural Argument, Related to the Proof — by Jonathan Emerson

What are "natural forces"?

Science strives to explain how the world works and to predict the outcome of complex situations. It does so by positing the existence of potent natural forces. But as magnificent as the current scientific models are, they do have limitations, and so it appears that there must exist other such natural forces beyond the scope of what is formally understood.

Define "Nature" as the totality of all natural forces, including the known ones and the unknown ones; the understood and the vast mystery. From this definition, it is clear that anything science explains or predicts is caused by Nature, i.e. by natural forces.

Define "x is Natural" to mean x is caused by Nature. By this definition, anything science explains or predicts is Natural; and if some mystery is not understood scientifically, then there still may exist natural forces that cause it, but we cannot know for sure (unless we simply assume that it is so).

Define "x is Supernatural" to abbreviate "x is not Natural", so that universally, every phenomenon is either Natural or Supernatural (but not both). By this definition, anything which is Supernatural is NOT caused by Nature — meaning there does not exist, and there can NEVER exist, any scientific explanation for it.

Again, any given thing is either Natural or Supernatural. But is there any reason to believe that something Supernatural exists?

Suppose that nothing is Supernatural, meaning everything is Natural. Then Nature is omnipotent: Nature causes everything, including itself. For if Nature does not cause itself, then Nature is Supernatural!

Finally, define God = Nature. By this definition, God must exist, unless science is completely wrongheaded and no natural forces exist. Assuming that natural forces indeed do exist, we conclude that God exists, and unless something Supernatural exists, God is both omnipotent and self-causing.

This definition of God is an ancient one, resembling Aristotle's. It is also referenced in the first sentence of the Declaration of Independence, where it says "the Laws of Nature and of Nature's God".

Just in the past decade, Antony Flew (now deceased), who was for most of his life a leading figure in atheistic philosophy, abandoned atheism and adopted Aristotle's God.

It is time to end the dichotomy between atheism and theism. Atheists typically put their faith in science, but as demonstrated above, God is the foundation of all science. Simply by clarifying the terminology — i.e. establishing a scientific definition of God — no religious conversion is necessary. On the other hand, theists must respect the scientific method as a valid mechanism for understanding God.

Consequences of A8 — by Jonathan Emerson

I only just now realized that Axiom 8 (Equipotency) is sufficient (together with Transitivity) to prove Axiom 3 (non-circularity).

Here is the proof:

Suppose there exists a counter-example to A3:

r and s are different, and each causes the other.

Since they are different, they both cause something else — that is, Cr and Cs (as in the stronger definition of C used in Version 2).

Since they cause each other, by Transitivity, whatever r causes, s also causes, and vice versa. That is, for any x, r causes x if and only if s causes x.

By A8, since r and s both have precisely the same effects (they cause all the same things), r = s. This contradicts the assumption that they are different.

This means that A3 has no counter-example, so it must be true.


Even so, it is useful to keep A3 as a separate axiom because it is so basic, essential, and intuitively obvious, whereas A8 requires a deeper analysis to understand. Also, since A8 is only used to prove uniqueness, even if it is rejected, the existence of an omnipotent, Absolute phenomenon still follows from the other axioms. Would anyone who accepts the conclusion that at least ONE such phenomenon exists seriously entertain the idea that perhaps TWO OR MORE exist? Imagine a pantheon of Gods where each one independently manages to cause everything in the universe — except for the other Gods! I have never heard of any actual polytheist philosophy suggesting this viewpoint. In fact polytheism is often just a particular version of monotheism where a single Absolute God delegates powers amongst His/Her forms, creations, and emanations, which constitute the deities, angels, or "gods". Ironically, there is a common misconception amongst Christians (many of whom believe in angels) that polytheism is inconsistent with monotheism.