Logic

An Overview of Kurt Gödel’s Ontological Argument

Perhaps I’ve not given it enough thought, but I must confess that I find Gödel’s ontological argument to be one of the most powerful and persuasive pieces of evidence for the existence of God — evidence that really is hard to ignore. I focus on Gödel, by the way, since his is the version I’ve looked at the most, though Leibniz offered a similar one in correction of Descartes’ which, along with Anselm’s, posited existence as a predicate and never wrestled with the question of whether or not the perfect being in question was itself logically possible. At any rate, it does make a good bit of sense, and the objection that positive properties are subjective and undefinable somehow doesn’t sit well with me, though it’s hard to say why. I’m sure there must be an answer, but it seems that we can all intuitively agree with his own explanation that appeals to the moral aesthetic of of such properties.

Formally, the argument is written thusly:

Ax. 1.    {[Pφ ⋄ ▢(∀x)(φx → ψx)] → Pψ}
Ax. 2.    {P~φ ↔ ~Pφ}
Th. 1.       ∴ {Pφ →♢(∃x)φx}
Df. 1.     {G(x) ↔ (∀φ)(Pφ → φx)}
Ax. 3.     P(G)
Th. 2.       ∴♢(∃x)G(x)
Df. 2.     {φ ess. x ↔ [φx ⋄ (∀ψ)(ψx → ▢(∀y)(φy → ψy)]}
Ax. 4.     {Pφ → ▢Pφ}
Th. 3.       ∴G(x) → G ess. x
Df. 3.     {E(x) ↔ (∀φ)(φ ess. x → ▢(∃y)φy)}
Ax. 5      P(E)
Th. 4.       ∴▢(∃x)Gx

All of these formulas can be restated in more colloquial English, however, so that one doesn’t need to know the symbolic language to understand the argument.

Ax. 1. If a property is positive and the instantiation of that property entails the instantiation of another property, that other property is also positive.
Ax. 2. A negation of a property is positive if and only if the property itself is not positive.
Th. 1. Therefore, if a property is positive, it is possible that such a property exists.
Df. 1. A thing is Godlike if and only if that thing has all positive properties and only positive properties.
Ax. 3. Being Godlike is a positive property.
Th. 2. Therefore, it is possible that something Godlike exists.
Df. 2. A property is an essential feature of something if and only if that property entails every other property that thing has.
Ax. 4. If a property is positive, it is necessarily positive.
Th. 3. Therefore, if something is Godlike, being Godlike is an essence of that thing.
Df. 3. Something necessarily exists if and only if every essential feature of that thing necessarily exists.
Ax. 5. Necessary existence is a positive property.
Th. 4. Therefore, something Godlike necessarily exists.

If that still seems like a bit much to handle, don’t worry. It took me a while to understand it too. Keep in mind, also, that philosophy has its own language so that those who haven’t been acquainted with that language might find some of the arguments difficult to grasp. That sounds a bit demeaning perhaps, but I only say it so that no one feels stupid if they’re having a hard time following along. Oftentimes philosophers use very common words in very precise ways so that most of us think their sentences are incoherent mumbo jumbo until we realize that they’re actually defining words differently than we expected. I’ve tried to be relatively straight forward in my interpretation of the argument that I wrote above while remaining true to the premises as they were written formally, and I think I’ve been successful, even though the argument itself is a bit dense and abstract. It might be worth keeping in mind, however, that the words “necessarily” and “possibly” carry with them certain connotations so that they refer to things that are logically necessary or logically possible in some possible world. Or, to put it another way, something is logically possible if, in some hypothetical world, it can exist without logical contradiction. For example, it is conceivable that in some possible world the continent of North America is shaped like a perfect triangle. Bizarre though it may seem, there is no inherent contradiction in this idea, meaning that it is logically possible and not an absolute impossibility. On the other hand, in no possible world is there a triangle that has any more or less than three angles, since a triangle is by definition something with three angles. If it has four angles it would be a quadrilateral and no longer a triangle. This is, then, a necessary truth since in all possible worlds triangles will always have three angles — no more; no less. And while we’re at it, I might as well clarify what is meant by the term “essence” or “essential.” The essence of something is that which makes a thing what it is. A triangle, to use the last example, is a geometrical figure of three angles, and these three angles are the essence of that geometrical figure called the triangle. It may be a blue triangle or a green triangle or even a metal triangle used to make noise, but these are not essential properties of the triangle since they don’t define what it actually is. So, if what I’ve written doesn’t make sense, try reading it with these ideas in mind and see if it doesn’t help.

To boil it all down, though, the core of the argument goes like this:

1. Something that is Godlike has all positive properties and only positive properties.
2. We can prove that these so-called “positive properties” are at least logically possible.
3. Furthermore, since being Godlike is a positive property, we can conclude that something Godlike possibly exists.
4. Now, if something is Godlike, then this Godlike thing entails all positive properties, since positive properties are the essence of being Godlike.
5. Necessary existence is a positive property.
6. Therefore, something Godlike necessarily exists. This is because being Godlike entails all positive properties and necessarily existence is a positive property, meaning that something Godlike necessarily exists.

As far as I can tell, all of this is a very solid piece of reasoning (far better than the moral argument, which simply doesn’t hold up, or even the Kalam Cosmological argument, which has some pretty clear weaknesses), making it hard for me to seriously deny the existence of some Godlike being.

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