Existence as a predicate and Godel's ontological argument

Question

I am referring to this paper https://github.com/FormalTheology/GoedelGod/blob/master/GodProof-ND.pdf which has formalized the ontological argument.

If I am not mistaken, watered down, the argument should be that we can define existence as a property by saying it's exemplification of essential properties of a thing, and then take a set of properties (here he takes 'positive properties') to include it. Once we accept existence as a property, the rest kind of naturally follows. Once we establish existence as a property we can simply in different terms define something in whose essence is to have that property, and then the very possibility of that thing would imply its necessary existence.

I'm aware why ontological arguments have historically failed, and why they must fail, but that doesn't show me the error of this argument, and that's what I'm interested in. I would prefer not to reach outside of the argument to search for the fact that there must be an error, but attempt to point at the error. Where does it exactly go wrong?

The only thing that crosses my mind right now is to refer to the equivalence made by the property of existence and the exemplification of essential properties made by the argument. If the complete sum of all essential properties is exemplified, then a thing has the property of existence. If we use existence defined in such a way as an essential property of something, we are inferring a complete sum of all properties, but we're doing it from within the essence of a thing, which is still not complete, so the reference to it is not justified. This is the only thing I could think of so far, aside from rejecting the notion of essence.

How do we answer this paper? Is there shakiness of some axiom that I'm missing?

Answer 1

The main attacks on Gödel's ontological argument are not found necessarily in attacking the logical structure of his argument, more so it is done by rejecting the axioms he provides. From the Stanford Encyclopedia of Philosophy's article on ontological arguments:

Given a sufficiently generous conception of properties, and granted the acceptability of the underlying modal logic, the listed theorems do follow from the axioms. ... So, criticisms of the argument are bound to focus on the axioms, or on the other assumptions which are required in order to construct the proof.

Additionally from Wikipedia:

Most criticism of Gödel's proof is aimed at its axioms: As with any proof in any logical system, if the axioms the proof depends on are doubted, then the conclusions can be doubted. This is particularly applicable to Gödel's proof, because it rests on five axioms that are all questionable. The proof does not say that the conclusion has to be correct, but rather that if you accept the axioms, then the conclusion is correct.

Christopher Small provides a paper outlining Gödel's ontological argument and some of the objections to it. To highlight some of the more publicized objections:

Sobel (1987) has objected that Gödel’s argument leads to modal collapse. So this objection needs to be examined in detail. A modal logic is said to suffer from modal collapse if every true statement in the system becomes necessarily true. While this is not in itself inconsistent, it does appear to undermine the point of the logic by negating any distinction between accidental and necessary truth.

C. Anthony Anderson has objected to axiom (P2) on the grounds that it does not permit neutral attributes. For example, being of average height would seem to be a neutral attribute. However, axiom (P2) demands that either it is positive or that not being of average height is a positive attribute.

The paper goes into much detail about the dubious status of the axioms, as does the SEP article above.

It's also interesting to note the history of existence as a valid predicate. Many philosophers disagree that existence can at all be a predicate. From Vacuous Names and Fictional Entities (note 6) by Saul Kripke:

Both Frege and Russell think that existence cannot be a predicate of individuals but identify it with the ‘higher level’ property expressed when we attach an existential quantifier to a one-place predicate. Frege said that the error of regarding existence as a predicate of individuals rather than (in his terminology) a second-level concept is the fundamental error of the ontological argument (see Frege 1997:146, note H). Russell’s view is in fact similar to Frege’s, though formulated in terms of his theory of descriptions, so that the illusion that there is a predicate of individuals can be connected to the illusion that the descriptions are terms referring to objects.

In general, most objections to a naive acceptance of existence as a property are due to the acceptance coming with some very undesirable or contradictory conclusions. For example, the second order predicate existence that Russell and Frege later formulated is based off of a descriptivist theory of reference, and the flaws of that theory are well documented. Ultimately, it is still very dubious as to whether or not anyone, including Gödel, have provided a sufficient and acceptable definition of an existence predicate. Even more so, many deny that existence is trivial for any case (see the paper by Kripke mentioned above.)