Perhaps I didn’t make myself clear enough: the cosmological argument is a family, a broad genre of arguments.
Part of it is the argument from contingency (the argument from a contingent being): every contingent entity requires a cause; therefore, there exists an entity that is a necessary existent (the being from which the contingent being derives).
That is what my question is focused on.

In that argument, indeed the conclusion is that there exists a necessary being. What’s the problem with that? And in my opinion, the question whether necessity is a property or not (I think it is) changes nothing at all.

The problem is that either way,
if an entity that “exists necessarily” is a property, then that is ruled out from the outset once we assume that existence is not a property.
And if “necessary existence” is not a property, then there is only one way to understand it: you are claiming that the concept of “necessary existence” emerges from the set of properties of the entity. If so, there is no necessity that existence should emerge in this way rather than another.
The claim that the entity simply exists and cannot cease to exist (or exist in another way) usually requires some kind of characterization and connection to the concept of the object. And that, as stated, does not exist.

2. The Rabbi surprised me here by writing that he thinks necessity is a property. If so, why do you reject the ontological argument? And why not posit “an island that necessarily exists” and “a demon that necessarily exists”?

I disagree. Existence indeed is not a property, but in my opinion necessity can be a property (of my existence, and therefore of me).
I didn’t understand the other side of it (if necessary existence is not a property).

I don’t reject it categorically; rather, I think it is based on assumptions that can be rejected. I claim that it is not merely conceptual analysis (as Anselm thought), but an argument based on assumptions.
In the updated formulation that will appear in the book, I really do reject Anselm’s argument, because one of the assumptions is not plausible. But that requires a detailed analysis, and this is not the place.

Is it specifically Anselm that you reject? Or the entire ontological argument?

My claim is that if necessity is not a property, then how can the conclusion be expressed in terms of an entity that exists necessarily, rather than simply an entity that exists? If it is not something in the thing’s properties, then what distinguishes that entity? It follows that there is no such entity.

I don’t know what the difference is. Anselm’s argument.

I still didn’t understand the last piece of pilpul.

Questioner, maybe you didn’t notice, but in the previous reply the Rabbi wrote that in his opinion necessity is indeed a property, so the whole pilpul falls away.

D, I understand the Rabbi to be speaking also according to the view of one who holds that necessity is not a property.

Rabbi, (regarding the cosmological argument)
My claim is that if necessity is not a property, then how can there be a being that exists necessarily? After all, this has to be expressed in some way in the being itself. If there is no way for this essence to be expressed in the being, then there is no reason to assume there can be an argument that proves the thing’s necessary existence. Because there cannot be necessary existence for a thing, only contingent existence.

You are basically challenging the very concept of “exists necessarily,” with no connection at all to the cosmological or ontological argument.
I don’t see the difficulty. A being that exists has the property that its existence is necessary. And if that is not a property, then it is like another kind of existence. So what? Just as one can say that the being exists, one can say that it exists necessarily.
As I said, I tend to think that necessity is indeed a property.

First of all, I am not looking for the answer by assuming that necessity is indeed a property. Let’s put that aside.
(For someone who holds that, then of course the question does not arise in the first place.) My question is only according to the view of one who does not hold that.

Now, I see here in the Rabbi’s words two lines of thought, but I didn’t understand either of them.
A. The Rabbi mentioned: “A being that exists has the property that its existence is necessary.” I understand you to be claiming that insofar as the being exists, some of its properties cause its existence to be necessary.
In effect, that the essence of necessity emerges from the set of the being’s properties. But how does that happen? For each and every property in it has no connection to existence (because, as stated, existence is the realization of the being and is not a property).
B. One could understand your words to mean that insofar as a being exists, there is another property called “necessity,” and it acts on the being so that it will always remain in existence. But it cannot compel the being to exist (as in the ontological argument; rather, only once it exists, it exists necessarily), but I still don’t really understand how that works. After all, if you admit that there is no connection between existence and the properties of the being, then how could a property affect the being’s existence?
C. Later the Rabbi wrote, “And if it is not a property, then it is like another kind of existence.” I didn’t really understand how that could be, since there is nothing that causes the difference between existence and necessary existence (neither a property nor an essence in reality).

A. I don’t know what emerges from what and why. There is necessary existence as distinct from existence (assuming necessity is not a property). That’s all. Why does something have to emerge from something else?
B. What does it mean that it cannot compel the being to exist? I don’t understand these word games.
C. It is a different kind of reality in essence. What makes two identical raindrops two different objects? The fact that they are two different ones. That’s all.
I don’t see any point in this discussion.

A. So how does that difference happen (between necessary existence and regular existence) if it is not expressed in any parameter?!?
B. That if they agree that properties do not affect existence (and not like the ontological argument), how does the property of necessity affect it (for one who holds that such a property exists)?
C. But how is it expressed if there is no difference between the two drops of water? Maybe the prior assumption is: what is existence, for one who holds that it is not a property?

There is great value in this discussion, regarding the cosmological argument.

A-C. I already asked you what distinguishes between two identical raindrops (that are even in the same place. In what sense are they two?). There are distinctions that do not stem from a difference but from non-identity. See what I wrote in Two Carts, second gate, regarding the principle of the identity of indiscernibles. You did not answer me about that.
B. I don’t understand a word. I don’t think there is any point in continuing.

That we are speaking of two different entities. Each entity contains the characteristics. Therefore, even when the characteristics are identical, there is not always an identical entity.
But I still didn’t understand what distinguishes an entity that exists necessarily from an entity that exists, if this is not expressed in any characteristic or essence.
C. I think this can be merged with what I just wrote above.

I’m not sure I understood what you wrote. You are saying that despite the identity in characteristics, these may be two different objects. If that is what you meant, then we are done. That is exactly what I am claiming, and therefore here too these are two different kinds of existence that differ not in a property but in their very essence.

How can there be existence that differs in its very essence? (That is question C.)
Either a thing exists or it does not exist. It cannot exist necessarily.

You are begging the question. That is exactly what the Rabbi is claiming: that there can be non-existence, existence, and necessary existence.

I agree that this is how it sounds from what I wrote, but that is not so, because the concept of existence is not a “property” (in the broad sense of the word), but rather the realization of the being in reality. So either there is realization or there is not. There is no concept of realization necessarily.