You edited your comment. I think I see the confusion. I thought you were referring to lines 3 and 4 of this post. But I guess you were talking about this.

For the record, I don't think the criticism, in the post you referred to, is successful. I developed why in my post last page.

I don't know which of my posts or comments you are commenting on.

In a recent post, I said that I don't understand the proof at the proof generator.

I'm not stating a criticism of Michael's posts. I'm just trying to figure them out.

So under these axioms, in S5, every possible positive property is exemplified in at least one being, meaning that necessarily there are innumerably many beings — every possible being with a certain set of positive properties necessarily exists. — Lionino

Trying to put this in logical terms, I think it would be ∀φ[□∃xφ(x)]. From source [3]:

It would seem to even follow that there are near-perfect, but defective, demi-gods and all matter of other theologically repugnant entities. Gaunilo concluded, reasonably enough, that something must be wrong with the argument.

Isn’t all of this just begging the question? I mean, are we not allowed to challenge the assertion that “necessary existence” is a “positive quality?” Isn’t it possible for necessary existence to be a negative quality? After all, human beings exist, and they are imperfect and mortal; they make mistakes, they sin, etc. So maybe “non-existence” (as opposed to “necessary existence”) is the (more) positive quality.

Consider this: “non-existent” beings don’t age, suffer and die. And because they transcend time and space, non-existent beings aren’t restricted by the laws of physics. In fact, non-existent beings are not adversely affected by anything in the universe – including hatred, discrimination, war, ignorance and greed. Taken one step further, if God exists (or even if only the idea or concept of God exists), then perhaps God (or the concept of God) values non-existence over (necessary) existence. Why does Anselmo or Descartes or Gödel get to decide what God (or the concept of God) values most?

This is what I’ve always found troubling about Pascal’s “Wager.” Pascal argued that belief in God will get you into heaven after you die if God does exist. And yet, Pascal continued, you won’t be worse off by believing in God if God doesn’t exist after all; your death will be met with the same fate whether you believe in God or not. So you might as well believe in God.

But Pascal is assuming (begging the question) that one of God’s characteristics is rewarding believers after death. But what if God rewards those who don’t believe? Maybe God prefers critical thinkers over those who dogmatically follow religious tenets. Why does Pascal’s assumption of “God-rewards-those-who-believe-God-exists” take precedence over someone else’s assumption of “God-rewards-those-who-don’t-believe-God-exists?”

And so it is with any ontological proof of God – whether it be valid or not, sound or not, or well-argued or not. Maybe existence is not the positive quality it’s cracked up to be. (?)

If I haven't made any mistakes here:

At least for me, this is more exact and clear:

(1) E!xFx ... premise

(2) pExAx ... premise

(3) pEx~Ax ... premise

(4) {(1), (2), (3)} is consistent

(5) pE!x(Fx & Ax) ... (1),(2)

(6) pE!x(Fx & ~Ax) ... (1),(3)

(7) pnEx(Fx & Ax) -> nEx(Fx & Ax) ... theorem

(8) pnEx(Fx & ~Ax) -> nEx(Fx & ~Ax) ... theorem

(9) {(1), (2), (3)} |/- pnEx(Fx & Ax) ... (1),(4),(6),(7)

(10) {(1), (2), (3)} |/- pnEx(Fx & ~Ax) ... (1),(4),(5),(8)


* But the inferences at (5) and (6) are invalid (according to the validity checker).

https://www.umsu.de/trees/#((~7x~6y(Fy~4x=y)~1~9~7xAx))~5~9(~7x(~6y((Fy~1Ay)~4x=y)))||universality

https://www.umsu.de/trees/#(~7x~6y(Fy~4x=y)~1~9~7xAx)~5~9~7x~6y((Fy~1Ay)~4x=y)

"There exists exactly one falcon, and it possible that there exists a non-falcon" doesn't entail "It is possible that there exists exactly one falcon that's a non-falcon".

and

"There exists a falcon, and it possible that there exists a non-falcon" doesn't entail "It's possible that there exists a falcon that's a non-falcon".

But

{(1), (2), (3)} |/- pnEx(Fx & Ax)

and

{(1), (2), (3)} |/- pnEx(Fx & ~Ax)

are correct anyway (according to the validity checker). Just not by your argument.

https://www.umsu.de/trees/#((~7x~6y(Fy~4x=y)~1(~9~7xAx~1~9~7x~3Ax)))~5~9~8~7x(Fx~1Ax)||universality

* I don't see the relevance of this to your specific argument:

https://www.umsu.de/trees/#~9~7xP(x)~5~9~8~7xP(x)||universality

What you want to prove is not just that that formula is invalid but to prove:

{(1), (2), (3)} |/- pnEx(Fx & Ax)

and

{(1), (2), (3)} |/- pnEx(Fx & ~Ax)

Those are correct (according to the validity checker). Just not by your argument.

It could be fixed this way:

(1) E!xFx ... premise

(2) pEx(Fx & Ax) ... premise

(3) pEx(Fx & ~Ax) ... premise

(4) {(1), (2), (3)} is consistent

(5) pnEx(Fx & Ax) -> nEx(Fx & Ax) ... theorem

(6) pnEx(Fx & ~Ax) -> nEx(Fx & ~Ax) ... theorem

(7) {(1), (2), (3)} |/- pnEx(Fx & Ax) ... (1), (3), (4), (5)

(8) {(1), (2), (3)} |/- pnEx(Fx & ~Ax) ... (1), (2), (4), (6)

https://www.umsu.de/trees/#((~7x~6y(Fy~4x=y)~1(~9~7x(Fx~1Ax)~1~9~7x(Fx~1~3Ax))))~5~9~8~7x(Fx~1Ax)||universality

I think you've misunderstood these:

1. ◇∃x(F(x) ∧ A(x))
2. ◇∃x(F(x) ∧ ¬A(x))

They say:

1. It is possible that there exists some X such that X is the only unicorn and is male
2. It is possible that there exists some X such that X is the only unicorn and is not male

They are not inferences but independent premises and might both be true.

My argument is that we cannot then infer these:

3. ◇□∃x(F(x) ∧ A(x))
4. ◇□∃x(F(x) ∧ ¬A(x))

Which say:

3. It is possibly necessary that there exists some X such that X is the only unicorn and is male
4. It is possibly necessary that there exists some X such that X is the only unicorn and is not male

Under S5 they cannot both be true.

This matters to modal ontological arguments because (3) and (4) are equivalent to the below:

3. It is possible that there exists some X such that X is the only unicorn and is male and necessarily exists
4. It is possible that there exists some X such that X is the only unicorn and is not male and necessarily exists

The switch from "possibly necessary that there exists some X" to "possible that there exists some X such that X necessarily exists" is a sleight of hand. It is used to disguise the fact that asserting the possible existence of God – where necessary existence is a property of God – begs the question.

They are not inferences but independent premises and might both be true. — Michael

You wrote in the argument:

If take A(x) to mean something like "x is male" then both (2) and (3) are true. — Michael

[emphasis added]

So in my first post I captured that implication.

And in my second post I gave a version in which instead they are premises:

(2) pEx(Fx & Ax) ... premise

(3) pEx(Fx & ~Ax) ... premise — TonesInDeepFreeze