## Gödel's ontological proof of God

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Presumably Godel is making the same sort of error, equivocating on "possibility."
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So we both agree that modal ontological arguments like Gödel's fail to prove the existence of God.
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- :roll:
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But regarding your answer (I'm using 'Q' rather than 'G' or 'U' to steer clear of theological or fictive connotations):

If I understand (I've not read subsequent posts to your answer to me), your argument starts with: Q is consistent and ~Q is consistent, so S5 proves ~pnQ v ~pn~Q.

I can see that argument if these are theorems of S5:

Q -> ~pn~Q

~Q -> ~pnQ

Are they? If not, then what is the argument that "Q is consistent and ~Q is consistent" implies that S5 proves ~pnQ v ~pn~Q?

Then you say, "Therefore, we cannot just assume that because some X is not a contradiction that it is possibly necessary."

I take that to mean: "Q is consistent" does not imply S5 |- pnQ.
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S5 has as an axiom that ◊□p ⊢ □p.

Therefore, under S5, these cannot both be true:

1. ◊□q
2. ◊□¬q

Therefore, under S5, this is not true:

3. ¬□¬p ⊢ ◊□p

This then relates to the post above.

Assuming that (a) means (b), (b) needs to be justified. Given that (3) is false, this is false:

4. ¬□¬∃xC(x) ⊢ ◊□∃xC(x)

So ◊□∃xC(x) must be justified some other way for a modal ontological argument to work.
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Imagine that some intelligent, all powerful, all knowing, creator of the universe actually does exist, but that because it doesn't necessarily exist then we refuse to call it God, as if the name we give it is what matters.

"Q"?

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I do see now that to show that (3) is not the case, we need rely only on pnQ -> nQ and the fact that it is not the case that pQ |- nQ.

But It is difficult to follow you as you jump around among very different formal formulations and among different English formulations and different kinds of examples. I started out trying to sort out your original argument as originally formulated but now you've twice jumped to different, though related, formulations. I'm giving up for now. It would help if you would give one self-contained argument with transparent inferences from start to finish.
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All we are left with is the claim that it is possibly necessary that there exists an X such that X is all powerful, is all knowing, etc.

The claim that ◊□∃xP(x) where P is every positive property besides necessary existence.

The claim comes T1 and D1. A God possesses all positive properties. Positive properties are possibly exemplified. Then we have C. Being that necessary existence is a positive property, and it is possibly exemplified, we end up with ◊□∃xG(x). Your argument seems to be that, to take advantage of S5, for ◊□p ⊢ p, we must pull the predicate NE out of the variable, so we end up with ◊□∃xP(x) where P is every positive property except necessary existence, and since necessary existence has been pulled out, we don't know whether ◊□∃xP(x) is true. Is that right?
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Some points.

I have seen elsewhere that some think that D1 is basically Zermelo-Russell's paradox. I would say it doesn’t imply the paradox. The set of all sets argument is paradoxical precisely because it’s universal. D1 however is restrictive, “all positive properties” isn’t the same as “all properties”. Having all of something isn’t an issue, an all-colourful being can be defined as “having all colours”, such a being is not only possible but also exist in real life. Stars emit all frequencies in the visible light spectrum.

Then that, for A3 to be valid, D1 cannot yield paradoxes (violation of LNC). I would say it doesn’t end up in paradox because of A1 — it makes sure that no positive properties conflict with each other. For example, being colourless is the denial of being colourful, only one of those is positive. Being transparent ←→ being colourless, therefore being transparent is also the denial of being colourful (this example ties in with A2).
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As far as S5 is concerned, if F is a formula, then AxF and ExF are both treated as if they are just a propositional letter. Is that not correct?
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The explanation of the argument here presents the problem more clearly.
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It would help if you would give one self-contained argument with transparent inferences from start to finish.

I think the previous argument did that? Perhaps you could let me know which line(s) you'd like me to explain further?
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The proof we are discussing is Gödel's proof. From what I've read, epistemic modality has nothing to do with the proof, the proof uses S5. On the contrary, Gödel is trying to prove exactly something it is clear we have no clue about.

what the operators are supposed to mean ('◊' and '□')

Many-worlds semantics.

But then:

Otávio Bueno and Scott Shalkowski (2015) adopt modalism about modality. They maintain that there is no reductive analysis of modality in terms of non-modal facts or properties. For arguably one cannot reduce the truth of “It is possible that P ” to “It is true in some world, w, that P” without wondering whether or not w itself is possible or impossible. — SEP

and as soon as they try to nail them down other logicians will disagree

Such is the life of the philosophuck.

Presumably Godel is making the same sort of error, equivocating on "possibility."

That would be out of scope just like discussing whether S5 obtains or whether existence is a predicate. We can make a thread for ontological arguments in general.
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Ok

Your transition to from 3 to 4 is basically ◊∃x(P(x)∧NE(x)) ↔ ◊□∃xP(x).
Your reply to Leontiskos fights with the definition of God including necessary existence or not. I don't think it has to include it. As you said:

Imagine that some intelligent, all powerful, all knowing, creator of the universe actually does exist, but that because it doesn't necessarily exist then we refuse to call it God, as if the name we give it is what matters.

I don't see any issues because, differently from others, for me your 3 and 4 are equivalent. Once we establish ◊∃x(P(x)∧NE(x)), we can then claim ◊□∃xP(x). The issue with this transition, as I pointed, is that it invites a Gaunillon counter-argument:
So, there would an infinite amount of lesser gods each having all positive properties except one, except two, and so on.
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P. Q. ¬(p → q):
0. 0. 0
0. 1. 0
1. 0. 1
1. 1. 0

P. Q. (p → ¬q)
0. 0. 1
0. 1. 1
1. 0. 1
1. 1. 0

The first table by the way is a NOR gate with an inverted P, and the second table a NAND gate. Physically different.
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I think the most remarkable and amusing part of Godel's argument, is in the beginning before the use of modal logic, in which he argues for the existence of a 'god term' by turning the principle of explosion on its head.

Constructively speaking, an existential proposition is proved by constructing a term that exemplifies the proposition, as per the Curry Howard Isomorphism. Classically speaking, an existential proposition can also be derived by proving that it's negation entails contradiction, as per the law of double negation.

In Godel's proof however, he defines a so-called Godliness predicate G, where as usual ~G(x) corresponds to the principle of explosion

G(x) --> B(x)
G(x) --> ~B(x)

where B is any predicate.

But in Godel's case, he defines G as only implying properties that satisfy a second-order predicate he calls "Positivity", which is a predicate decreeing that G(x) --> B(x) and G(x) --> ~B(x) cannot both be true.

So in effect, Godel crafted a non-constructive proof-by-absurdity that implies the existence of a god term on the basis that non-existence otherwise causes an explosion! this is in stark contrast to the normal constructive situation of proofs-by-absurdity in which a term exemplifying a negated existential proposition is constructed in terms of a function that sends counterexamples to explosions.

The rest of Godel's proof is unremarkable, since he defined G as implying it's own necessity, meaning that if G is said to be true in some world, then by definition it is said to be true of adjacent worlds, which under S5 automatically implies every world.
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It seems to me you are thoroughly confused.
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My questions were here:

https://thephilosophyforum.com/discussion/comment/914470

Your response was to switch to a different description of your idea.

Am I correct that by "we cannot assume pEx(nPx) is true for any logically consistent Px" you mean "For all consistent Px, we have that pEx(nPx) is not logically true"?

We can assume anything. So I take it "cannot assume" is colloquial for something more logically definite. Thus my question above.

Also, you have a modal operator after a quantifier. I don't think S5 can do anything more with that than to regard the quantified formula as just a sentence letter, so S5 sees pEx(nPx) as just pQ.
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It seems to me you are thoroughly confused

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Also, you have a modal operator after a quantifier.

Yes, good catch. I should have used ◇□∃xP(x).

Am I correct that by "we cannot assume pEx(nPx) is true for any logically consistent Px" you mean "For all consistent Px, we have that pEx(nPx) is not logically true"?

What I am saying is that ◊∃xP(x) ⊬ ◊□∃xP(x), i.e "it is possible that X exists" does not entail "it is possibly necessary that X exists".
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I hope it won't be too long that I'll have time to resume going over your argument with the emendations.
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I don't see where that is implied in the argument.

P(ψ)≡¬N(ψ) — sime

If N is supposed to mean necessary existence, that is a rejection of axiom 5.

N was supposed to mean the possibility modality (N standing for Negative Properties, in order to stand for the opposite of Positive Properties). The question here I was interested in, is how to give a syntactical definition of Positive Properties such that the resulting argument follows as a valid tautology in some modal logic. This was partly in order to help clarify the the definitions Godel provided, even his assumptions need to be altered slightly and the resulting argument and its conclusion aren't quite the same.

For example, taking Positive properties to refer to what is necessarily true of all individuals in every possible world, turns Axiom A2 into the definition of a functor, which is rather tempting. It also makes the possibility of god follow as a matter of tautology.

Also, Godel's definition of essences seems close to the definition of the Categorical Product. So why not take the essence of an individual to be the conjunction of his properties?

One thing I overlooked was that God was defined as referring to the exact set of positive properties, which would mean that according to my definition of P, all individuals would be identical. But then supposing we weaken the definition of "Godliness" to refer to a set that contains all the positive properties and possibly some of the negative (i.e contigent) ones?

I think there is quite a few pedagogically useful questions here.
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Also, you have a modal operator after a quantifier. I don't think S5 can do anything more with that than to regard the quantified formula as just a sentence letter, so S5 sees pEx(nPx) as just pQ.

If it is worth something, ◇∃x(□Px) → □∃x(Px) is valid in S5.
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I don't understand that proof.

Where can I see a specification of S5 extended to a deduction calculus with quantifiers?

I don't know what deduction in S5 permits:

inferring line 4 from line 3. (~nQ does not imply ~Q)

inferring line 5 from line 2. (pQ does not imply Q)

line 6 from line 5 is existential instantiation applied to a modal formula, but S5 is only a modal propostional logic.
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It is not from S5 but from Gödel's definitions. When you go to D3 in the OP, NE(x) is basically the exemplification of all properties of x, so ◊∃x(P(x)∧NE(x)) ↔ ◊□∃xP(x) where NE is necessary existence and P is all positive properties except necessary existence.
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I don't understand. You said a certain formula is valid in S5. The proof generator shows a deduction of the formula. But I can't make sense of the deduction at the lines I mentioned. The proof generator makes no mention of exemplification and positive. Bringing in exemplification and positive does not address my points. And I'm not even talking about Godel. I'm just looking at certain claims about what is derivable in S5, as those claims don't invoke exemplification or positive.
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I hope it won't be too long that I'll have time to resume going over your argument with the emendations.

Are you waiting on me for something else or are you saying that you're currently too busy to examine what I've said?
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I'm saying that I'll take your latest note and incorporate it as I go over your argument again. Not waiting on you.
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Cool. Well if it helps, I've re-written that first comment to correct the typos and to hopefully be clearer.
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Thanks.
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