## Gödel's ontological proof of God

• 2.1k
It seems there is no thread around Gödel's ontological proof of God, so, prompted by another thread, I will make one for such as it is high time.

If you don't know the history around the argument, here it is:

Ontological arguments in their modern sense go back to Sant'Anselmo d'Aosta. Monk Gaunillon of Tours makes a rebuttal often deemed as succcessful, parodying the argument to imply "the perfect island exists". Gaunillon argues we cannot go from idea to reality. Ontological arguments evolved through times.

Descartes writes his own argument, where necessary existence is part of the definition of God. When Descartes's argument is formalised, we find out that, if God's existence is possible, it is necessary; however the argument falls short of proving that God's existence is possible [7].

Leibniz somewhat accepts Descartes' argument, but identifies a critical issue — it assumes the being is conceivable or possible [8]. Leibniz attempts to patch the argument by proving God's existence is possible.

Gödel writes his ontological argument, based on Leibniz's, which comes from Descartes' [7]. A manuscript can be seen here:

Anticipating an imminent death, Gödel shared his proof with Dana Scott. Although Gödel lived for 8 more years, he never published the proof. The details of the proof became known through a seminar led by Scott, whose slightly modified version was eventually made public.

As logic was not well-developed in Gödel's time as it is today, it is unsure what kind of logic Gödel had in mind when writing the proof, so others took it upon themselves to put the proof on firm grounds. Sobel plays Gaunillon to Gödel's Anselmo, wanting to show the argument could be applied to prove more than one would want [7].

Sobel also proves that the argument ends in modal collapse:

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For the proof of modal collapse, let Q be some arbitrary truth. We will show that □Q. We know, from Gödel’s theorem 3, that a Godlike being exists: call it j again. So, we know G(j). We also know, from theorem 2, that G is the essence of j. This means that G necessitates all of j’s actual properties. Since Q is true, j has the property of being such that Q (i.e., from (Q&j = j), we can deduce that j has the property x[Q&x = x]). Thus, being G must necessitate being such that Q. Since G is instantiated in every world, it follows that something is such that Q is true in every world. Hence, □Q.

Scott raises that Gödel's proof really amounts to an elaborate begging of the question. The proof itself is correct (if God is possible, it is necessary), but it breaks down at the same point as Descartes: proving that God is possible [7].

Software is developed that is able to verify arguments in higher-order logic. Gödel's ontological argument fails a consistency test.

Scott's is confirmed consistent for including a conjunct in D2, but it implies modal collapse (everything that is possible is also necesssary, and it has been argued this implies no free will) [1][9]. Here is Dana Scott's version of the argument:

Gödel's original axioms (without the conjunct) are proven to be inconsistent [4][5][6].

C. Anthony Anderson and Melvin Fitting then provide different versions of the consistent proof, to avoid modal collapse [7][10]. Fitting's version works with extensional properties, rather than intensional properties as is believed that Gödel had in mind [7].

Anderson along with Michael Gettings argues that his own emendation may be refuted by the same objection as Gaunilo raised against St. Anselmo [3].

Both Anderson's and Fitting's proofs are computer-verified to avoid modal collapse. Both proofs are verified as consistent [9].

Note: Gödel's original "positive properties" is to be interpreted in a moral-aesthetic sense only [2].

[1]
"Automating Gödel's Ontological Proof of God's Existence with Higher-order Automated Theorem Provers", Christoph Benzmüller and Bruno Woltzenlogel Paleo (2014).
[2] K. Gödel, Appx.A: Notes in Kurt Gödel’s Hand, 144–145
[3] "Gödel's Ontological Proof Revisited", C. Anthony Anderson and Michael Gettings (1996)
[4] "Experiments in Computational Metaphysics: Gödel’s Proof of God’s Existence", Christoph Benzmüller and Bruno Woltzenlogel Paleo (2015)
[5] "An Object-Logic Explanation for the Inconsistency in Gödel’s Ontological Theory (Extended Abstract)", Christoph Benzmüller and Bruno Woltzenlogel Paleo (2016)
[6] "The Inconsistency in Gödel's Ontological Argument: A Success Story for AI in Metaphysics", Christoph Benzmüller and Bruno Woltzenlogel Paleo (2016)
[7] "Types, Tableaus, and Gödel's God", Melvin Fitting (Trends in Logic, Volume 12)
[8] "Two Notations for Discussion with Spinoza", Gottfried Leibniz (1676)
[9] "Types, Tableaus and Gödel’s God in Isabelle/HOL", David Fuenmayor and Christoph Benzmüller (2017)
[10] ""Some Emendations of Gödel's Ontological Proof", C. Anthony Anderson (1990)

Reveal
The post will be updated eventually to include objections from scholars and replies.
1. Do you accept Scott's version of Gödel's axioms? (5 votes)
Yes
60%
No
40%
2. Do you believe Gödel's proof proves that God exists if the axioms are correct? (5 votes)
Yes
40%
No
60%
• 14.8k
Consider this:

1. ∃xF(x) → ∃x∀y(F(y) ↔ (x = y))

If we take F(x) to mean something like "x is the only unicorn" then (1) is true.

Now consider these:

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

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

Now consider these:

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

Under S5, ◇□p ⊢ □p, and so these entail:

6. □∃x(F(x) ∧ A(x))
7. □∃x(F(x) ∧ ¬A(x))

(6) and (7) cannot both be true, and so therefore (2) does not entail (4) and (3) does not entail (5):

8. ◇∃xP(x) ⊬ ◇□∃xP(x).

This is where modal ontological arguments commit a sleight of hand. To claim that it is possible that God1 exists, where necessary existence is one of God1's properties, is to claim that it is possibly necessary that God2 exists, where necessary existence is not one of God2's properties.

The claim that it is possibly necessary that God2 exists isn't true a priori, and so the claim that it is possible that God1 exists isn't true a priori. As it stands it begs the question.

Or we have to reject S5, but if we reject S5 then modal ontological arguments are invalid because “possibly necessary” wouldn’t entail “necessary”.
• 301
The true value of Gödel's work is not that it manages to reduce the belief in God to a belief in 5 complex axiomatic expressions in higher-order modal logic. The true value of Gödel's work is that it manages to prove that atheists will reject a mathematically unobjectionable proof if it proves something that they disagree with. Gödel was truly a genius.
• 2.1k
Scott argued that Gödel's argument begs the question — which is to say it is circular. I had the same feeling when first reading the argument two years ago:

1 Being god-like is the essence of a god-like being
2 A god-like being has all positive properties
3 Being god-like implies having all positive properties
4 Necessary existence is a positive property
5 Being god-like implies having necessary existence
6 God exists because he has the property of necessary existence

We give a property to God (existence) before his existence is established. It reminds me of Descartes, simply defining God into existence. The reports on automated results however don't bring that up.

In fact, it is precisely at the present point in the argument that Scott's claim can be localized. Godel's assumption that the family of positive properties is closed under conjunction turns
out to be equivalent to the possibility of God's existence, a point also made in [SobOl]. We will see, later on, Godel's proof that God's existence is necessary, if possible, is correct. It is substantially different from that of Descartes, and has many points of intrinsic interest. What is curious is that the proof as a whole breaks down at precisely the same point as that of Descartes: God's possible existence is simply assumed, though in a disguised form.
— [7]

Besides, "essence" here is used strangely. It does not mean "the thing that defines X" or "the thing without which X is not itself", but seems to be "the thing from which all other features of X sprawl". Perhaps it is due to being a translation from German.
• 9k
Anyone reading the language of the proof will - ought to - have questions as to what exactly is being affirmed in each step of the proof. For example:
A1) What is "positive" and why not both?
A2) How does that work?
D1) Martin Luther apparently did not think so.
A3) Depends on who you ask.

And so forth. I cannot tell if the form of the argument is valid: if I convert it to truth tables, it is not. And what is meant here by "exist."
• 587
Just to add to Tim Wood's previous question,what is meant by the words "God" (or "God-like") and "positive"
• 8.9k
'No sequence of words or of logical symbols, however cunningly arranged, can oblige the world to be thus and not so.'

Thus saith the unenlightened.

This is simply a sad fact of life for me, though God can famously speak, and it is so. God's words are infinitely more puissant than mine. He can speak me into existence, allegedly, but I cannot return the favour, and nor can Gödel.
• 14.7k
God's words are infinitely more puissant than mine. He can speak me into existence, allegedly, but I cannot return the favour, and nor can Gödel. — unenlightened
:fire: :up:
• 3.4k
Sorry for the sidetrack, this is just about Anselmian ontology:

If we suppose that existence and non-existence (the negation) can be properties of something, X, then what does it mean to say that X does not exist?
What was that X in the first place, then? :chin:
Either it's nonsense, or such a property already presupposes existence (implicitly) in some way, i.e. that X we spoke of that so happens to not exist.
As a starting point, I'm guessing that failure to differentiate imaginary/fictional and real can lead to reification; that certainly holds elsewhere.

By the way, in mathematics, a proper existential quantification form can be:
p = ∃x∈S φx
where p is the proposition, x is a (bound) variable, S a set, and φ a predicate.
Note that x is bound by S, and ∃ and φ aren't quite interchangeable.
Less confusion invited.
• 2.1k
A1) What is "positive" and why not both?

See note. If being all-knowing is positive, being not-all-knowing is not positive. Beautiful, not beautiful.

A2) How does that work?

Everything that necessary follows from a positive property is also a positive property.

On A3, there is this from [3]:

And he might just maintain that the less evident axioms, for example that a conjunction of positive properties is positive, is an assumption which he adopts on grounds of mere plausibility and is entitled to accept until some incompatibility between clearly positive properties is discovered.

As a starting point, I'm guessing that failure to differentiate imaginary/fictional and real can lead to reification; that certainly holds elsewhere.

Anselmo did reply to Gaunillon by basically saying the latter misses the point. The perfect island may be thought of as non-existant, while God, which is exactly the greatest being, may not. On the other hand, if God is complex, we may think of him as having all the attributes he does, besides existence. Then however, Anselmo affirmed the doctrine of divine simplicity, so on that point the counter does not work.

On divine simplicity:

"The doctrine of Divine simplicity [quite nonsensical], according to which God is absolutely simple, has been out of favour for a while now in both Christian theology and philosophy. It is accused of being inconsistent with the doctrine of the Incarnation (Hughes 1989: 253–64), with that of the Trinity (Moreland and Craig 2003: 586) and of being incoherent in its own right (Plantinga 1980: 46–61)." — Collapsing the modal collapse argument: On an invalid argument against divine simplicity (Christopher Tomaszewski)

Though I am not convinced by C.T. that divine simplicity has been out of favour, especially because of the sources given.
• 2.1k
I don't understand.
• 9k
A1) What is "positive" and why not both?
— tim wood

See note. If being all-knowing is positive, being not-all-knowing is not positive. Beautiful, not beautiful.
Which overlooks - ignores - the conditional. At least two problems, then: the logic, and the definitions. I'm not much interested in the details of the argument. If you want to go through them step-by-step I reckon I won't be the only person to learn from the exercise. But categorical statements are a bear. If they're important, then they need to be proved to be true - and that can be difficult to impossible.

Let's try an example to see if and how it works. Winning WW2 was positive (presumably). The firebombing of Dresden is a necessary implication of winning WW2. So the firebombing of Dresden is positive???
• 2.1k
Winning WW2 was positive (presumably)

That is not a property, is it?
• 2.1k
I agree with this. Gaunillon also said in his "Liber pro Insipiente" regarding the ontological argument that "there is nothing that may bring something from possibility to actuality".

http://www.ptta.pl/pef/haslaen/a/anselm.pdf
• 9k
I don't know. Do we have a standard? If winning WW2 was positive, may we not say that the winning was a positive property? Not arguing, here. More asking what a property is. In the notation in A2, a bit formidable, P is abstract, so any value assigned to P would seem to require the same qualities as P itself, in this sense to be free of ambiguity or equivocation. Sense?
• 9k
Collingwood took on Anselm's proof directly, characterizing it (my words, not his) as preaching to the choir, and as such unassailable (An Essay on Metaphysics, pp. 189 - 190, accessible online).
• 2.1k
When we talk of properties naturally we are talking of things like big, small, nice, kind, smart, lazy, etc, not actions like winning this or that war. But let's say somehow "winning WW2" is a property. Is it a positive property or not? So far, we don't know. There are some ways to know if it is not a positive property, which are whether it necessarily implies or is implied by a not positive property, and whether it is contingently positive. So the doubt around a property being positive or not does not help us rejecting the axiom.
• 301
And so forth. I cannot tell if the form of the argument is valid: if I convert it to truth tables, it is not. And what is meant here by "exist."

Say that the following is provable from theory T:

xx and yy and zz --> rr

With xx, yy, zz the axioms of T.

What does that mean about rr?

In and of itself, such rr means nothing at all. It's just string manipulation.

The semantics, i.e.the truth about rr, lies elsewhere than in any of the syntactic consequences provable from T. Furthermore, it requires a specific mathematical process to unveil such semantics.

First of all, you must have some model-existence (or even soundness) theorem in T that guarantees that any provable theorem rr is indeed true in such models of T.

What is a model of T or even just a universe of T? How does it harness the truth of T?

From any (even arbitrarily) chosen metatheory, you need to construct a structure M, which is a set along with one or more operators. Every such structure M represents an alternative truth of T, i.e. a legitimate interpretation of T.

In other words, unveiling the truth cannot be done on the fly, between lunch and dinner. You also had better avoid non-mathematical methods of interpretation. They simply don't work.

It would cost an inordinate amount of work to correctly harness the truth of Godel's theorem.

This work has not yet been done at this point. The researchers have currently only spent time on investigating the consistency of his axioms and the issue of a possible modal collapse.

With this groundwork out of the way, it will still take quite a bit of time and work to develop a legitimate interpretation for Godel's theorem.

I can personally certainly not do the work, because I am familiar only with PA's truth in its ZFC models. I actively avoid trying to interpret anything else, because these interpretations tend to be extremely confusing. When I accidentally get to see some advanced model theory, I run away.
• 2.1k
And so forth. I cannot tell if the form of the argument is valid: if I convert it to truth tables, it is not. And what is meant here by "exist."

Ignore the schizophrenic above. He has been shown to be an ill-informed sophist in another thread several times. The argument as shown in the OP is verified as valid. You can't easily convert to truth tables.

what is meant by the words "God" (or "God-like") and "positive"

The definition is given in D1, and see the note in OP.
• 301
Ignore the schizophrenic above.

You do not understand enough mathematics to interpret the semantics of Godel's theorem. I have merely pointed out that you are clearly not even aware of that.
• 2.1k
You do not understand enough mathematics

Oh, really?

Go ahead and solve the following operation:

$\bigtriangledown \times (2 x y, 2 y z, 2 x z)$

Give the answer with the unit vectors specified, no parenthesis in the notation. On the left is the nabla operator, so we are clear.

Edit: The crank, despite online and active, hides when pressed to give an answer to an extremely basic vector calculus (an undergraduate subject for everyone in science) that one could do in one's head. That is all it takes to show the cranks claiming to know "mathematics" do not have surface knowledge of what they are babbling about.
• 9k
You can't easily convert to truth tables.
Let's try A1. You can tell me where/how this truth table fails. I render A1 as

(p -> ~q) <=> ~(p -> q)

I can't put in the table - it won't format. But it's a 90 second exercise to see that it doesn't hold when p is false - if p is false, the left side is true but the right side is false.

which are whether it necessarily implies or is implied by a not positive property,
Maybe I'm reading in too much. By positive/negative do you mean purely affirmation and negation. - which really won't do for your, "Gödel's original "positive properties" is to be interpreted in a moral-aesthetic sense only."
• 2.1k
A1 is an axiom, so it is not tautological, you won't get anything out of putting it in a proof checker.

The (p → ¬q) ↔ ¬(p → q) is invalid by the way, the left side is not the same as the right side.

Maybe I'm reading in too much.

Yes, moral-aesthetic sense. What you quoted is me translating A2.
• 9k
Yes, moral-aesthetic sense. What you quoted is me translating A2.
Which is not-so-amenable to pure affirmation/negation.
A1 is an axiom, so it is not tautological,
Then maybe it should be expressed in a different form - excluding the "both"? You did not object to my rendering of it.
• 9k
The semantics, i.e.the truth about rr, lies elsewhere than in any of the syntactic consequences provable from T. Furthermore, it requires a specific mathematical process to unveil such semantics.
Agreed - except that I do not see a mathematical process unveiling meaning - how could it?
• 2.1k
Which is not-so-amenable to pure affirmation/negation.

Correct.

You did not object to my rendering of it.

I did. The logical rendering of A1 is as it is written in the image. Your rendering is invalid because it can entail contradictions.
• 2.1k
(p → ¬q) ↔ ¬(p → q):

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

Therefore (p → ¬q) ↔ ¬(p → q) is false
• 9k
Ok. I see it. Thank you.
• 14.8k

So, the first question to consider is:

1. If something is possibly necessary, is it necessary?

Under S5 (one type of modal logic), the answer is "yes". Ontological arguments depend on this. They all reduce to the claim that because God is possibly necessary, God is necessary.

If we reject S5 then the answer is "no" and all ontological arguments fail.

But let's assume S5 and that the answer is "yes".

The next questions are:

2. Is it possible that there necessarily exists a God who is unique and performs miracles?
3. Is it possible that there necessarily exists a God who is unique and does not perform miracles?

If we accept S5 and if (2) and (3) are both true then it is both the case that there necessarily exists a God who is unique and performs miracles and that there necessarily exists a God who is unique and does not perform miracles.

This is a contradiction. Therefore, (2) and (3) cannot both be true.

Therefore, either:

4. It is not possible that there necessarily exists a God who is unique and performs miracles, or
5. It is not possible that there necessarily exists a God who is unique and does not perform miracles

Even though "God is unique and performs miracles" is not a contradiction, it might not be possibly necessary, and even though "God is unique and does not perform miracles" is not a contradiction, it might not be possibly necessary.

Therefore, one cannot claim that because some definition of God is consistent then it is possibly necessary.

Therefore, the claim that God is possibly necessary begs the question, and as such all ontological arguments fail.
• 2.5k
Pretty clear why it wasn't published.
• 1k
S5 is the logic of epidemics in which every possible world is infected by a virus whose transmission is symmetric and transitive.

As for Godel's argument, if we take the special case of his argument in which the positive properties P are taken to be the properties that are true for every possible individual, i.e by taking

$P(\psi) := \Box \forall x, \psi (x)$

and if we replace axiom A1 above with

$P(\psi) \equiv \neg N(\psi)$

where

$N(\psi) := \Diamond \exists x, \neg \psi (x)$

Then i expect that the resulting argument reduces to a trivial tautology of S5 in which all individuals are infected by the godliness virus.
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