Gödel's logical proof of God

[Jesus_Is_God_1]

Mathematician, logician, and philosopher, Kurt Gödel, provided his ontological proof of God using modal logic. His proof is provided below.

View attachment 23681

(Sorry if the quality is not that great.)

Any thoughts, comments, or discussions?

 

[JapaneseKappa]

I mentioned in another thread that some researchers used computer logic systems to verify this proof. They found and corrected a few issues with Godel’s original formulation. They did end up successfully verifying the proof. However, their computer logic systems also quickly found a corollary: there are no contingent truths. That is, all true statements are true necessarily, which has troubling implications for free will.

 

[Jesus_Is_God_1]

I mentioned in another thread that some researchers used computer logic systems to verify this proof. They found and corrected a few issues with Godel’s original formulation. They did end up successfully verifying the proof. However, their computer logic systems also quickly found a corollary: there are no contingent truths. That is, all true statements are true necessarily, which has troubling implications for free will.

Can you link me to their verification of the proof?

 

[JapaneseKappa]

github.com/FormalTheology/GoedelGod

 

[Jesus_Is_God_1]

github.com/FormalTheology/GoedelGod

Thanks.

 

[dmar198]

I mentioned in another thread that some researchers used computer logic systems to verify this proof. They found and corrected a few issues with Godel’s original formulation. They did end up successfully verifying the proof. However, their computer logic systems also quickly found a corollary: there are no contingent truths. That is, all true statements are true necessarily, which has troubling implications for free will.

A conditional statement is one like this: “If I drink a lot of soda, I’ll need to use the restroom.” When they “found” that there are no conditional statements, does that mean the sentence I just mentioned isn’t real? It’s not actually there in front of you? I’m confused. I’d like more data.

 

[Charlemagne_III]

decodedscience.org/modal-logic-proved-godel-right-god-exists/38801

 

[JapaneseKappa]

A conditional statement is one like this: “If I drink a lot of soda, I’ll need to use the restroom.” When they “found” that there are no conditional statements, does that mean the sentence I just mentioned isn’t real? It’s not actually there in front of you? I’m confused. I’d like more data.

What they found was that every true statement is true necessarily. That is, denying a true statement entails a logical contradiction: like denying that cats are mammals. Cats are by definition mammals, so denying that cats are mammals is a contradiction; it is not logically possible for cats to be not-mammals. Cats have claws, but claw-having is not something that is in the definition. That’s why we say that cat’s claw-having is contingent: it is logically possible for cats to not have claws, even if cats don’t actually have claws.

In the same way, we would expect that it is logically possible for your statement to be false, even if it was true. That is, we would think that proposing “If I drink a lot of soda, I won’t need to use the restroom” might end up being false, but we shouldn’t be able to know that just by looking at the definitions of drinking soda, using restrooms, and I. Their proof said we are wrong. If your statement is true, denial of your statement is equivalent to arguing that a circle is square.

 

[dmar198]

What they found was that every true statement is true necessarily. That is, denying a true statement entails a logical contradiction: like denying that cats are mammals. Cats are by definition mammals, so denying that cats are mammals is a contradiction; it is not logically possible for cats to be not-mammals. Cats have claws, but claw-having is not something that is in the definition. That’s why we say that cat’s claw-having is contingent: it is logically possible for cats to not have claws, even if cats don’t actually have claws.

In the same way, we would expect that it is logically possible for your statement to be false, even if it was true. That is, we would think that proposing “If I drink a lot of soda, I won’t need to use the restroom” might end up being false, but we shouldn’t be able to know that just by looking at the definitions of drinking soda, using restrooms, and I. Their proof said we are wrong. If your statement is true, denial of your statement is equivalent to arguing that a circle is square.

Thank you for these additional details. Is there a particular link from which you figured out how to summarize this point so well? (Something tells me you are able to summarize this information so well because you have a broad range of experience in logic rather than familiarity with a particular text involving Godel’s theorem, but there’s no harm in asking.) I’d like to collect some resources on this so that someday I’d be able to look more into these issues (for example, can free will exist if all true statements are true necessarily?).

I am reminded of a book by a priest where he was answering a series of objections to the Catholic doctrine of free will. A bunch of the objections came from an atheist. Here’s part of the exchange:

“How would you answer the assertion that what has to be, has to be?”

By granting it. If a thing has to be, it has to be. But I deny that everything has to be.

“If a man riding a bicycle is run down by a car and killed accidentally, did that have to be?”

If such a thing happens, then it happens. And it could not “happen” yet “not happen” simultaneously. At the same time, granted that it does happen, it does not happen by any absolute necessity. The man need not have chosen to go for a ride on that particular day. The car driver was under no compulsion to be on the road at the time. So we cannot say that the accident was foreordained by God and inevitable. There was no determining force outside and independent of those two men compelling them to come into collision, with fatal consequences for the rider of the bicycle.

“Is the hour of one’s death appointed by God at the moment of one’s birth?”

No; for a man is physically free even where his own life is concerned. Keep in mind this principle. Although God’s providence extends to all things from a universal or general point of view, that same providence has willed that within the universe there should be a vast series of secondary causes affecting each other immediately. Now amongst these secondary causes is man, a free agent. The moment of a suicide’s death is obviously not determined by God, for God forbids suicide; and if the suicide had obeyed God’s law he would have lived longer. The same thing is true of murder, where a man’s life is terminated by the rebellion of another man’s will against the will of God.

source

 

[Perplexity]

It does not seem to me that Gödel’s argument is one for theism. What Gödel calls “God” seems to be a unity of properties, and thus a creature. Moreover, it is said to exist in or at possible worlds, not to precede or cause them. Finally, existing in or at every possible world may very well indicate that one’s existence is necessary, but it does not indicate why one’s existence is necessary: it could just be because one is caused to exist necessarily. So, proving something exists in or at every possible world is not sufficient to prove that it is divine.

However, suppose that Gödel’s argument is not for a unity of anything, but rather for a unit. At least we have an argument for theism now, right? Not necessarily. The problem now lies in conceiving of this argument as being one for “God.” What’s wrong with that? Well, use of the English term “God” without the indefinite article carries monotheist connotations with it, and there is no such thing as monotheism: theism just is polytheism.

See, monotheism claims that only one God exists. Now, whatever else “God” refers to here, it is either a unity or a unit. If it is a unity, then it is a creature and monotheism is actually non-theistic. If it is a unit, then it has no parts of any kind, and is therefore utterly peculiar and individual; having nothing in common with anything. Saying that a unit exists is to say that a specific, individual deity exists. Monotheism would end up saying “there is only one [YHWH, Poseidon or Odin, etc.]”, which claim is trivially compatible with polytheism.

So, whether monotheism affirms a unity or a unit, it is either non-theistic or trivially compatible with polytheism. In either case, it is not – as it is commonly understood to be – an alternative to or contradiction of polytheism.

With this background in place, we may return to Gödel’s argument and say that if it is indeed an argument for theism, it is an argument for a God.

 

[tafan]

I mentioned in another thread that some researchers used computer logic systems to verify this proof. They found and corrected a few issues with Godel’s original formulation. They did end up successfully verifying the proof. However, their computer logic systems also quickly found a corollary: there are no contingent truths. That is, all true statements are true necessarily, which has troubling implications for free will.

This is a misrepresentation of what they computer scientists did. Gödel’s proof does not need verified by a computer program, it can be well understood (both its strength and weaknesses) by any logician who has studied it. What they did was try to write a computer program that could perform proofs traditionally done by mathematicians . They simply picked Gödel’s ontological proof as one to test their program. It was more an accomplishment of AI and says nothing either way about Gödel’s proof.

Your confusion is understandable because this has been heavily misrepresented in the media accounts.

 

[tafan]

Mathematician, logician, and philosopher, Kurt Gödel, provided his ontological proof of God using modal logic. His proof is provided below.

View attachment 23681

(Sorry if the quality is not that great.)

Any thoughts, comments, or discussions?

I think any version of Anslem’s ontological proof is very limited in usefulness. I quote Fr Ronal Knox’s “Belief of Catholics” as my reasoning:

This attempt to prove the existence of God, or to declare the proof of it unnecessary, without reference to the effects of his power which we experience in his visible creation, is a permanent temptation to the human mind. Intellects as far removed from one another as those of Anselm, Descartes, and de Bonald have umdertaken it, and it is probable that they will never lack successors. Protestant thought, in our day, is much wedded not to these but to similar speculations. Thus, you will seldom read any piece of non-Catholic apologetic without coming across some reference to man’s sense of his need for God, or man’s notion of holiness, a notion which can only be perfectly realised in God. The implication of all such language is that it is possible to argue directly from the existence of concepts in our own mind to the existence of real objects, to which those concepts correspond.
The Catholic Church discountenances all such methods of approach to the subject; some of them, at the Vatican Council, she has actually condemned. She discountenances them, at least, if and in so far as they claim to be the sole or the main argument for the existence of God. The main if not the sole, argument for the existence of God–so she holds, and has always held- -is the argument which proves the Unseen from the seen, the existence of the Creator from his visible effects in Creation.

…

If the existence of God was one of the first principles of all our mental process, then the contrary idea, that there is no God, should be unthinkable-- but is it unthinkable? People think it every day. “But at least,” St. Anselm would retort, “it is impossible to think of an imperfect God, and therefore it is impossible to think of a non-existent God.” To which the atheist replies with some justice that, since God does not exist, it is not necessary to think about him at all. You cannot argue from the ideal to the real order of things.

 

[Bradski]

I think any version of Anslem’s ontological proof is very limited in usefulness. I quote Fr Ronal Knox’s “Belief of Catholics” as my reasoning:

I take you are saying that Anselm could have said: ‘It’s impossible to think of an imperfect God’. Which has me scratching my head.

That argument only works if you define God, in the first instance, as being perfect in every way. Which, when it is pointed out that there is no logical necessity for God to be, for example, omniscient, results in the trite response: ‘Well, that’s not how we define God’.

A definition is a description of what something is, not what you decide it must be. You are defining God by comparing Him to the definition you have already decided He must have. It really is the smallest radius circular argument in existence.

Notwithstanding that the biblical God, the original God of the Jews, was far from being perfect. Obtuse, capricious, angry, surprised, frustrated…

It was only Anselm, amongst others, and much later, who thought that it might be a difficult gig getting people to worship someone who had the same faults and character defects as everyone else.

 

[tafan]

I take you are saying that Anselm could have said: ‘It’s impossible to think of an imperfect God’. Which has me scratching my head.

That argument only works if you define God, in the first instance, as being perfect in every way. Which, when it is pointed out that there is no logical necessity for God to be, for example, omniscient, results in the trite response: ‘Well, that’s not how we define God’.

A definition is a description of what something is, not what you decide it must be. You are defining God by comparing Him to the definition you have already decided He must have. It really is the smallest radius circular argument in existence.

Notwithstanding that the biblical God, the original God of the Jews, was far from being perfect. Obtuse, capricious, angry, surprised, frustrated…

It was only Anselm, amongst others, and much later, who thought that it might be a difficult gig getting people to worship someone who had the same faults and character defects as everyone else.

No, I am not saying that at all. I am saying that any of these so called ontological arguments/proofs of God’s existence, ie proofs based on reason that does not consider the world around us, are of limited usefulness.

We can through our reason know that God exists with certainty, but only by considering His creation. That is the difference between Aquinas and Anselm. I find Aquinas’s arguments for God convincing, I do not find Anslem’s.

 

[JapaneseKappa]

This is a misrepresentation of what they computer scientists did. Gödel’s proof does not need verified by a computer program, it can be well understood (both its strength and weaknesses) by any logician who has studied it. What they did was try to write a computer program that could perform proofs traditionally done by mathematicians . They simply picked Gödel’s ontological proof as one to test their program. It was more an accomplishment of AI and says nothing either way about Gödel’s proof.

Your confusion is understandable because this has been heavily misrepresented in the media accounts.

Of course no proof “needs” to be verified. But the fact that they found problems with Godel’s formulation by checking the logic using a computer proved that computer verification was a worthwhile thing to do.