Gödel's ontological proof

[Aetixintro]

Gödel's ontological proof can be questioned, however I've contributed with a version that makes it stand out as splendid and at the same time being accepted without a question outside the logical soundness objections to a "necessary God".

Here is:

UoD: Everything.

Gx: x is God-like
Ex: x has essential properties.
Ax: x is an essence of A.
Bx: x is a property of B.
Px: property x is positive.
Nx: x is a General property.
Xx: x is Positive existence.
Cx: x is consistent.

The final argument by my interpretation is presented below in 4 parts:

1.

1 │ □Ex ≡ □Px ≡ □Gx A (A is Assumption)
2 │ □Ex A
3 │ ◊Px ≡ □Px A
4 │ ◊Px A
------------------
5 │ □Px ≡ □Gx 1, 2 ≡E
6 │ □Px 3, 4 ≡E
------------------
7 │ □Gx 5, 6 ≡E

...

4.

1 │ □Bx ≡ □Gx A (A is Assumption)
2 │ ◊Ax ≡ □Bx ≡ (◊Ax ⊃ □Bx) A
3 │ ◊Ax A
------------------
4 │ □Bx 3, 2 ≡E
------------------
5 │ □Gx 4, 1 ≡E

Note for the 4th part: Consider (◊Ax ⊃ □Bx) as “added explanation�!
Also, line 2 of the 4th part is Definition 2 from the original argument of Gödel.
Note2: The following lines are taken out for having no use in this interpretation of the argument.
8 │ □Gx ⊃ □Px A
16│ □Gx ⊃ □Cx A
17│ □Gx ⊃ □Ax A.

From:
Definition 1: x is God-like if and only if x has as essential properties those and only those properties which are positive
Definition 2: A is an essence of x if and only if for every property B, x has B necessarily if and only if A entails B
Definition 3: x necessarily exists if and only if every essence of x is necessarily exemplified
Axiom 1: Any property entailed by—i.e., strictly implied by—a positive property is positive
Axiom 2: If a property is positive, then its negation is not positive.
Axiom 3: The property of being God-like is positive
Axiom 4: If a property is positive, then it is necessarily positive
Axiom 5: Necessary existence is positive
Axiom 6: For any property P, if P is positive, then being necessarily P is positive.
Theorem 1: If a property is positive, then it is consistent, i.e., possibly exemplified.
Corollary 1: The property of being God-like is consistent.
Theorem 2: If something is God-like, then the property of being God-like is an essence of that thing.
Theorem 3: Necessarily, the property of being God-like is exemplified.

http://en.wikipedia.org/wiki/G%C3%B6del ... ical_proof
http://en.wikipedia.org/wiki/Universe_of_Discourse - UoD from above.
http://en.wikipedia.org/wiki/Logical_consequence - "Entailment"/"entails".

Some of the text is from http://whatiswritten777.blogspot.no/201 ... on-of.html.

[Nilloc James]

Nobody is attacked here, so please, relax. ... Gödel's argument here, in its logical beauty, is the DESCRIPTION, that he adds to the description for "necessary-God" in the tradition of the other "Ontological Arguments", see fx. St. Anselm etc., please.

Finally,
To Nilloc James. These are not my premises, they are Gödel's premises. I'm merely restating his deduction for better logical defences so that the criticism of his Argument can STOP!

Alright, all? Take care!

You are not being clear and concise here, as a matter of fact, you are being the opposite.

And, yes, that is exactly what Godel is doing, fancied up in modal logic, saying 'God implies god'.

No. This is wrong, directly. Accounting 1 - 4 plus 2 alt. to 1st part, 'God implies god', '□Gx → □Gx', = Zero results! Correct?

Cheers!

Im too stupid for formal logic. Can you give a gestalt in plain english?

[keithprosser3]

It's not worth the effort. If it really proved God's existence we'd have heard before now.

[Divine Insight]

It's not worth the effort. If it really proved God's existence we'd have heard before now.

Truly.

Even so it would only be proof of "God" based on how it defines God by its axiomatic assumptions. I don't even necessarily agree with all of the axioms used.

Finally, even if it did prove these conditions there are still two very important things to realize.

1. Is there really any point in calling this situation "God"?

I didn't see any axioms about sentience, consciousness, or any other traits that would suggest that this situation actually implies any sort of living deity.

So that's highly questionable anyway.

And secondly:

2. Even if this definition of "God" were shown to be true wouldn't that just suggest that some potential spirituality may exist?

In other words, it wouldn't point to any particular religion or "God". It certainly wouldn't be proof of Zeus. And for the same reasons, it also wouldn't be proof of the Hebrew God either.

In fact, this kind of "abstract proof" loans itself far more to supporting the abstract notions of the "God" of Eastern Mysticism which many people argue should be spelled with a lower case "g" as "god" because it doesn't necessarily imply that god has an ego like the Biblical God necessarily must have.

So even if these purely logical reasoning leads to the idea that some type of "god" must exist, it would actually place something like Taoism at the top of the list of most likely spiritual philosophies. And something like the Biblical picture would be way down in the mud with things like Greek Mythology, Thor, Odin, and all the other mythological egotistical Gods.

[nayrbsnilloc]

[Replying to post 13 by Divine Insight]

The biggest gripe I have with any ontological argument, especially one presented formally like Gödel's to seem like pure logic, is that it uses a subjective value objectively. It uses "good", or in this case its synonym "positive", like it is a well-defined, objective fact, which is far from the case.

Specifically when presented the way it is in the OP, Axiom 3 and 5 are entirely subjective claims that shouldn't fit into the logical format.

An example to illustrate the problem, replace every instance of "good" or "positive" with some other adjective of your choosing.

You can say that the God property of being god-like is bad (or for the sake of absurdity blue) and the statement would hold as much truth. The "logic" will still work out.

There are other problems with ontological arguments, but this is by far the most glaring to me.

[EduChris]

...The biggest gripe I have with any ontological argument, especially one presented formally like Gödel's to seem like pure logic, is that it uses a subjective value objectively. It uses "good", or in this case its synonym "positive", like it is a well-defined, objective fact, which is far from the case...

Your objection is easily overcome by using "necessary" properties as a foundation for Godelian "positive" properties. "Necessary" properties are those which are inherent in any and all possible universes. Since such properties cannot be lacking in any possible universe, it becomes pointless to describe them as "arbitrary" or "subjective."

In my previous posts, I referred to "necessary" properties as "superpositive" properties. Godelian "positive" properties (which may or may not be "necessary") can be determined by whether they are compatible with the superpositives, whether they pass Godel's filter (his axioms 2 through 4) and whether they are evident in at least one possible universe.

Existence, Differentiation, and Relationality are superpositives, since they are inherent in all possible universe. Consciousness, Volition, Creativity, and Love are Godelian positive properties, since 1) they are compatible with the superpositives, 2) they pass through Godel's filter, and 3) they are observed in at least one possible universe (viz, our own universe).

Thus, Godel's theorem tells us that God can be described in terms of Existence, Differentiation, Relationality, Consciousness, Volition, Creativity, and Love. And this sounds very much like the Christian God.

[Divine Insight]

[Replying to post 13 by Divine Insight]

The biggest gripe I have with any ontological argument, especially one presented formally like Gödel's to seem like pure logic, is that it uses a subjective value objectively. It uses "good", or in this case its synonym "positive", like it is a well-defined, objective fact, which is far from the case.

Specifically when presented the way it is in the OP, Axiom 3 and 5 are entirely subjective claims that shouldn't fit into the logical format.

An example to illustrate the problem, replace every instance of "good" or "positive" with some other adjective of your choosing.

You can say that the God property of being god-like is bad (or for the sake of absurdity blue) and the statement would hold as much truth. The "logic" will still work out.

There are other problems with ontological arguments, but this is by far the most glaring to me.

I totally agree. The logic can only be as strong as it's axioms, and in this case the axioms are arbitrary and clearly nothing more than subjective opinions.

Moreover, the very introduction of the term "God" is absurd. We could have just as easily used "Dog" and we would have proven that a "Dog" must exist.

I don't accept some of the other axioms as well. For example axiom 2 states:

Axiom 2: If a property is positive, then its negation is not positive.

That appears on the surface to be a tautology, but it may not be depending on precisely how this property of "positive" is defined. It may very well be that the negation of positive could simply be the absence of positive. Therefore it's negation would not be "Not Positive" but instead it would simply be "The Absences of a Positive Trait".

This is important in logic because here we can have a property that can have three states; Positive, Negative, Non-existent. But we're applying this to a binary system of logic where we only have two states; TRUE, or FALSE.

So these types of details need to be watched very carefully.

One problem with using symbolic logic is that this short-hand method of notation can easily be used to hide and obscure the finer details.

So in addition to a this list of axioms and definitions I would need to also see detailed definitions for the terms being used (such as the term "Positive")

Definition 1: x is God-like if and only if x has as essential properties those and only those properties which are positive
(need to rigorously define the intended meaning of the term Positive here)
Definition 2: A is an essence of x if and only if for every property B, x has B necessarily if and only if A entails B
Definition 3: x necessarily exists if and only if every essence of x is necessarily exemplified
Axiom 1: Any property entailed by—i.e., strictly implied by—a positive property is positive
(need to rigorously define the intended meaning of the term Positive here)
Axiom 2: If a property is positive, then its negation is not positive.
(need to rigorously define the intended meaning of the term Positive here)
Axiom 3: The property of being God-like is positive
(need to rigorously define the intended meaning of the term Positive here)
Axiom 4: If a property is positive, then it is necessarily positive
(need to rigorously define the intended meaning of the term Positive here)
Axiom 5: Necessary existence is positive
(need to rigorously define the intended meaning of the term Positive here)
Axiom 6: For any property P, if P is positive, then being necessarily P is positive.
(need to rigorously define the intended meaning of the term Positive here)
Theorem 1: If a property is positive, then it is consistent, i.e., possibly exemplified.
(need to rigorously define the intended meaning of the term Positive here)
Corollary 1: The property of being God-like is consistent.
Theorem 2: If something is God-like, then the property of being God-like is an essence of that thing.
Theorem 3: Necessarily, the property of being God-like is exemplified.

Just look at all the red ink here! And it's all over the same obscure definition of the term "Positive".

I wouldn't even bother to move on toward any logical arguments until a very rigorous definition of what "positive" means in this context is given in a way that can satisfactorily justify all these axioms that rely upon it.

To move forward without this rigorous definition in hand would be foolish.

Just look at how much of this argument depends upon this single concept.

This single concept must be defined in a way that is rigorous enough to assure the truth of every one of these axioms.

That would need to be done in detail first before it's even worth moving forward to any further arguments.

Using symbolic logic is not an excuse for a lack of rigorous definitions of the terms being used within axioms.

So these fundamental axioms must be squared away first. And clearly most of them are highly dependent upon the meaning of this concept of "Positive".

So that's what I would focus on first. Tell me precisely what "positive" means, in a way that I can justify all of these axioms first.

Thanks.

[Furrowed Brow]

Don't think we need to ask for further definitions to see how the argument works. It is assumed that if necessarily x is essential that this is logically equivalent to necessarily x being positive which in turn is logically equivalent to necessarily x being God like. So as soon as it is assumed that necessarily x is positive and it is also assumed possibly x is positive is the same as saying necessarily x is positive, it is quite legitimate to conclude necessarily x is Godlike. But look at all the assumptions!!!!!

Here is the same argument reinterpreted and this interpretation is equally valid:

Gx: x is God-like
Ex: x is extravagant.
Px: property x is pretentious.

1 │ □Ex ≡ □Px ≡ □Gx A (A is Assumption)
2 │ □Ex A
3 │ ◊Px ≡ □Px A
4 │ ◊Px A
------------------
5 │ □Px ≡ □Gx 1, 2 ≡E
6 │ □Px 3, 4 ≡E
------------------
7 │ □Gx 5, 6 ≡E

The question is how meaningful is this argument form and how much light does it really throw on necessary properties and God? And none I'd say if just about any interpretation can be made to fit the logical notation.

I think it is worth noting how "essential" is used as a predicate when it is also a synonym for "necessary". It is a tad slippery introducing one meaning of necessity as a modal operator and another as a predicate. I take this as a signal that something is being obscured here. What that is, I'm not sure....it is obscure.

If we look at the dictionary meaning of the word "positive" and just about any reasonable interpretation of what that might mean it is clear this word is not equivalent to "essential". It is vaguely plausible though far from air tight to argue some thing essential must be positive (carbon is essential to life but pumping too much into the atmosphere may not be a positive), it is much less assured that everything positive must be essential. I'm sure it is a positive thing if I lost 14lbs and reduce my waist by 2 inches, but this is not essential....well not yet anyway. If you can think up any counterfactual in which a positive property is not essential or an essential thing is at times not positive then the assumption □Ex ≡ □Px is not axiomatic. But of course there is a third term to consider along with the modal operator.

The assumption □Ex ≡ □Px ≡ □Gx tells us □Ex mean the same thing as □Px which means the same as □Gx. which is to say that what is necessarily essential is necessarily positive is necessarily God-like. But what is that thing? The problem with a word like "essential" is that it forces us to ask "essential for what? ?". This is not very well defined. It is obvious to think in terms of essential for life or essential for existence. But as a couple of comments I have just made show it is possible to invoke things like oxygen, water, carbon which are essential for life but these things are not always positive as they have a down side. So essential for what exactly? And positive in what context? That these words are relative shows that the underlying semantics on which the argument relies is a bit of a muddle; one I think that is impossible to unpick unless there are universal examples of a thing that is always essential in every context and always positive in every possible context. I'd be interest to see what that thing might be. Given the amount of effort that does into the debate in these forums we can't even reach agreement on the notion that the idea of God is a positive thing or essential. In which case DC&R is evidence that the assumptions on which this argument relies are a best highly debatable at worst false.