Godel's Ontological Theorem.

[LiamOS]

This thread is both for discussion of Godel's Ontological Theorem and a continuation of a debate which was in another thread.

Godel's Ontological Argument is expressed symbolically as:

For those unfamiliar with modal-logic, there is an article on the general Ontological Argument here.


With respect to the theorem's axioms, WikiPedia tells us the following:

We first assume the following axiom:

Axiom 1: It is possible to single out positive properties from among all properties. Gdel defines a positive property rather vaguely: "Positive means positive in the moral aesthetic sense (independently of the accidental structure of the world)... It may also mean pure attribution as opposed to privation (or containing privation)." (Gdel 1995)

We then assume that the following three conditions hold for all positive properties (which can be summarized by saying "the positive properties form a principal ultrafilter"):

Axiom 2: If P is positive and P entails Q, then Q is positive.
Axiom 3: If P₁, P₂, P₃, ..., P_n are positive properties, then the property (P₁ AND P₂ AND P₃ ... AND P_n) is positive as well.
Axiom 4: If P is a property, then either P or its negation is positive, but not both.

Finally, we assume:

Axiom 5: Necessary existence is a positive property (Pos(NE)). This mirrors the key assumption in Anselm's argument.

Now we define a new property G: if x is an object in some possible world, then G(x) is true if and only if P(x) is true in that same world for all positive properties P. G is called the "God-like" property. An object x that has the God-like property is called God.

For debate:
-Is the Ontological Theorem logically valid?
-Are all the axioms of the theorem valid?
-Can the argument hold without the axioms being valid, if they are not necessarily so?

[EduChris]

...You do realize I was agreeing with you, right?...

I realized you were agreeing with this:

If our universe is the only conceivable universe, then existence, relationality, and differentiation hold for all conceivable universes--and hence, existence, relationality, and differentiation are Godelian "positive properties."

But I did not (do not) believe that you were (or are yet) agreeing to this:

On the other hand, if there are conceivable universes other than our own, then those other universes would all need the conditions of existence; differentiation from our universe (and from any other conceivable universes); and relationality in terms of being conceivable. So again, existence, differentiation, and relationality must be Godelian "positive properties," since they hold for all conceivable universes.

Anyway,

I took away that you were assuming this universe being the only possible universe for the thought experiment. Please correct me if this is not so...

My only assumption is that "something exists." Everything else follows logically. Either our universe is the only conceivable universe, or it is not. I handled both cases, and in each case it follows that existence, differentiation, and relationality are necessary conditions for all conceivable universes--which is to say, they are not accidental properties, but instead are Godelian positive properties.

...Granted, and comparing what is possible is a much stronger argument for you. The problem is I see no reason it couldn't have been possible for it to be impossible for any but an empty universe to exist. Barring some kind of underlying cosmic unity or structure, it seems what is possible and what is not in reality now seems to have an element of contingency. Suppose, hypothetically, something changes irreversibly in the superstructure of reality, making this universe, and any but a single, completely empty universe possible. Given that nothing else is in existence or ever can be, would things like differentiation still be applicable?

I believe what you are saying is this:

It might have been the case that an empty universe had been the only conceivable universe. In such a case, differentiation and relationality would not necessarily have turned out to be Godelian positive properties.

I agree that such a situation might have been the case--and if that were in fact the case then your logic would hold. However, your "might have been" is contrary to fact. "What actually is the case" is a much stronger argument than "what might have been the case, but isn't." Given our current actuality, we cannot unconceive that which we have already shown to be conceivable. That being the case, all three conditions--existence, differentiation, and relationality--must be Godelian positive properties for all conceivable universes.

If you agree to this, then I believe the case is settled. Godel's Ontological Proof is valid. More than that, it is compatible with the Triune God of the Christian faith--a God that might be reasonably (if minimalistically) described as "a non-contingent, positive-property 'condition set' involving existence, differentiation, and relationality."

[Zeeby]

All three conditions--existence, differentiation, and relationality--must be Godelian positive properties for all conceivable universes.

I agree that existence is a positive property (although in this situation perhaps you should specify whether you mean necessary or contigent existence), but am confused by what you mean by "differentiation" or "relationality" being properties. No object has the property "differentiation" - if you are referring to a property that could be described as "differentiationability" then I would be interested to hear your definition, as a statement such as "X is differentiationable if it can be differentiated from things that are different to X" is circular (use of 'different').

Also notice Axiom 2 says "either a property is positive, or its negation is". This is giving the God object rather a lot of properties - as the God object has all positive properties and any property it has is positive. For example, in the Christian belief system it would be necessary that "created horses in this universe" is a positive property but that "created unicorns in this universe" is not a positive property. Similarly "is believed to not exist by some people" would be a positive property.

If you agree to this, then I believe the case is settled. Godel's Ontological Proof is valid. More than that, it is compatible with the Triune God of the Christian faith--a God that might be reasonably (if minimalistically) described as "a non-contingent, positive-property 'condition set' involving existence, differentiation, and relationality."

I think the proof is valid logically - after all, Godel was a logician. Saying that the proof supports the notion of the Christian God is, I feel, unjustified (which is why Godel did not consider it a 'proof of God' - merely a formal structuring of the ontological argument). Indeed, if you have only shown that existence is a positive property (attainable by an object) then you have shown only that something necessarily exists. Perhaps this is what you would describe as God, perhaps it is the structure of the universe.

Do you have a way to determine which of P or P is the positive property? Can you justify each of the axioms in the proof? Until such are presented, the argument has the same weaknesses and counterarguments as the original ontological argument from which it was derived.

[EduChris]

...I agree that existence is a positive property (although in this situation perhaps you should specify whether you mean necessary or contigent existence)...

For more background, you should go back and read the entire thread (or at least start on page seven). But I will attempt to summarize our findings here as well.

The "Godelian positive properties" I have put forward--Existence, Differentiation, and Relationality--constitute a "condition set" which must necessarily apply if anything at all exists. Without this minimal set of initial conditions, nothing could conceivably exist. But since we are assuming that something exists, then the set of initial conditions must necessarily apply.

...but am confused by what you mean by "differentiation" or "relationality" being properties. No object has the property "differentiation" - if you are referring to a property that could be described as "differentiationability" then I would be interested to hear your definition, as a statement such as "X is differentiationable if it can be differentiated from things that are different to X" is circular (use of 'different')...

You have admitted that Existence is a Godelian positive property. What is Existence except that which allows something to exist? In the same way, what is Differentiation except that which allows one thing to be differentiated from any and all other things? And what is Relationality except that which allows multiple things to relate to one another in some fashion?

Existence is a positive property; its negation is oblivion, non-existence, nothingness, and so on. Differentiation is a positive property; its negation is uniformity, sameness, indistinctness, unidentifiability, and so on. Relationality is a positive property; its negation is isolation or acontextuality.

...Also notice Axiom 2 says "either a property is positive, or its negation is". This is giving the God object rather a lot of properties - as the God object has all positive properties and any property it has is positive. For example, in the Christian belief system it would be necessary that "created horses in this universe" is a positive property but that "created unicorns in this universe" is not a positive property. Similarly "is believed to not exist by some people" would be a positive property...

We are concerned only about "Godelian positive properties." An "accidental property"--e.g., "horse"-ness vs. "unicorn"-ness--cannot be a "Godelian positive property" since we can conceive of universes with or without horses or unicorns. "Accidental properties" pertain only to some conceivable universe(s) but not to all conceivable universe(s). "Godelian positive properties," on the other hand, pertain or apply to all conceivable universe(s).

...I think the proof is valid logically - after all, Godel was a logician...

Yes, I believe that you and I and AkiThePirate and Abraxas are all in agreement here.

...Saying that the proof supports the notion of the Christian God is, I feel, unjustified (which is why Godel did not consider it a 'proof of God' - merely a formal structuring of the ontological argument)...

The properties of Existence, Differentiation, and Relationality are integral to the Christian concept of "Trinity." Whether or not Godel saw the similarity between our three "positive properties" and the Christian "Triune God" is beside the point.

...Indeed, if you have only shown that existence is a positive property (attainable by an object) then you have shown only that something necessarily exists. Perhaps this is what you would describe as God, perhaps it is the structure of the universe...

Not merely Existence, but also Differentiation and Relationality have been shown to be Godelian positive properties. These three properties pertain to all conceivable universes, and not just to our own universe.

...Do you have a way to determine which of P or P is the positive property? Can you justify each of the axioms in the proof? Until such are presented, the argument has the same weaknesses and counterarguments as the original ontological argument from which it was derived.

We are assuming that something exists. Everyone agrees that Existence is a positive property. Given that something exists, our universe instantly becomes a conceivable universe, since our universe could be that "something" which we have assumed to exist.

If our universe were the only conceivable universe, then all properties entailed by our universe (and Existence, Differentiation and Relationality are only three of a multitude of properties entailed by our universe) would end up being positive properties, since a positive property applies to all conceivable universe(s).

But in fact our universe is probably not the only conceivable universe, so we must see if there are any other properties of our universe, other than Existence itself, which must necessarily apply to all other conceivable universes. So, if there are other conceivable universes, these other universes must be differentiable from our universe, or else they would not be conceptually different from our own universe. Thus, the property of Differentiation applies to all conceivable universe(s). Similarly, the property of Relationality also applies to every conceivable universe, since all of these conceivable universes are related by virtue of their conceivability.

And since we can conceive of an empty universe--or at least some minimally interesting near-empty universe--it seems that Existence, Differentiation, and Relationality are the only positive properties that can pertain to all conceivable universes. Coincidentally or not, these three are the hallmarks of Christian trinitarian tradition.

[Zeeby]

...I agree that existence is a positive property (although in this situation perhaps you should specify whether you mean necessary or contigent existence)...

For more background, you should go back and read the entire thread (or at least start on page seven). But I will attempt to summarize our findings here as well.

The "Godelian positive properties" I have put forward--Existence, Differentiation, and Relationality--constitute a "condition set" which must necessarily apply if anything at all exists. Without this minimal set of initial conditions, nothing could conceivably exist. But since we are assuming that something exists, then the set of initial conditions must necessarily apply.

Perhaps we should assume that more than one thing exists to give differentiation and relationality between objects. In any case, I am happy to assume quite a lot about the structure of the universe.

...but am confused by what you mean by "differentiation" or "relationality" being properties. No object has the property "differentiation" - if you are referring to a property that could be described as "differentiationability" then I would be interested to hear your definition, as a statement such as "X is differentiationable if it can be differentiated from things that are different to X" is circular (use of 'different')...

You have admitted that Existence is a Godelian positive property. What is Existence except that which allows something to exist? In the same way, what is Differentiation except that which allows one thing to be differentiated from any and all other things? And what is Relationality except that which allows multiple things to relate to one another in some fashion?

My definition for existence is something like 'An object x can be said to exist if there is a subset of the universe (for some universe) labelled by x'. Thus 'exists' is a property. I consider properties as conditions which describe an object, of which 'exists' is one, and 'differentiation' or 'relationality' is not. Do you have a definition for the properties you are referring to?

Existence is a positive property; its negation is oblivion, non-existence, nothingness, and so on. Differentiation is a positive property; its negation is uniformity, sameness, indistinctness, unidentifiability, and so on. Relationality is a positive property; its negation is isolation or acontextuality.

'An object x has the property "sameness" if...'
'An object x has the property "isolation" if...'

These are the types of statements that I am asking you to complete.

...Also notice Axiom 2 says "either a property is positive, or its negation is". This is giving the God object rather a lot of properties - as the God object has all positive properties and any property it has is positive. For example, in the Christian belief system it would be necessary that "created horses in this universe" is a positive property but that "created unicorns in this universe" is not a positive property. Similarly "is believed to not exist by some people" would be a positive property...

We are concerned only about "Godelian positive properties." An "accidental property"--e.g., "horse"-ness vs. "unicorn"-ness--cannot be a "Godelian positive property" since we can conceive of universes with or without horses or unicorns. "Accidental properties" pertain only to some conceivable universe(s) but not to all conceivable universe(s). "Godelian positive properties," on the other hand, pertain or apply to all conceivable universe(s).

That is why I phrased the property to refer to this universe. "Created horses in this universe" is a property - accidental or not - and so from Axiom 2 either that or its negative is positive. Do you have a different interpretation of Axiom 2?

...Saying that the proof supports the notion of the Christian God is, I feel, unjustified (which is why Godel did not consider it a 'proof of God' - merely a formal structuring of the ontological argument)...

The properties of Existence, Differentiation, and Relationality are integral to the Christian concept of "Trinity." Whether or not Godel saw the similarity between our three "positive properties" and the Christian "Triune God" is beside the point.

That there are three "properties" you have identified does not make it similar to the idea of the Trinity. Saying they are integral to the concept is obvious because as you mentioned, they are integral to anything conceivable.

...Indeed, if you have only shown that existence is a positive property (attainable by an object) then you have shown only that something necessarily exists. Perhaps this is what you would describe as God, perhaps it is the structure of the universe...

Not merely Existence, but also Differentiation and Relationality have been shown to be Godelian positive properties. These three properties pertain to all conceivable universes, and not just to our own universe.

Still waiting on those definitions of differentiation and relationality.

...Do you have a way to determine which of P or P is the positive property? Can you justify each of the axioms in the proof? Until such are presented, the argument has the same weaknesses and counterarguments as the original ontological argument from which it was derived.

We are assuming that something exists. Everyone agrees that Existence is a positive property. Given that something exists, our universe instantly becomes a conceivable universe, since our universe could be that "something" which we have assumed to exist.

If our universe were the only conceivable universe, then all properties entailed by our universe (and Existence, Differentiation and Relationality are only three of a multitude of properties entailed by our universe) would end up being positive properties, since a positive property applies to all conceivable universe(s).

But in fact our universe is probably not the only conceivable universe, so we must see if there are any other properties of our universe, other than Existence itself, which must necessarily apply to all other conceivable universes. So, if there are other conceivable universes, these other universes must be differentiable from our universe, or else they would not be conceptually different from our own universe. Thus, the property of Differentiation applies to all conceivable universe(s). Similarly, the property of Relationality also applies to every conceivable universe, since all of these conceivable universes are related by virtue of their conceivability.

And since we can conceive of an empty universe--or at least some minimally interesting near-empty universe--it seems that Existence, Differentiation, and Relationality are the only positive properties that can pertain to all conceivable universes. Coincidentally or not, these three are the hallmarks of Christian trinitarian tradition.

I have been aware for some time that you think Existence, Differentiation, and Relationality are positive properties. My problems with this are outlined above. The questions I asked in the quoted passage were not referring these issues, rather the problems Axiom 2 generates in choosing positive properties, and that it is necessary to explain why the axioms are valid.

[EduChris]

...Perhaps we should assume that more than one thing exists...I am happy to assume quite a lot about the structure of the universe...

The goal for "assumptions" is that they should be kept as minimal as possible. The single assumption, "Something exists," is more minimal than "Two or more things exist."

...to give differentiation and relationality between objects...

There is no need to assume differentiation and relationality. The mere fact that we assume "something exists" is sufficient to allow us to conceive that our universe might be that "something" which exists. Our universe might be that "something," or not, but unless we can conclusively rule out the existence of our universe, our universe remains conceivable. Therefore, since our universe entails existence, relationality, and differentiation, we do not need to assume those properties at the outset.

At this point in our argument, we know that Existence is a positive property, but we have not yet determined whether relationality and differentiation are positive properties or accidental properties. But as we proceed further from here we will see that Existence, Differentiation, and Relationality are indeed the only positive properties in the Godelian sense--that is, Existence, Differentiation, and Relationality are entailed by all conceivable universes, not merely some conceivable universes.

...My definition for existence is something like 'An object x can be said to exist if there is a subset of the universe (for some universe) labelled by x'. Thus 'exists' is a property. I consider properties as conditions which describe an object, of which 'exists' is one, and 'differentiation' or 'relationality' is not. Do you have a definition for the properties you are referring to?...

In addition to the definitions I already gave, how about this: For any proposed set of two "things," apply the labels A and B to those proposed "things." Item A entails the property of differentiation if it is possible to affirm that A is not B. Similarly, Item A entails the property of relationality if it is possible to affirm that A belongs in some conceptual category in which B also belongs.

Now we do not at the outset need to assume that A and B actually exist. It is sufficient to note that our universe might be that "something" which we have assumed to exist, and in our universe we have an abundance of "things" which exhibit the properties of differentiation and relationality.

...That is why I phrased the property to refer to this universe. "Created horses in this universe" is a property - accidental or not - and so from Axiom 2 either that or its negative is positive. Do you have a different interpretation of Axiom 2?...

Yes I do. You are making the same mistake that Abraxas had been making initially. You are assuming that Godel is claiming that all properties conform to his axioms. But in fact, that is not what Godel is claiming at all. Godel is using his axioms to define the ways in which "positive properties" differ from other properties which are "accidental" rather than "positive." We will all have to become very clear about this, or we will never be able to understand what Godel is doing with his argument.

...That there are three "properties" you have identified does not make it similar to the idea of the Trinity. Saying they are integral to the concept is obvious because as you mentioned, they are integral to anything conceivable...

Existence, Differentiation, and Relationality are indeed integral to anything conceivable--that is why they are Godelian positive properties, and more than that, I believe they are the only conceivable Godelian positive properties. I simply note that the set of Godelian positive properties bears all the hallmarks of traditional Christian trinitarian thought, and that no other religion can make the same claim for their conception of God.

Now the reason why Existence, Differentiation, and Relationality are the only Godelian properties stems from the following facts:

1) Something exists (this is our only assumption)

2) Since we cannot logically rule out the existence of our universe, it follows that our universe could be that "something" which exists; therefore, our universe is conceivable

3) As we conceive it, our universe entails the properties of Existence, Differentiation, and Relationality (in addition to numerous other properties, which we shall see are accidental rather than positive properties)

4) If our universe is the only conceivable universe, then all of the properties entailed by our universe would be positive properties; however, we can conceive of at least one other universe: an empty universe (or at least some minimally interesting near-empty universe)

5) This empty universe constitutes that universe which entails the fewest possible positive properties

6) Since in Godelian terms, positive properties must be shared in common by all conceivable universes, it follows that the empty universe entails the maximum upper limit for the number of Godelian positive properties.

7) The only properties entailed by the empty universe are: a) it exists; b) it is conceptually different from our universe; and c) it is related to our universe by the fact that our universe and the empty universe both belong to the set of conceivable universes.

8) The empty universe entails the fewest possible number of positive properties: Existence, Differentiation, and Relationality

9) These three properties are similarly shared by all other conceivable universes

10) Therefore, these three properties are the only Godelian positive properties.

[Zeeby]

...That is why I phrased the property to refer to this universe. "Created horses in this universe" is a property - accidental or not - and so from Axiom 2 either that or its negative is positive. Do you have a different interpretation of Axiom 2?...

Yes I do. You are making the same mistake that Abraxas had been making initially. You are assuming that Godel is claiming that all properties conform to his axioms. But in fact, that is not what Godel is claiming at all. Godel is using his axioms to define the ways in which "positive properties" differ from other properties which are "accidental" rather than "positive." We will all have to become very clear about this, or we will never be able to understand what Godel is doing with his argument.

Could you state which properties phi it is valid to substitute in P(phi) <-> P(phi)? I think that will clarify this issue.

[EduChris]

...Could you state which properties phi it is valid to substitute in P(phi) <-> P(phi)? I think that will clarify this issue.

If it is conceivable in some universe that P(phi), and that in some other universe P(phi), then the labeling of (P vs. P) or (phi vs. phi) is arbitrary or accidental.

In other words, if the choice in labeling does not apply in all conceivable universes, it is ruled out as a "positive property" in the Godelian sense.

[omniscience darkness]

Hi

...That there are three "properties" you have identified does not make it similar to the idea of the Trinity. Saying they are integral to the concept is obvious because as you mentioned, they are integral to anything conceivable...

Existence, Differentiation, and Relationality are indeed integral to anything conceivable--that is why they are Godelian positive properties, and more than that, I believe they are the only conceivable Godelian positive properties. I simply note that the set of Godelian positive properties bears all the hallmarks of traditional Christian trinitarian thought, and that no other religion can make the same claim for their conception of God.

hi

with all the very classic and logical notions behind trinity i am just wonder why (see the verses below) they are not in trinitarian formula in the first place i'm sure they exist in the universe before your Godel was born so....

Mark(48-55) 1:8
Lk(57-62) 3:16
Mt(65-70) 3:11
Jhn(90-100) 1:33
Acts(62-63) 1:5


The above Baptism verses are the example of trinity fraud ;none of the early gospels had ever baptized by the trinity formula ; the holy spirit ,father and the son.

the verses above in general are of John baptized you with .......& then jesus will come to baptized you with ........(over here should be in trinitarian formula but its not) I hope you notice they lack the properties of trinity

base on the "Godel" they should be written in the trinitarian formula but why they were not ?

best regards
enjoy reading

[Zeeby]

...Could you state which properties phi it is valid to substitute in P(phi) <-> P(phi)? I think that will clarify this issue.

If it is conceivable in some universe that P(phi), and that in some other universe P(phi), then the labeling of (P vs. P) or (phi vs. phi) is arbitrary or accidental.

In other words, if the choice in labeling does not apply in all conceivable universes, it is ruled out as a "positive property" in the Godelian sense.

That is not exactly answering my question. We have a statement involving a free variable phi, and to use the statement it is necessary to know what is valid to substitute into it. From the quote in the article "Axiom 4: If P is a property, then either P or its negation is positive, but not both.", I interpret that Axiom 2 is valid for substitution of any property, but that does not appear to be what you interpret.

[EduChris]

...From the quote in the article "Axiom 4: If P is a property, then either P or its negation is positive, but not both.", I interpret that Axiom 2 is valid for substitution of any property, but that does not appear to be what you interpret.

If you look at the sentence that immediately precedes Axioms 2 through 4, you will see that "the following three conditions hold for all positive properties." It is not claimed that the three conditions (Axiom 2, Axiom 3, and Axiom 4) hold for any property; rather, the axioms serve to filter out accidental or arbitrary properties.