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 cannot have two without evenness, two entails it...

What is "evenness" except the property of being "divisible by two without remainder"? Saying that we can't have "2" without "divisibility" by "2" is rather circular, don't you think? But anyway, how is any of this relevant? We're not just talking about "any properties at all," but rather about "Godelian properties," which have certain and specific requirements according to the axioms of his argument.

...the only thing that seems to qualify is existence itself, which is ruled out ala Kant...

It seems to me that Godel, who was very much aware of Kant's objections, formulated his own argument so as to preclude Kant's objection.

...Everything else, as far as we can determine, is an accident of the universe...

Not only is this speculative, it is also circular--you are assuming that which needs to be proven. It isn't at all clear to me how there could be any universe at all without the properties that I have put forward for numbers--i.e., Relationality, Distinctiveness, Logic, Order, Significance, etc.

...I would also request you defend the idea that "existence-affirming" properties, as Godel is using it, are required for anything to exist, as opposed to the more common use of the word being sufficient to fit the bill...

Shouldn't we discuss Godel's argument as it is, rather than how we think it ought to be? Sure, the fact that the existence of numbers entails relationality is probably equivalent to saying that the existence of numbers would be sufficient to demonstrate the existence of relationality, but I'm not sure what we will have gained by using one formulation over the other. Why not stick with Godel's formulation?

...usefulness is a relationship that may or may not exist. For something to be useful, something must be able to use it. This is not a property of an object, but rather a descriptor of how other incidental things can relate to the object...

It seems to me that "usefulness" (or "utility") and "relationality" must exist then, since it is possible to observe instances of such.

...The difference lies in your equivocation, numbers have all kinds of observed properties, they simply don't conform to "positive properties" or, as you keep using "existence-affirming properties" as you define them...

If numbers can't be observed as inherently and necessarily bearing certain properties, then in what sense can you say that numbers exist? If "X" can't be observed, and if "X" entails no necessary or inherent properties, in what way does "X" exist?

...If we cannot define 2 as "exactly that number which lies between 1 and 3," then how do we define two in a matter that distinguishes it from 1 or 3?

1.7 is 2 under your definition...

Are you saying that 1.7 lies exactly between 1 and 3?

...This conversation can only end in a debate on what constitutes identity which is rather off topic, but two is the number that has the identity two. Two is the name of a specific object. You are treating it like a concept to be defined rather than an object to be described, which, in my view, is a category mistake...

It seems to me that the very nature of numbers entails relationality. Two is not merely a specific "object," as you claim. Two is the identity of the number which lies exactly between the numbers 1 and 3 in the set of all numbers. That is a more adequate definition than simply the tautological statement that 2 is 2.

...They are forms of categorical thinking, yes, but, being systematic rather than instantiated, they are not properties or defining characteristics of individual items but rather descriptors of relationships between items as we perceive them...

So "forms of categorial thinking" and "relationships" and "perceptions" all exist?

...and, they seem to be accidental rather than necessary at that. They would not qualify as positive properties (or really, properties at all) as the term is being used...

Again, this is speculation and circular reasoning.

...Two is that object which has the identity two. Four is that which has the identity four. There are a number of properties in the traditional sense that apply, but not in the "existence affirming" sense. Otherwise it just boils down to the identity of the object...

You are simply repeating yourself. We've already addressed this (see above).

...On this topic, I consider Kant to have the stronger argument...

Okay, personal opinion noted. But you haven't shown why, and in fact we haven't really come to a definition of "positive property" yet, though I think we're at least close to putting forward a tentative proposal.

I'm hoping to get a bit more input from others here... Anyone else care to chime in?

[Abraxas]

...the only thing that seems to qualify is existence itself, which is ruled out ala Kant...

It seems to me that Godel, who was very much aware of Kant's objections, formulated his own argument so as to preclude Kant's objection.

Tried to, certainly, but I think he failed to do so. From where I sit, there are no positive properties as the definition is narrow to the degree it causes any potential property to be removed. It further runs into problems with relying on an assumed, unprovable moral aesthetic that may or may not exist. The only thing it leaves, then, is existence, which falls right back into Kant.

...Everything else, as far as we can determine, is an accident of the universe...

Not only is this speculative, it is also circular--you are assuming that which needs to be proven. It isn't at all clear to me how there could be any universe at all without the properties that I have put forward for numbers--i.e., Relationality, Distinctiveness, Logic, Order, Significance, etc.

No, it is the default position. If you are going to claim something is necessary for universes, the onus is on you to show it is. As for your examples, relationality is unnecessary for a universe that is empty, same with distinctiveness. Logic and order would not necessarily apply to a completely random universe, as one might get from multiple intersecting time dimensions. Significance is just something exists, again, which isn't a property per Kant.

...I would also request you defend the idea that "existence-affirming" properties, as Godel is using it, are required for anything to exist, as opposed to the more common use of the word being sufficient to fit the bill...

Shouldn't we discuss Godel's argument as it is, rather than how we think it ought to be? Sure, the fact that the existence of numbers entails relationality is probably equivalent to saying that the existence of numbers would be sufficient to demonstrate the existence of relationality, but I'm not sure what we will have gained by using one formulation over the other. Why not stick with Godel's formulation?

Numbers yes, a number, no. However, that was not my point. My point was that you keep insisting these "existence affirming" properties, are necessary properties from the existence of things and that they must be or else we could not know the item in question. I am asking you to justify your insistence that necessary properties, "existence affirming properties", "positive properties", are required to identify and define as opposed to mere accidental properties being sufficient. If accidental properties are sufficient to define objects, then it casts strong doubt as to whether there can be "positive properties", as Godel and yourself define them to begin with.

...usefulness is a relationship that may or may not exist. For something to be useful, something must be able to use it. This is not a property of an object, but rather a descriptor of how other incidental things can relate to the object...

It seems to me that "usefulness" (or "utility") and "relationality" must exist then, since it is possible to observe instances of such.

Must exist or do exist? It is not sufficient that they do exist, in order for them to be a positive property things must not be able to exist without them. Otherwise, once again, it is an accident of the universe and thus not a property as it is being used.

...The difference lies in your equivocation, numbers have all kinds of observed properties, they simply don't conform to "positive properties" or, as you keep using "existence-affirming properties" as you define them...

If numbers can't be observed as inherently and necessarily bearing certain properties, then in what sense can you say that numbers exist? If "X" can't be observed, and if "X" entails no necessary or inherent properties, in what way does "X" exist?

Through incidental, accidental properties instead, which gets back to what I said earlier. That I happen to have ten fingers, a characteristic of me, does not mean I necessarily have that property. This is where you are getting caught up, you keep wanting to declaring features necessary, thus properties under Godel's definition, without providing cause they must be necessary.

...If we cannot define 2 as "exactly that number which lies between 1 and 3," then how do we define two in a matter that distinguishes it from 1 or 3?

1.7 is 2 under your definition...

Are you saying that 1.7 lies exactly between 1 and 3?

Yes. An infinite number of numbers lie between 1 and 3. 1.7 is one of them.

...This conversation can only end in a debate on what constitutes identity which is rather off topic, but two is the number that has the identity two. Two is the name of a specific object. You are treating it like a concept to be defined rather than an object to be described, which, in my view, is a category mistake...

It seems to me that the very nature of numbers entails relationality. Two is not merely a specific "object," as you claim. Two is the identity of the number which lies exactly between the numbers 1 and 3 in the set of all numbers. That is a more adequate definition than simply the tautological statement that 2 is 2.

It happens it happens to be between 1 and 3, however, if numbers are only defined in relation to one another they are inherently valueless. If two, as an object, is to have any meaning at all it must have meaning (identity) independent of other objects.

...They are forms of categorical thinking, yes, but, being systematic rather than instantiated, they are not properties or defining characteristics of individual items but rather descriptors of relationships between items as we perceive them...

So "forms of categorial thinking" and "relationships" and "perceptions" all exist?

Yes, thinking things create them. In universes without thinking things they do not, except relationships, though those only exist in systems with more than one item. None of those things are necessary or "positive" under the definition though.

...and, they seem to be accidental rather than necessary at that. They would not qualify as positive properties (or really, properties at all) as the term is being used...

Again, this is speculation and circular reasoning.

Not at all. I can consider possible universes that lack those entirely. It isn't circular reasoning in the slightest. What might be, though, is how you have changed the definition of property to include only those things that fit your argument.

...Two is that object which has the identity two. Four is that which has the identity four. There are a number of properties in the traditional sense that apply, but not in the "existence affirming" sense. Otherwise it just boils down to the identity of the object...

You are simply repeating yourself. We've already addressed this (see above).

Yes, I repeated myself in response to you repeating yourself. What would be silly is if I replied differently each time.

...On this topic, I consider Kant to have the stronger argument...

Okay, personal opinion noted. But you haven't shown why, and in fact we haven't really come to a definition of "positive property" yet, though I think we're at least close to putting forward a tentative proposal.

I'm hoping to get a bit more input from others here... Anyone else care to chime in?

Why? Because how properties are applied logically follows the form "There exists an X that Y", if Y is "exists", the form becomes "There exists an X that exists", a circular argument. We cannot prove something exists by asserting it exists up front, as such, X exists is built into the subject. There is no cause to make it a property.

[EduChris]

...From where I sit, there are no positive properties as the definition is narrow to the degree it causes any potential property to be removed...[the completely accidental nature of the universe] is the default position...relationality is unnecessary for a universe that is empty, same with distinctiveness. Logic and order would not necessarily apply to a completely random universe, as one might get from multiple intersecting time dimensions. Significance is just something exists, again, which isn't a property per Kant...I am asking you to justify your insistence that necessary properties, "existence affirming properties", "positive properties", are required to identify and define as opposed to mere accidental properties being sufficient. If accidental properties are sufficient to define objects, then it casts strong doubt as to whether there can be "positive properties", as Godel and yourself define them to begin with...This is where you are getting caught up, you keep wanting to declaring features necessary, thus properties under Godel's definition, without providing cause they must be necessary...We cannot prove something exists by asserting it exists up front, as such, X exists is built into the subject. There is no cause to make it a property.

When all is said and done, we can either say that nothing at all exists, or that something--let's call it X--exists. If nothing at all exists, then we are done. We don't exist, our universe doesn't exist, no universe exists, etc. This is not a fruitful option for discussion.

The remaining option is that something, X, does exist in some universe U. The question then becomes, what must necessarily hold in order for X to exist. Whatever that set of necessary "properties" or "conditions" may be--and let's refer to it as N--they might be accidental or not. Let's refer to accidental properties as A, and non-accidental properties as P.

Thus we have the following: given X, either (N,A) or (N,P). That is, the set of conditions N that must hold in order for X might be P ("positive" per Godel's terms) or A ("accidental" as you have proposed).

Now for any given X, the set of N will be A if there is some conceivable universe U for which the N of X does not hold. Conversely, the set of N for that X will be P if for every conceivable universe U, the set of N holds.

Now it is clear that our universe is either the only possible universe, or else it is not the only possible universe. If there are no other possible universes, then whatever N must hold for any X in our U is, by definition, P.

On the other hand, if other universes do exist or could conceivably exist, then the following must apply:

1) Other universes either do exist or conceivably could exist
2) These other universes can be differentiated from our universe (and from each other, if there is more than one other conceivable universe)
3) These other universes must have some relation to our universe, if for no other reason than they are conceivable.

In other words, for all conceivable universes, existence, relationality, and differentiation must apply. Therefore, existence, relationality, and differentiation are necessarily the case for all conceivable universes. They are thus positive properties per Godel's terminlogy.

Actually, Existence, Relationality, and Differentiation sound a lot like the Christian concept of Trinity.

At any rate, Godel's Ontological Proof is sound.

[EduChris]

Oops. Delete duplicate.

[Abraxas]

In other words, for all conceivable universes, existence, relationality, and differentiation must apply. Therefore, existence, relationality, and differentiation are necessarily the case for all conceivable universes. They are thus positive properties per Godel's terminlogy.

Actually, Existence, Relationality, and Differentiation sound a lot like the Christian concept of Trinity.

At any rate, Godel's Ontological Proof is sound.

The problem is here you assume the necessity of our own universe. If it were possible our universe would not exist, and possible that an empty universe would, relationality and differentiation no longer apply, leaving only existence existing necessarily, which I granted already. That relationality and differentiation exist could conceivably be an accident of our universe existing, not necessarily a necessary condition for existence as a whole.

[EduChris]

...The problem is here you assume the necessity of our own universe...

No, I am assuming merely that something exists, since the alternative is to assume that nothing exists (and that would be an unproductive assumption, at least for purposes of discussion).

...If it were possible our universe would not exist, and possible that an empty universe would, relationality and differentiation no longer apply, leaving only existence existing necessarily, which I granted already. That relationality and differentiation exist could conceivably be an accident of our universe existing, not necessarily a necessary condition for existence as a whole.

Given that something exists, our universe would seem to be at least as conceivable as any other conceivable universe. Our universe entails the conditions of existence, relationality, and differentiation. 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."

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.

Godel's Ontological Proof is valid; not only that, it accords very nicely with the Christian concept of Trinity.

[Abraxas]

...The problem is here you assume the necessity of our own universe...

No, I am assuming merely that something exists, since the alternative is to assume that nothing exists (and that would be an unproductive assumption, at least for purposes of discussion).

...If it were possible our universe would not exist, and possible that an empty universe would, relationality and differentiation no longer apply, leaving only existence existing necessarily, which I granted already. That relationality and differentiation exist could conceivably be an accident of our universe existing, not necessarily a necessary condition for existence as a whole.

Given that something exists, our universe would seem to be at least as conceivable as any other conceivable universe. Our universe entails the conditions of existence, relationality, and differentiation. 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."

If we take your assumption that this is the only possible universe, yes.

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.

Godel's Ontological Proof is valid; not only that, it accords very nicely with the Christian concept of Trinity.

What you are failing to recognize is that if our universe is contingent, then the other universe, conceived of as empty for this thought experiment, only needs relationality and differentiation accidentally, not necessarily. Suppose, hypothetically there are only two possible universes, ours and one completely empty, both universes existing contingently. If our universe were to cease to exist, or never had existed to begin with, relationality and differentiation no longer have any meaning to an empty universe in isolation. Thus, to me, relationality and differentiation are contingent features resulting from a reality that just so happens to have things that can be related and differentiated, but it is logically possible that there could be or could have been a state where this is not so. In other words, in order to make the claim relationality and differentiation are necessary, you have to make the claim at least two things exist necessarily because both require more than one thing.

I still think we have only made it as far as existence exists.

[Cathar1950]

I 'pologize if I'm spamming or wrongly promoting my own, but...

The proof can summarized as:

IF it is possible for a rational omniscient being to exist THEN necessarily a rational omniscient being exists.

Is Wikipedia's summary accurate? If no, please help me out.

If the summary is accurate, then the question remains...

Is the possibility that something may exist sufficient to declare that that something does exist?

I say no.

That's not what the argument says. It's not talking about just any old "something".

The way I read it is: if it is logically possible for a being to possess all positive properties, then that being exists necessarily since necessary existence is also a positive property. Meaning that it is not possible that the being does not exist.

I would read it as : if a being that has all perfections would also have existence as an attribute as something that exists or is real is more perfect then something that doesn't, so for God to be perfect God has to exist or God isn't perfect.
A perfect God has to exist while a non-existing God could not be perfect.
If God doesn't exist then it is irrelevant.
Only if God exists is it necessary for God to exist, and if God doesn't exist then God doesn't need to exist.
If God doesn't exist then God would need to be perfect and wouldn't be depending on how you define God.
You could move the goal or change the definition as EduChris does when he calls God the ground of all being instead of the greatest or most perfect.

[EduChris]

...If we take your assumption that this is the only possible universe, yes...

I do NOT assume that this is in fact the only possible universe. Here is my argument point-by-point:

1. I assume only that something exists. This assumption establishes the condition of existence, which is required in order to conceive that anything might actually exist.

2. Given the condition of existence, it follows that our universe is conceivable.

3. Other universes may also be conceivable.

4. That a universe is conceivable says nothing about whether it actually exists, or whether it exists contingently or non-contingently.

5. If we can identify certain conditions which must apply for all conceivable universes, then those conditions are Godelian "positive properties."

6. Conditions that apply only for some conceivable universes are "accidental properties."

7. Either our universe is the only conceivable universe, or else other universes besides ours are also conceivable.

8. If our universe is the only conceivable universe (and I do not claim that it is) then whatever "condition set" that must apply in order for some X to exist in our universe must be a Godelian positive property, since this "condition set" is entailed for X in all conceivable universes (remember that we are here considering what would be the case if our universe were the only conceivable universe; also remember that our universe does entail differentiation and relationality).

9. On the other hand, if there are conceivable universes other than our own (as I believe there are) then these other universes automatically entail the conditions of existence, relationality, and differentiation (because these other universes are different from our universe, and because they are all conceivable).

10. Thus, whether our universe is the only conceivable universe, or whether other universes are also conceivable, the conditions of existence, differentiation, and relationality apply for all conceivable universes; these conditions are therefore Godelian positive properties.

...If our universe were to cease to exist, or never had existed to begin with, relationality and differentiation no longer have any meaning to an empty universe in isolation. Thus, to me, relationality and differentiation are contingent features resulting from a reality that just so happens to have things that can be related and differentiated, but it is logically possible that there could be or could have been a state where this is not so. In other words, in order to make the claim relationality and differentiation are necessary, you have to make the claim at least two things exist necessarily because both require more than one thing...

Our universe must be a conceivable universe, since we are in fact conceiving it. The cat is out of the bag, the train has left the station. We cannot now claim that our own universe is inconceivable. Our universe belongs in the set of conceivable universes--whether or not it actually exists, and whether or not its existence were contingent or non-contingent. And I would say that your "empty universe" belongs in that set as well. Thus we have at least two conceivable universes in our set of all conceivable universes. And of course I could add more to the set--for example, a universe like ours in every respect, except without platypuses.

[Abraxas]

...If we take your assumption that this is the only possible universe, yes...

I do NOT assume that this is in fact the only possible universe. Here is my argument point-by-point:

1. I assume only that something exists. This assumption establishes the condition of existence, which is required in order to conceive that anything might actually exist.

2. Given the condition of existence, it follows that our universe is conceivable.

3. Other universes may also be conceivable.

4. That a universe is conceivable says nothing about whether it actually exists, or whether it exists contingently or non-contingently.

5. If we can identify certain conditions which must apply for all conceivable universes, then those conditions are Godelian "positive properties."

6. Conditions that apply only for some conceivable universes are "accidental properties."

7. Either our universe is the only conceivable universe, or else other universes besides ours are also conceivable.

8. If our universe is the only conceivable universe (and I do not claim that it is) then whatever "condition set" that must apply in order for some X to exist in our universe must be a Godelian positive property, since this "condition set" is entailed for X in all conceivable universes (remember that we are here considering what would be the case if our universe were the only conceivable universe; also remember that our universe does entail differentiation and relationality).

9. On the other hand, if there are conceivable universes other than our own (as I believe there are) then these other universes automatically entail the conditions of existence, relationality, and differentiation (because these other universes are different from our universe, and because they are all conceivable).

10. Thus, whether our universe is the only conceivable universe, or whether other universes are also conceivable, the conditions of existence, differentiation, and relationality apply for all conceivable universes; these conditions are therefore Godelian positive properties.

You do realize I was agreeing with you, right?

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."

To which my reply was:

If we take your assumption that this is the only possible universe, yes.

Now, given the underlined portion of that paragraph, 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.

...If our universe were to cease to exist, or never had existed to begin with, relationality and differentiation no longer have any meaning to an empty universe in isolation. Thus, to me, relationality and differentiation are contingent features resulting from a reality that just so happens to have things that can be related and differentiated, but it is logically possible that there could be or could have been a state where this is not so. In other words, in order to make the claim relationality and differentiation are necessary, you have to make the claim at least two things exist necessarily because both require more than one thing...

Our universe must be a conceivable universe, since we are in fact conceiving it. The cat is out of the bag, the train has left the station. We cannot now claim that our own universe is inconceivable. Our universe belongs in the set of conceivable universes--whether or not it actually exists, and whether or not its existence were contingent or non-contingent. And I would say that your "empty universe" belongs in that set as well. Thus we have at least two conceivable universes in our set of all conceivable universes. And of course I could add more to the set--for example, a universe like ours in every respect, except without platypuses.

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?