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]

...the next step would be to determine if there are positive properties or only super-positives...

Agreed.

...Information...it is an inherent metaproperty of whatever framework universes exist in...

I agree with you here, and I think what we're really trying to do is to identify the metaproperties that inhere in the "framework" within which universes are conceivable. As such, I see no difference between a "superpositive," as we have been using this term, and something that is identified as an "inherent metaproperty" of the "framework." If you believe otherwise, please explain why.

...Only if we assume meaning does not exist independently of observers, which I submit it does...

Please tell me how "meaning" can exist independently of any real or potential observer.

...if an observer, actual or potential is required for information to exist, I then submit information is not a super-positive, only data is.

What is data? Does the meaning strictly inhere in the data, or is it at least partially a property of the observer?

I submit that "information" or "meaning" is realized only in the (potentially) fruitful exchange between a source and some actual or potential observer.

[Abraxas]

...the next step would be to determine if there are positive properties or only super-positives...

Agreed.

...Information...it is an inherent metaproperty of whatever framework universes exist in...

I agree with you here, and I think what we're really trying to do is to identify the metaproperties that inhere in the "framework" within which universes are conceivable. As such, I see no difference between a "superpositive," as we have been using this term, and something that is identified as an "inherent metaproperty" of the "framework." If you believe otherwise, please explain why.

Such a difference would be trivial and not important to the project, the only distinction I see is that superpositives, as we have been using them inherently exist within both the framework and any subject universe, where a metaproperty inherently exists as part of the framework, but is entirely optional inside any given universe, Information would qualify as a metaproperty as an empty universe holds no information, but from an external view information can be gathered about it.

...Only if we assume meaning does not exist independently of observers, which I submit it does...

Please tell me how "meaning" can exist independently of any real or potential observer.

A thought experiment. Suppose we have only two universes, this one and one that is, at the moment, completely empty. Suppose the only way to connect the two is a wormhole that naturally occurs once in the lifetime of a universe (impossible to induce artificially too) and, it just so happens to appear in a library under a copy of the Bible sucking it into this empty universe and sealing it off forever.

Now, the book is cut off from any actual observer, there exists no potential of any observer ever finding it again, does the book's meaning change? Does it suddenly stop having information? I would submit information is a property of the object and that something incidental, like the location of the object, does not change the information in it but can render it impossible for an observer to get that information.

...if an observer, actual or potential is required for information to exist, I then submit information is not a super-positive, only data is.

What is data? Does the meaning strictly inhere in the data, or is it at least partially a property of the observer?

I submit that "information" or "meaning" is realized only in the (potentially) fruitful exchange between a source and some actual or potential observer.

From your link earlier about GDI:

1.3 A definition of data

According to GDI, information cannot be dataless but, in the simplest case, it can consist of a single datum. Now a datum is reducible to just a lack of uniformity (diaphora is the Greek word for difference), so a general definition of a datum is:

The Diaphoric Definition of Data (DDD):
A datum is a putative fact regarding some difference or lack of uniformity within some context.


If you want to say information requires an observer, we can go that route, but the consequence I think is that it also requires data, thus making data the super positive and observers (thus information) being a contingent property if it just happens to be so.

[EduChris]

...Such a difference would be trivial and not important to the project, the only distinction I see is that superpositives, as we have been using them inherently exist within both the framework and any subject universe, where a metaproperty inherently exists as part of the framework, but is entirely optional inside any given universe...

If we say that a superpositive is a property which is entailed by all conceivable universes, then perhaps we will be free to ignore distinctions which are unimportant to our project.

...A thought experiment...the book is cut off from any actual observer, there exists no potential of any observer ever finding it again, does the book's meaning change? Does it suddenly stop having information?...

The question would be, "Did that book ever have any impact, actual or potential, on any observer?" If so, the book conveys information, regardless of its current unavailability to other potential observers.

...if an observer, actual or potential is required for information to exist, I then submit information is not a super-positive, only data is...

Not to belabor the point, but the "empty-set universe" has no data in it, but nevertheless it entails data by the mere fact that it is a unique universe within the context of all conceivable universes.

...A datum is a putative fact regarding some difference or lack of uniformity within some context...If you want to say information requires an observer, we can go that route, but the consequence I think is that it also requires data, thus making data the super positive and observers (thus information) being a contingent property if it just happens to be so.

It seems to me that information is entailed by all conceivable universes, and information entails data, context, and at least some potential observation.

[EduChris]

...It seems to me that information is entailed by all conceivable universes, and information entails data, context, and at least some potential observation.

It appears that no challenges are forthcoming, so either we are in agreement or else the challengers have lost interest. So I will sum up my position thus far and attempt to extend the debate.

The "super-positive" properties of Existence, Differentiation, Relationality, and Information are entailed by all conceivable universes.

In order to move the debate forward, we now need to consider if there are any "positive properties" (i.e., non-accidental properties) which are not super-positives themselves, but which nevertheless are at least compatible with the super-positives and which pass through the filter of Godel's Axioms 2 through 4.

It seems to me that the properties of Consciousness, Volition, Creativity, and Love are all entailed by our universe. They are compatible with the super-positives, and they pass the filter of Godel's Axioms 2 through 4. Thus they are "positive properties."

I am not currently aware of any other "positive properties," although I am willing to consider any proposals. I will also attempt to define the particular "positive properties" that I have put forward, if anyone wishes.

[Grumpy]

EduChris

It seems to me that the properties of Consciousness, Volition, Creativity, and Love are all entailed by our universe.

You cannot evidence such outrageous assumptions. You seem to think if you believe it, it must be so. But then, you think if you can imagine it, it exists as well. No one wants to argue with anyone who simply will not follow the rules as to be presenting evidence for their statements.

Godel and Anselm are both cloud castle constructors. My son and me used to lay in the grass pointing out clouds that looked like things(say, a rabbit), but neither of us claimed there are actually rabbits in the air

Grumpy

[EduChris]

...You cannot evidence such outrageous assumptions...

Rather than simply assert your beliefs, why don't you attempt to show that the positives I have adduced--Consciousness, Volition, Creativity, and Love--are incompatible with the superpositives of Existence, Differentiation, Relationality, and Information? Or failing that, why not attempt to demonstrate their failure to pass through the filter of Axioms 2 through 4?

[scourge99]

So I will sum up my position thus far and attempt to extend the debate.

The "super-positive" properties of Existence, Differentiation, Relationality, and Information are entailed by all conceivable universes.
...

Is the finer points of this "proof" something that you invented/"discovered" or is is professed by others of repute in the field of philosophy?

If this theory is so "bullet-proof" then why does debate continue?

Furthermore, there seems to be considerable complexity. Terms such as Differentiation, Relationality, Information , properties of Consciousness, Volition, Creativity, and Love are tossed about abundantly. One would think a god would make a "proof" of his existence far more simple and obvious than this.

To the "uneducated" it would appear that such a theory is an attempt to baffle others with bull-poop.

[sickles]

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?

For debate:
-Is the Ontological Theorem logically valid?

It seems to follow a logical pattern. Some might call this logic itself. I would call it a "symptom of logic". The premise that the pattern expresses is inane, however. It relies completely upon subjective experiences of the person completing the formula , and this is not logical as each individual person that completes the pattern will arrive at a different answer.

-Are all the axioms of the theorem valid?

no. Ax 1 relies upon subjective information. "Gdel defines a positive property rather vaguely: "Positive means positive in the moral aesthetic sense (independently of the accidental structure of the world). "

first positive in the moral aesthetic sense? isnt both morality and aestheticism subjective? How would you define something as independent of the structure of the world? Isnt the mind a structure of the world?

-Can the argument hold without the axioms being valid, if they are not necessarily so?

you havent translated the non axioms into english from the gobbledygook, so i cant say. To make this work, we require knowledge of absolute good and evil, which we do not posses.

[EduChris]

...Ax 1 relies upon subjective information. "Gdel defines a positive property rather vaguely: "Positive means positive in the moral aesthetic sense (independently of the accidental structure of the world). "

first positive in the moral aesthetic sense? isnt both morality and aestheticism subjective? How would you define something as independent of the structure of the world? Isnt the mind a structure of the world?...

The bulk of this thread has been devoted to pinning down "positive properties" in some meaningful and non-arbitrary way. I have done so, with Abraxas's and Aki's help. If you have any objections, you'll need to go back and read the thread (starting with this post) to see where we've gone off track (in your opinion).

[Grumpy]

scourge99

To the "uneducated" it would appear that such a theory is an attempt to baffle others with bull-poop.

To the educated as well. If you can't dazzle them with wit, bury them...well you know the rest.

Anselm and Godel make the same basic mistake, and it is in the very first assumptions they make. They assume that if you can conceive it, it is real. Then they go round and round about the most perfect being that can be conceived of, and the...well, the BS gets deeper and deeper from there. The point is not one of these axioms have the least bit to do with anything real. No wonder scientists have so little regard for philosophical sophistry. Pretending, as we have seen in this thread, that putting it in symbolic notation makes it smell any better is it's own form of lunacy.

I call it building cloud castles. Even if you succeed you still can't move into it BECAUSE IT DOESN'T REALLY EXIST. Of, course, it is all they have to work with, having absolutely no valid evidence that any supernatural entity or event is real, thus the cloud castle construction.

Grumpy