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. Gödel 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)." (Gödel 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?

[LiamOS]

...Ax 1 relies upon subjective information. "Gödel 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).

You've made a somewhat compelling case for their existence being necessary(Philosophically), and a commendable effort in attempting to show that they are positive properties, but thus far I cannot hold them to be 'positive' in the manner that the argument requires.
Your labelling of 'positive' seems to arbitrary and inconsistent with the necessary property of 'good', etc. which is required for the argument's conclusion.

[EduChris]

...No wonder scientists have so little regard for philosophical sophistry...

What is your evidence for this assertion? Is this just a personal opinion? Obviously no one has much regard for "sophistry," but there is no reason to refer to Godel--a world class logician, philosopher, and mathematician--as a sophist.

Regarding everything else you wrote, there have been three atheists, two of who are among the most intelligent on this forum, in addition to one theist (myself) who have looked at the mathematics and found the actual proof sound. The problem has been not the proof itself, but rather the axioms--particularly Axiom 1 and the seeming arbitrariness it initially suggests.

My move was to show that Existence, Differentiation, Relationality, and Information are entailed by all conceivable universes--that is what is meant by the term "superpositive." This eliminates the "arbitrary" element from the Axiom, and makes it possible to define those properties which may not be "superpositive," but which are at least "positive" (i.e., non-arbitrary) on the basis of their congruence with the superpositives and on the basis of their passing through the filter of Axioms 2 through 4. I have identified these "positive" properties as Consciousness, Volition, Creativity, and Love. I probably could also have added Logic (or Reason) as a separate property, though I believe it is inherent in the property of Information.

At this point in the thread, the question is, are there any other "positive" properties in addition to those I've mentioned? Or, are there any objections to the ones which I have proposed? Can you show any other properties which are consistent with the superpositives and which pass through the filter of Axioms 2 through 4?

[EduChris]

...Your labelling of 'positive' seems to arbitrary and inconsistent with the necessary property of 'good', etc. which is required for the argument's conclusion.

Godel coined the "positive" label, though in my mind "non-arbitrary" would have been a better label, a label that does a better job at capturing what he's trying to do.

What are you talking about with your "necessary property of 'good'"? And are you forgetting about "pure attribution as opposed to privation"?

Can you show that any of my "positive" (i.e., non-arbitrary) properties violate the superpositives, or that they fail to pass through the filter of Axioms 2 through 4? It's pretty easy to show how arbitrary properties (e.g., units of measurement, as Abraxas started out with) fail the axiomatic filter, but I don't see how my properties can be shown to fail that filter.

[LiamOS]

Essentially what I'm saying is that although these properties may necessary for existence, there appears to be no reason to deem them to be 'positive'.

Even Godel's definition of a positive property appears to entail subjective insight: "Positive means positive in the moral aesthetic sense [...] It may also mean pure attribution as opposed to privation."

That your chosen properties do not violate the axioms is not proof that there exists a supreme being, as what supremacy entails has been defined in an arbitrary, subjective way. At least, that would be my opinion on the matter.

Also, I don't actually have the time to follow up on this(Yet), but I was told that Immanuel Kant even disputed the idea that 'Existence' was a property.

[EduChris]

...there appears to be no reason to deem them to be 'positive'...

You are far too hung up on a particular slant on "positive." In Godel's usage, the term should be "non-arbitrary."

...Even Godel's definition of a positive property appears to entail subjective insight...

That objection is handled by my move to superpositives, which are entailed for all conceivable universes.

...that your chosen properties do not violate the axioms is not proof that there exists a supreme being, as what supremacy entails has been defined in an arbitrary, subjective way...

I agree that my properties do not violate the axioms, but the "arbitrariness" and "subjectivity" is addressed by my move to the superpositives in conjunction with the axiomatic filter. If something is entailed by all conceivable universes, it is not arbitrary at all.

...I was told that Immanuel Kant even disputed the idea that 'Existence' was a property.

Kant was objecting to earlier ontological formulations, and Godel's Proof avoids Kant's objection.

[Grumpy]

EduChris

My move was to show that Existence, Differentiation, Relationality, and Information are entailed by all conceivable universes--that is what is meant by the term "superpositive." This eliminates the "arbitrary" element from the Axiom

No, it does not. You problem is you think reality CARES what you think it should be. When you assign properties to "all conceivable universes" it is an act of hubris, not definition. That is my point. Hubris does not dictate reality, and just like what we can conceive(the basis for both Godel and Ansem)has nothing to do with what is real(rather, doesn't NECESSARILY have anything to do with reality), your sidestep has no effect on the axioms of the theorum. You still have not dealt with the arbitrary nature of the value judgements(good)in them. Everything that is logical(IE "those properties which may not be "superpositive," but which are at least "positive" (i.e., non-arbitrary) on the basis of their congruence with the superpositives and on the basis of their passing through the filter of Axioms 2 through 4. ")is not necessarily real.

I have identified these "positive" properties as Consciousness, Volition, Creativity, and Love. I probably could also have added Logic (or Reason) as a separate property, though I believe it is inherent in the property of Information.

And here you have completely left the reservation. Building a cloud castle out of nothing but YOUR assignment of "superpositives" that are not supported by the Axioms in the first place. This is you projecting your need or desire for "Consciousness, Volition, Creativity, and Love", evidently in the form of a god, on reality. It just does not compute, so you have shown nothing of the kind.

So, to reiterate, this statement by St.Ansem...

"God, by definition(who gave it that definition? G), is that for which no greater(an undefined and undefinable concept G)can be conceived(imagined). God exists in the understanding(how does one understand imaginary things? G). If God exists in the understanding, we could imagine Him to be greater by existing in reality. Therefore, God must exist(er...no G)."

Or, to shorten and simplify, "What we imagine must exist in reality"

...is nonsense. So is any, more convoluted argument(Godels' or yours)based upon it. A proof in math and logic is based on a set of stated axioms, if the axiom has nothing to do with reality, then neither does the proof. To insist it does is called "sophistry" and that is what this whole thread has been. God will never be shown by logic if that logic is unsupported by reality, and what we can imagine may or may not have anything to do with reality. I do not have to go through the headache of disproving Ansem, Godel or you if your house is made out of clouds which appear to have substance but in reality are nothing but mist, blown away by the least bit of reality.

Grumpy

[EduChris]

...YOUR assignment of "superpositives" that are not supported by the Axioms in the first place...

Nonsense. You haven't even read the previous posts where this is discussed.

...to reiterate, this statement by St.Ansem...

This is not the place to discuss Anselm's argument. We are discussing Godels Ontological Proof, and you are simply commenting off the cuff without even having read the previous discussions.

[Grumpy]

EduChris

...YOUR assignment of "superpositives" that are not supported by the Axioms in the first place...

Nonsense. You haven't even read the previous posts where this is discussed.

You are assigning properties to all conceivable Universes based on nothing but your imagination. There is not any possible way for such assignment to be anything but your LACK of imagination, nor is there any possible way for you to verify your "superpositives. It's just sophistry.

This is not the place to discuss Anselm's argument. We are discussing Godels Ontological Proof

Which is just an itineration of Anselm's. They are both cloud castle construction projects.

you are simply commenting off the cuff without even having read the previous discussions.

You would be surprised how much of them I have read, they are tedious and boring SOPHISTRY. Logic without reality is exercise for the mind, NOTHING ELSE. I took enough logic and philosophy to know it gives me headaches, I prefer a much more direct and concise approach. But you, Godel and Anselm cannot get past the first axiom before you leave reality behind. That fatal flaw in your arguments absolves me of having to wade through the mounds of BS built upon it.

All you have to do is show how what we conceive(believe, imagine)is real or describes reality IN ALL CASES. Failing that, I prefer to excersize my mind dealing with the reality we CAN evidence. Show evidence that "greatest" can be objectively quantified. Show that the "greatest" is more than a concept(IE real). Your whole basis is not reality, our imagination(or conceptions)have no effect on reality. Your assigned "superpositives" are arbitrary and subjective to your standards and the Emperor has no clothes(the moral of that story is that no matter how much BS you pile on top of nothing, you still have nothing).

Grumpy

[EduChris]

...You are assigning properties to all conceivable Universes based on nothing but your imagination. There is not any possible way for such assignment to be anything but your LACK of imagination, nor is there any possible way for you to verify your "superpositives. It's just sophistry...

I guess you could say that all human speech, ultimately, amounts to nothing more than sophistry. But within the paramaters of normal discourse, and within discourse here on this forum, I have argued my case in detail, and your posts amount to nothing more than fatuous "I don't buy it" one-liners strung together by someone who gives no evidence of having even read the previous discussion.

...just an itineration of Anselm's. They are both cloud castle construction projects...

If they are the same, then please stay on topic and confine your discussion to Godel. None of us has been talking about Anselm's formulation.

...You would be surprised how much of them I have read, they are tedious and boring...

Put up or shut up. I challenge you to come up with one property which is:

1) consistent with my super-positive properties of Existence, Differentiation, Relationality, and Information

2) able to filter through Axioms 2 through 4

3) either counterintuitive or damaging to my case or not already implicit within the superpositives or within my other "positive" (i.e., non-arbitrary) properties of Consciousness, Volition, Creativity, and Love.

If you can do that, it will either strengthen my argument or undermine it. I welcome any advance in either direction.

[sickles]

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).

thats my point. You may pin down many of "somethings" positive properties, but some of those properties that you may mention are infallible, while other properties that you may think of as negative are actually positive. As i said, to answer this we require the knowledge of good and evil. As we dont possesss this knowledge, everthing else is so much mental masterbation.