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 proof can summarized as: IF it is possible for a rational omniscient being to exist THEN necessarily a rational omniscient being exists...

I don't believe that summary really gets at the heart of Godel's theorem. A physicist summarized the theorem for me as, "If anything at all exists, something must necessarily--i.e., non-contingently--exist."

Here is another link and a highlighted section from the paper. To the extent that the author is an "expert witness" himself, one should not be too quick to claim "fallacy" when presented with Godel's argument:

The ontological argument attempts to ground all existing things in a necessary being, namely God....It takes a certain amount of courage to state publicly that you believe Gods existence can be proved. Many scholars who admire Kurt Gdel as the greatest logician of the twentieth century have found themselves a bit embarrassed by his ontological argument. His Incompleteness Theorems are undoubtedly masterpieces of mathematics. His rotating universe is respectable in physics because its properties can be explained without reference to his belief that Becoming is in some sense illusory. By showing that the rotating universe is conceptually consistent with general relativity, he helped make academic discussion of time travel acceptable for later researchers such as Kip Thorne. As a proof theorist, he helped make modal logic respectable by demonstrating the utility of modal operators in the foundations of mathematics. By establishing the consistency of the continuum hypothesis with Zermelo-Fraenkel set theory, he helped make Cantors transfinite arithmetic respectable. Yet the ontological argument remained on the margin of intellectual thought. It is now emerging from this intellectual wilderness to a more prominent place where it belongs. Gdels research forms a unified body of thought in the Platonic tradition. Within that body of thought, his ontological argument is no aberration, but an essential part of one of the most remarkable thinkers of modern times (pp. 27-28).

[EduChris]

...I am not sold on the existence of objective meaningfulness. If we recognize the pleasure principle may be what is telling us to seek meaningfulness then we would have to step outside of this human condition to seek the existence of objective meaning...we need more proof outside of human feelings like trust and fear that could be passed off as a larger psychological condition of our species...

Which is precisely what we cannot do. The choice available to us is, do we choose "the fear of meaninglessness," or do we choose "the hope of meaningfulness."

Given that neither assumption can be proven in any objective fashion, and neither is inherently implausible, I cannot imagine why anyone would choose fear over hope.

[cnorman18]

...I am not sold on the existence of objective meaningfulness. If we recognize the pleasure principle may be what is telling us to seek meaningfulness then we would have to step outside of this human condition to seek the existence of objective meaning...we need more proof outside of human feelings like trust and fear that could be passed off as a larger psychological condition of our species...

Which is precisely what we cannot do. The choice available to us is, do we choose "the fear of meaninglessness," or do we choose "the hope of meaningfulness."

Given that neither assumption can be proven in any objective fashion, and neither is inherently implausible, I cannot imagine why anyone would choose fear over hope.

Not everyone casts the choice(s) in those terms. One may also choose one's own meaning, without claiming that it applies to everyone or is an objective truth.

[ScotS]

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.

[JoeyKnothead]

From Post 37:

I don't believe that summary really gets at the heart of Godel's theorem. A physicist summarized the theorem for me as, "If anything at all exists, something must necessarily--i.e., non-contingently--exist."

Here is another link and a highlighted section from the paper. To the extent that the author is an "expert witness" himself, one should not be too quick to claim "fallacy" when presented with Godel's argument:
...
...

I 'preciate the reply, but right now time and the old lady's got me hemmed up like a hog in a bushel basket. I've downloaded the links and'll get right back here directly.

[EduChris]

...Not everyone casts the choice(s) in those terms. One may also choose one's own meaning, without claiming that it applies to everyone or is an objective truth.

Where do you find me "claiming that it applies to everyone or is an objective truth"?

I don't think I said that or implied it, and I certainly don't expect that everyone's notion of meaningfulness will necessarily be the same. I think it is more likely that the Ground of All Existence, to the extent that it cares enough about us to be concerned about our desire for meaning, will relate to each us is according to our needs and deepest hopes.

[LiamOS]

I wish you were right. In my experience, I've had to repeat myself numerous times, saying essentially the same thing each time, and still it seems that the accusers aren't reading my posts at all--everything I say is being filtered through the lense of their preconceived notions of what they think theists always say, rather than what I'm actually saying.

I guess this just supports my assertion that one's presuppositions and prejudices will always affect one's understanding and one's conclusions--sometimes even more than the evidence and argument itself.

That's true on both sides of the divide. I'll not name names, but some Theist debaters still active and somewhat respected have often ignored entire sections of my posts and criticised my argument in a manner already covered.
Also, some other Theists will repeatedly use the same arguments despite them being invalid and having repeatedly been shown as such.

I don't think I said that or implied it, and I certainly don't expect that everyone's notion of meaningfulness will necessarily be the same. I think it is more likely that the Ground of All Existence, to the extent that it cares enough about us to be concerned about our desire for meaning, will relate to each us is according to our needs and deepest hopes.

As an example, we have this post.
Here, you have implied that a God exists and that it is capable of empathising.
Normally this would be open to challenge, but given the context of the post doing so would be a bit harsh.

Also, given that you've just admitted that meaningfulness is subjective(You did, right?), have you not by default just conceded that the Ontological Argument is not in fact a valid proof?

[cnorman18]

...Not everyone casts the choice(s) in those terms. One may also choose one's own meaning, without claiming that it applies to everyone or is an objective truth.

Where do you find me "claiming that it applies to everyone or is an objective truth"?

I don't think I said that or implied it, and I certainly don't expect that everyone's notion of meaningfulness will necessarily be the same. I think it is more likely that the Ground of All Existence, to the extent that it cares enough about us to be concerned about our desire for meaning, will relate to each us is according to our needs and deepest hopes.

My mistake; I stand corrected. In that case, I suppose I agree with you -- with the proviso that the Ground of all Being, if any, may or may not be a personal Entity.

[Board]

Not everyone casts the choice(s) in those terms. One may also choose one's own meaning, without claiming that it applies to everyone or is an objective truth.

As usual, cnorman18 says plainly what I was attempting to mean. Can I borrow your mad skills?

As with any psychological problem, step 1 is recognizing it. Step 2 is washing it out.

I'm not saying that the pleasure principle is a psychological problem but I think if we ignore the implications it has on our decision making process then we are doing ourselves a disservice.

I think we can acknowledge our human condition and selectively choose the "pain" response without fear of pain. It is only pain in light of our condition if we take our condition to be true. Outside of that condition it may be pleasure.

For Example:

The search for objective meaningfulness may end in either objective (pleasure response) or subjective (pain response) meaningfulness.

Personally I find joy in defining my own meaning in life. I think we have become too programed as to what is pleasure and what is pain and we need to re-wire society.

[EduChris]

...That's true on both sides of the divide...

That should go without stating, though I will go ahead and state that I agree with you anyway.

...Here, you have implied that a God exists and that it is capable of empathising. Normally this would be open to challenge, but given the context of the post doing so would be a bit harsh...

It should not be necessary for the theist to stop after every few words of every sideline (i.e., not even the main topic of the thread) and produce pages of argument defending his/her decision to adopt one unprovable assumption (that there is a necessarily existing entity) over another unprovable assumption (i.e., that there is no necessarily existing entity). I appreciate your foregoing such behavior here, as I would consider it uncivil and disruptive. (hee..hee... maybe I should start challenging non-theists to produce pages of argumentation every time they suggest or imply or assume that there is no necessarily existing entity--you know, the old goose and gander thing... ).

...given that you've just admitted that meaningfulness is subjective(You did, right?), have you not by default just conceded that the Ontological Argument is not in fact a valid proof?

I don't think I indicated that "meaningfulness" in itself is subjective; rather, my intent was to say that different folks will have different needs and desires. The "meaningfulness" is a constant, but the particular form it will take will be different from one person to another.