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?

[EduChris]

...Is [Godel's] Ontological Theorem logically valid?

It appears to be.

...Are all the axioms of the theorem valid?

In general, axioms are not "valid"; instead, they are (in best cases) reasonable and not disproven. Although Axiom 1 seems hazy (or vague, as the article itself states) all of the axioms appear reasonable to me.

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

All arguments rest on assumptions (axioms) which have not been proven. If the axioms are reasonable (and not disproven) and if the form of the argument is logically sound, then the argument is valid. Whether or not it is "true" is a moot point, since no arguments can ever "prove" anything in an ultimate sense.

[LiamOS]

-Is the Ontological Theorem logically valid?

In my analysis of the argument, it appears to maintain consistency and coherence and maintains logical integrity.
I would qualify this as a logically valid argument.

-Are all the axioms of the theorem valid?

Ax. 1. is vague at best, and does not qualify an objective means for determining a positive property from something which is merely a property. As both the existence of and definition of positive properties are central to the theorem, the fact that they are not defined does not allow us to determine if the argument can be applied to the existence of something.

Ax. 2, 3, 4 although speculative, do not give any reason to doubt them, and could arguably be defined as such depending on Ax. 1.

Ax. 5. is not necessarily true as a positive property has not been appropriately defined, and until this has been done to a standard which would allow us to determine the nature of existence as a property, this axiom cannot be held.


Interestingly, these axioms combine to commit a bare assertion fallacy in that and are circular in that they rely on the conclusion of the argument.

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

It may be true, but the argument is not sound.



EduChris
As was being discussed in the last thread, for what reason are my critiques of the argument irrelevant?

[JoeyKnothead]

From the OP:

:
...
The proof can summarized as:

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

It's possible I'm highly intelligent. I think we've all seen how that'n turned out.

That something is possible to exist says little to nothing about that something actually existing.

[EduChris]

...EduChrisAs was being discussed in the last thread, for what reason are my critiques of the argument irrelevant?

Axiom 5 would be reasonable to the extent that Axiom 1 is reasonable. The other three Axioms are perfectly and independently reasonable, in my opinion.

Axiom 1, though vaguely formulated, corresponds to my notion of humans as "meaning-makers." Positive properties, to me, correspond to "meaningfulness." To the extent that "meaningfulness" is a reasonable concept, Axiom 1 is also reasonable (and Axiom 5 as well).

If "meaningfulness" is not a reasonable concept, then Axiom 1 is unreasonable and therefore the entire argument fails. But in that case, the near-universal human penchant for "meaning-making" is also invalidated. Which makes all debate irrelevant, and we should just close down this forum and go stick our heads in the sand.

[LiamOS]

I apologise for the imminent onslaught of questions, but they are necessary to a degree.

Axiom 1, though vaguely formulated, corresponds to my notion of humans as "meaning-makers." Positive properties, to me, correspond to "meaningfulness." To the extent that "meaningfulness" is a reasonable concept, Axiom 1 is also reasonable (and Axiom 5 as well).

So a positive property is a meaningful property? How did you conclude this?
How does one define a property which is meaningful?
Is existence meaningful? Why?

If "meaningfulness" is not a reasonable concept, then Axiom 1 is unreasonable and therefore the entire argument fails.

Delightful. So how is meaningfulness defined?

But in that case, the near-universal human penchant for "meaning-making" is also invalidated.

I don't think that meaning can be shown or argued to be necessary, but I'm open to being wrong on this.

we should just close down this forum and go stick our heads in the sand.

DC&R trip to the beach, anyone?

[Board]

If "meaningfulness" is not a reasonable concept, then Axiom 1 is unreasonable and therefore the entire argument fails. But in that case, the near-universal human penchant for "meaning-making" is also invalidated. Which makes all debate irrelevant, and we should just close down this forum and go stick our heads in the sand.

Is it possible that the human penchant for meaning-making exists because there is no universal meaning to be found? The only reason we seek meaning is because it does not exist? I don't think that makes all debate irrelevant. I think that makes all debate essential for us to define our meaning in lack of the universal existence of positive properties. Otherwise we would all have our heads in the sand...

[EduChris]

...Is it possible that the human penchant for meaning-making exists because there is no universal meaning to be found? The only reason we seek meaning is because it does not exist? I don't think that makes all debate irrelevant. I think that makes all debate essential for us to define our meaning in lack of the universal existence of positive properties. Otherwise we would all have our heads in the sand...

It may be that there is no absolute, ultimate meaning. But I am not constrained to believe this--there is no proof (or even any real evidence) that supports such a conclusion. So for me, our human desire for meaning is like our desire for food, and water, and air: we desire these things because they do exist, not because they don't exist.

[LiamOS]

It may be that there is no absolute, ultimate meaning. But I am not constrained to believe this--there is no proof (or even any real evidence) that supports such a conclusion. So for me, our human desire for meaning is like our desire for food, and water, and air: we desire these things because they do exist, not because they don't exist.

We desire such things because it's helpful to us to get them.

[EduChris]

...So how is meaningfulness defined?...

I don't think we can define, or even need to define, our deeply held, properly basic intuitions. At some point we have to ask, How deep can we go with our language? No matter how deep we go with our definitions, at some point we have to stop and say, "It is what it is, I can't define it, but I know it when I see it." This phenomenon is not unique to theology or philosophy, but to every aspect of human knowledge (including science).