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]

What about such a hypothesis are you requesting I consider?

Differentiation (spatial vs. temporal dimensions) and Relationality.

Also, how about Quantum entanglement? Doesn't that also demonstrate differentiation (two distinct particles, separated by distance) and relationality (what happens with one particle affects the other particle, even when the particles are separated by some arbitrary distance)?

Abraxas has said he is willing to accept (for sake of argument) not just Existence, but also Differentiation and Relationality as super-positives. You have indicated a reluctance to accept Differentiation and Relationality. I'm trying to bring up examples that will move you toward Abraxas's position.

Basically, if I can establish Differentiation and Relationality in our universe, then those properties are automatically super-positives if our universe is the only conceivable universe. But if there are other conceivable universes (as I believe there are) then I get Differentiation on the basis that those other conceivable universes are conceptually different than ours, while still related on the basis of their shared conceivability. Either way, it appears to me that Existence, Differentiation, and Relationality are super-positives, entailed for all conceivable universes (given the initial assumption that a universe such as ours is conceivable).

[Zeeby]

...observe that Axioms 2 and 3 begin with "if P is a positive property" whilst Axiom 4 does not...If you are suggesting that Axiom 4 is only valid when P is a positive property, then that does not suggest an explanation of "either P or its negation is positive"...

I believe Axiom 4 can be restated as follows: "If P is a positive property, then its negation cannot be positive."

Thanks for the clarification. Symbolically, you are stating the second axiom as P(¬phi) -> ¬P(phi), rather than what is stated in the Wikipedia article (P(¬phi) <-> ¬P(phi)), which is a lot stronger (as I claimed).

However, on further investigation, the article here gives the axiom (G1 in the article) as you have stated. I will need a little time to identify whether I have an issue with the proof presented there.

On the topic at hand, I still think your idea of differentiation is circular: "conceive of a universe different from ours; then it is different"?

[EduChris]

...I still think your idea of differentiation is circular: "conceive of a universe different from ours; then it is different"?

Actually, my argument is this:

1) A universe such as ours is conceivable

2) Other universes may also be conceivable

3) If there are no other conceivable universes, then whatever holds for our universe also holds for all conceivable universes (since in this case, ours would be the only conceivable universe)

4) Other "conceivable universes" (if there are any others) may or may not be conceptually different from ours

5) If all of the other universes are conceptually the same as our universe, then whatever holds for our universe (Existence, Differentiation, Relationality) also holds for all conceivable universes

6) If some other universe is conceptually different from ours, then at minimum, such other universe(s) will entail the properties of Existence; Differentiation (it is conceptually different from ours); and Relationality (it belongs to the set of conceivable universes)

I see no circularity here. However you look at it, the "super-positives" of Existence, Differentiation, and Relationality hold for all conceivable universes.

[Zzyzx]

.
When one resorts to philosophy and metaphysics to "prove" their point, there must be little actual evidence to support what they say -- only mental constructs or mental gymnastics. Philosophy and metaphysics don't prove anything except what one THINKS about a topic (i.e., opinion).

This is not to say that there is no place for philosophy, but rather to say that it emphasizes conceptualization rather than evidence. The term "philosophy" is defined as: a quest for truth through logical reasoning rather than factual observation.

Bold added for emphasis

One can "logically reason" about fairies, leprechauns and goblins (or how many angels can dance on the head / point of a pin) -- with absolutely no tie to reality and no indication that such things exist.

[Angel]

.
When one resorts to philosophy and metaphysics to "prove" their point, there must be little actual evidence to support what they say -- only mental constructs or mental gymnastics. Philosophy and metaphysics don't prove anything except what one THINKS about a topic (i.e., opinion).

This is not to say that there is no place for philosophy, but rather to say that it emphasizes conceptualization rather than evidence. The term "philosophy" is defined as: a quest for truth through logical reasoning rather than factual observation.

Bold added for emphasis

One can "logically reason" about fairies, leprechauns and goblins (or how many angels can dance on the head / point of a pin) -- with absolutely no tie to reality and no indication that such things exist.

The scientific method in large part stems from philosophy so if you say that metaphysics is assumption, then to be consistent, so is the metaphysics that scientists assume every day, naturalism and materialism. Scientific observations and logical reasoning aren't necessarily mutually exclusive. I mean they can be but I usually find scientific observations in the premises of logical arguments, especially when it comes to inductive and deductive reasoning. I would be reluctant to use an argument that's based 'solely' on metaphysics/ontology to conclusively establish the reality of some matter (not that I'd outrule it).

[EduChris]

...The scientific method in large part stems from philosophy so if you say that metaphysics is assumption, then to be consistent, so is the metaphysics that scientists assume every day, naturalism and materialism...

Agreed. If science is (by definition) concerned only with physical reality, it is illogical to expect that "scientific" (physical) evidence must be adduced for that which has always been conceived of as non-physical.

[McCulloch]

Agreed. If science is (by definition) concerned only with physical reality, it is illogical to expect that scientific evidence must be adduced for that which has always been conceived of as non-physical.

What evidence, aside from logical deduction, do you propose to support claims made about the alleged non-physical reality? Aesthetic? Intuition? Telepathy? Emotion?

I presume that since your objection to the use of science in the non-physical realm is based on illogic, that logic can be at least assumed to be an over-arching framework for all realms. Recent discussions around Trinity seem to belie this presumption, but I personally must abandon any debate without logic.

[EduChris]

...I presume that...logic can be at least assumed to be an over-arching framework for all realms...I personally must abandon any debate without logic.

I share your concerns. Accordingly, I have proposed Rationality (or Reason, or Logic) as a Godelian "positive property," since Rationality/Reason/Logic seems entailed by the super-positives of Differentiation (i.e., "X is not Y" and "X is Y" cannot both be true in the same sense at the same time) and Relationality (i.e., 2 is less than 3 within the set of all integers).

I'm still waiting for a response to my proposal.

[McCulloch]

"X is not Y" and "X is Y" cannot both be true at the same sense at the same time

Can "X is Z", "Y is Z" and "X is not Y" all be true in the same sense at the same time, where Z is a single entity not a set?

For example, substitute "1" for X, "2" for Y and "an integer" for Z.

• 1 is an integer, 2 is an integer, 1 is not 2.

This is a true and valid statement because integer refers to a set, not an individual item.
Second example, substitute "Barak Obama" for Z, "The President of the United States" for X and "the commander-in-chief of the United States armed forces" for Y.

• The President of the United States is Barak Obama.
  The commander-in-chief of the United States armed forces is Barak Obama.
  The President of the United States is not the commander-in-chief of the United States armed forces.

This is a false and invalid statement, because Barak Obama is a single entity.
Third example, substitute "God" for Z, "Jesus" for X and "the Holy Spirit" for Y.

• Jesus is God.
  The Holy Spirit is God.
  Jesus is not the the Holy Spirit.

This is a false and invalid statement, because God is a single entity, God is one. Unless Trinitarians accept polytheism, or that God is a set, they must abandon logic.

[EduChris]

...Unless Trinitarians accept polytheism, or that God is a set, they must abandon logic.

According to Christian tradition, the Triune God is one Divinity comprised of three persons. This claim sounds strange to us, but no more strange than a two-dimensional "flatlander" would find in her attempt to grasp the mysteries of a "cube" consisting of multiple two-dimensional "squares."

But in any case, the traditional hallmarks of the Christian Triune God have always been Existence, Differentiation, and Relationality--which (quite suggestively, in my opinon) comprise the trio of Godelian "super-positives" which is entailed by all conceivable universes.