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?

[Goat]

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?

I think what is as important as it being 'logically valid' is that is it 'logically sound'.

To show that it is logically sound, you have to show that all the axioms are valid.

It looks to me that , like all ontological arguments, it is trying to define God into place by making unsupported and unsupportable assumptions.

What you have to do is show define what a 'God like' property is.. and show that 'like' is actually 'equal to' , and show that it is 'possible' in any possible world.

[EduChris]

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

If they didn't exist, they wouldn't be very much help.

[Lux]

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.

Kind of an understatement. We "desire" food, water and air because we need them to survive, which is our primary instinct. However, it's pretty clear that humans can survive just fine without knowing whether or not their lives have objective meaning or what that meaning is. "Meaning" is not comparable to water, food and air at all.

DC&R trip to the beach, anyone?

I'll go but I'm not sticking my head in the sand.

[McCulloch]

It may be that there is no absolute, ultimate meaning.

I am pleased to find you admit your agnosticism on this one point.

But I am not constrained to believe this--there is no proof (or even any real evidence) that supports such a conclusion.

In other words there is no disproof of an absolute ultimate meaning, therefore you believe that such a thing exists. Right?

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.

Food, water and air really do exist.

You wish that there should be some kind of absolute ultimate meaning, therefore you will believe that there is, in spite of the evidence that death is final, the universe is uncaring and unjust. OK.

[EduChris]

...I am pleased to find you admit your agnosticism on this one point.

I am pleased to find that you have finally figured out what I have been saying over and over to you and to many others here on the forum. What took you so long?

...In other words there is no disproof of an absolute ultimate meaning, therefore you believe that such a thing exists. Right?...

In the absence of absolute objective truth, which is the common human experience, one must decide which avenues, which logical pathways, will be most fruitful.

...Food, water and air really do exist...

So you say. It could be all just a particularly persistent illusion--you know, like the "illusion" that we as agents can actually make choices that determine outcomes.

...in spite of the evidence that death is final, the universe is uncaring and unjust...

Um, what evidence do you have that death is final? Have you yourself died and personally discovered that death is final? Or are you just stating a conjecture?

[LiamOS]

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).

Funny thing; I don't think that a positive aspect is intuitive.
People differ, hence the need for a definition.

Are you essentially qualifying that this argument rests upon an appeal to emotion?

[EduChris]

...Are you essentially qualifying that this argument rests upon an appeal to emotion?

I guess in the sense that you are using "emotion" (i.e., an unprovable assumption that one feels inclined towards) then yes, every human argument ultimately depends on this sort of "emotion"--including your own arguments. This is the common human experience, wherein none of us is privy to absolute and objective proof for anything.

[LiamOS]

Kind of an understatement. We "desire" food, water and air because we need them to survive, which is our primary instinct. However, it's pretty clear that humans can survive just fine without knowing whether or not their lives have objective meaning or what that meaning is. "Meaning" is not comparable to water, food and air at all.

Indeed.

I'll go but I'm not sticking my head in the sand.

I'll do that for you, then.
I'm that nice.

If they didn't exist, they wouldn't be very much help.

In the case of 'meaning', is its actual existence necessary to fulfil what it would?

So you say. It could be all just a particularly persistent illusion--you know, like the "illusion" that we as agents can actually make choices that determine outcomes.

It could. The mind makes you think things that aren't necessarily true.

In the absence of absolute objective truth, which is the common human experience, one must decide which avenues, which logical pathways, will be most fruitful.

Appeal to consequences?

[LiamOS]

I guess in the sense that you are using "emotion" (i.e., an unprovable assumption that one feels inclined towards) then yes, every human argument ultimately depends on this sort of "emotion"--including your own arguments. This is the common human experience, wherein none of us is privy to absolute and objective proof for anything.

Rather than resorting to emotion, I resort to simplicity in the things I assume. Some time ago, I concluded that assuming as little possible would increase the likelihood of basing decisions and beliefs on that which is as close to objective reality as possible.

I'd also have to ask that if those are really your beliefs, how have you become a Christian?

[EduChris]

...Appeal to consequences?

No, absolutely not, as I have stated over and over again in response to numerous uncivil and unfounded accusations. I am using the Hypothetico-deductive method, which according to The Philosopher's Toolkit "is very widely applicable" (p. 41).

This incessant barrage of unfounded and uncivil accusations from you and others is beyond tiresome. Please cease the incivility.