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]

Suppose we conceive of a universe consisting entirely of liquid water. Suppose we then conceive of another universe consisting entirely of solid gold. We cannot then conceive of a universe consisting entirely of liquid water and entirely of solid gold (although of course we can have a universe which consists of some part liquid water and a remainder of solid gold).

What if in this universe the properties of gold and water are not mutually exclusive?

This tells us either that "entirely consisting of liquid water" is not a positive (non-arbitrary) property or that "entirely consisting of solid gold" is not a positive, or perhaps neither are positive (non-arbitrary) properties. The conceptualization helps us to see which properties cannot be "positive" in the Godelian sense.

I also have something of a problem with this(Who'd have thought? ) as it assumes far too much about the nature of a universe. Assuming that standard logic applies beyond our universe is something I cannot justify.
It's probably just me on this one, but I also fail to see how your aforementioned properties can be argued to be non-arbitrary without applying them. It seems to me that in order to ascribe non-arbitrariness to them you must assume those properties are always applicable.

Again, I've never seen any particles at all, so I can't disagree. But can you say whether there any physical forces (e.g., gravity, attraction, repulsion, etc) or any concepts or any persons which are distinct?

The idea of distinction is certainly meaningful at a macroscopic level, but whether this is a result of distinction at a fundamental level or a emergent property is an open question.

What aren't you following?

I don't understand what you are trying to convey with that sentence.

The computer seems to compel certain familiar electrical impulses in my brain, and it seems to produce similar effects in other people--to the extent that I can believe that other people exist independently of my own mind.

But isn't this the same problem we all face, regardless of our worldview? This sort of question isn't any more damaging to Godel's theorem than it is to any matter involving human cognition and epistemology. We simply have to face the uncertainty of all human knowing, and move forward as best we can with the tools we have.

I am more than aware that defining existence is extremely difficult, but I don't think the argument can be meaningful without actually defining the properties it is bestowing.

[EduChris]

...What if in this universe the properties of gold and water are not mutually exclusive?...

Such a case would be difficult to conceive. We might as well dispense with all logic and all physics and just admit that any word salad we can devise is just as valid as any other.

...Assuming that standard logic applies beyond our universe is something I cannot justify...

There are any number of things we cannot justify, but which we assume on the basis of some reason or another. We have no good reason to suppose that standard logic (or at least a superset of our standard logic) would fail to apply beyond our universe. If standard logic were denied outside of our own universe, then there could be no other universes besides our own which we could "conceive of" in any meaningful way. For our purposes here, I suppose we could simply explicitly state the assumptions that: 1) a universe such as ours is conceivable, 2) other universes may also be conceivable, and 3) we will assume that standard logic applies beyond our universe.

...It's probably just me on this one, but I also fail to see how your aforementioned properties can be argued to be non-arbitrary without applying them. It seems to me that in order to ascribe non-arbitrariness to them you must assume those properties are always applicable...

Not following you here. Can you explain what you mean?

...The idea of distinction is certainly meaningful at a macroscopic level, but whether this is a result of distinction at a fundamental level or a emergent property is an open question...

Is the number "one" distinct from the number "two"?

...I don't understand what you are trying to convey with that sentence...

Can you be more specific?

...I am more than aware that defining existence is extremely difficult, but I don't think the argument can be meaningful without actually defining the properties it is bestowing.

Whether we're talking about Godel's Ontological Proof or whether we're talking about the existence of anything at all, for any reason at all, we're all in the same boat. We can't hold one theory hostage on "existence" while we give all other theories and worldviews a free pass. I admit I don't know any foolproof way to prove whether one item belongs in the set of existent things or the set of non-existent things. I can't prove it, but as a pragmatic matter to which we are forced to submit in all of our theorizing, to me there is a difference between an existent thing and a non-existent thing, and somehow we can usually tell the difference.

How about this: "Existence is that property of an item which grants it membership in the set of existent things. Non-existence is that property of an item which grants it membership in the set of non-existent things." This may seem circular, but nevertheless membership within a set certainly is a property. Perhaps we need to stipulate the assumption that our commonly held intuitions regarding existence or non-existence will be deemed reliable enough for our purposes--especially since any alternative assumption would be ruinous to all theorizing.

[LiamOS]

Such a case would be difficult to conceive. We might as well dispense with all logic and all physics and just admit that any word salad we can devise is just as valid as any other.

There are any number of things we cannot justify, but which we assume on the basis of some reason or another. We have no good reason to suppose that standard logic (or at least a superset of our standard logic) would fail to apply beyond our universe. If standard logic were denied outside of our own universe, then there could be no other universes besides our own which we could "conceive of" in any meaningful way. For our purposes here, I suppose we could simply explicitly state the assumptions that: 1) a universe such as ours is conceivable, 2) other universes may also be conceivable, and 3) we will assume that standard logic applies beyond our universe.

My problem is that we haven't even defined what conceivability entails or means and that 'standard logic' hasn't even been shown to apply to our universe.

Not following you here. Can you explain what you mean?

I mean that in your conception of other universes you are giving them properties implicitly. Of course, it's impossible not to with human nature, but that's still something of a problem to me as it seems those implicit properties are giving you your super-positives.

Is the number one distinct from the number two?

That depends on a whole lot of things.

If we use inductive logic to perceive our surroundings, we'll probably conclude that they're distinct, though.

Can you be more specific?

Not really. The entire concept that the sentence is attempting to convey is just going over my head.

Whether we're talking about Godel's Ontological Proof or whether we're talking about the existence of anything at all, for any reason at all, we're all in the same boat. We can't hold one theory hostage on "existence" while we give all other theories and worldviews a free pass. I admit I don't know any foolproof way to prove whether one item belongs in the set of existent things or the set of non-existent things. I can't prove it, but as a pragmatic matter to which we are forced to submit in all of our theorizing, to me there is a difference between an existent thing and a non-existent thing, and somehow we can usually tell the difference.

I don't have much of a problem with people assuming certain things exist, but when we're dealing with things with which we have no experience or knowledge, it's imperative to define what existence in this context is and what is entails.

[EduChris]

...My problem is that we haven't even defined what conceivability entails or means and that 'standard logic' hasn't even been shown to apply to our universe...

Try telling that to Joey or Zzyzx. As far as I'm concerned, if standard logic doesn't even apply to our universe, then any word salad makes as much sense as any other. If the objection to Godel's argument is reduced to this level, then either Godel's argument is valid or else all arguments are invalid.

...I mean that in your conception of other universes you are giving them properties implicitly. Of course, it's impossible not to with human nature, but that's still something of a problem to me as it seems those implicit properties are giving you your super-positives...

Yes, I am using standard logic and set theory on universes other than our own, in order to make them conceivable. I am willing to stipulate that standard logic applies to our universe and to any other universes within the set of conceivable universes. Again, the "other conceivable universes" are merely a tool to help us conceptualize which properties of our own universe are non-arbitrary; we're not using the other universes to help us sneak non-existent properties into our set of existent properties.

Is the number one distinct from the number two?

That depends on a whole lot of things...If we use inductive logic to perceive our surroundings, we'll probably conclude that they're distinct, though...

That's good enough as far as I'm concerned, especially since to assume otherwise would be ruinous to all of our theorizing. (nice cartoon, by the way)

Can you be more specific?

Not really. The entire concept that the sentence is attempting to convey is just going over my head...

All I can say is that whenever we are faced with the unknown or the lesser known, we always have to start by describing in terms of things which are better known--if only by way of comparison and contrast.

...I don't have much of a problem with people assuming certain things exist, but when we're dealing with things with which we have no experience or knowledge, it's imperative to define what existence in this context is and what is entails.

We're simply deciding whether to include something in the set of existent things, or in the set of non-existent things. If we place something in the set of existent things, that thing will have the property of having been recognized in reality. Conversely, if something is placed into the set of non-existent things, that thing will have the property of not having been recognized in reality.

[LiamOS]

Try telling that to Joey or Zzyzx. As far as I'm concerned, if standard logic doesn't even apply to our universe, then any word salad makes as much sense as any other. If the objection to Godel's argument is reduced to this level, then either Godel's argument is valid or else all arguments are invalid.

I am already highly sceptical of most arguments.

Some logic does certainly apply to the universe, but our inductive reasoning can only give us so much of the story. Modern physics is made up largely of things that challenge what we're used to and redefine what is possible and what is not.
I'm not advocating throwing logic out the window, but I advocate care in where and how it is applied in the same way one should take care when using Maxwell's Equations.

Yes, I am using standard logic and set theory on universes other than our own, in order to make them conceivable. I am willing to stipulate that standard logic applies to our universe and to any other universes within the set of conceivable universes. Again, the "other conceivable universes" are merely a tool to help us conceptualize which properties of our own universe are non-arbitrary; we're not using the other universes to help us sneak non-existent properties into our set of existent properties.

As long as you are entirely aware of what is entailed in the assumptions you are making, you are as justified in accepting the argument as I am in rejecting it if not more so.

That's good enough as far as I'm concerned, especially since to assume otherwise would be ruinous to all of our theorizing.

To do otherwise is to end up like me, so you're probably making a good move.

I could never really be content with something that just 'usually works', especially when it's the areas in which it doesn't that are of interest.

(nice cartoon, by the way)

It's from the webcomic Saturday Morning Breakfast Cereal.
There are some rather funny comics on it, many in relation to science, philosophy and theology.

All I can say is that whenever we are faced with the unknown or the lesser known, we always have to start by describing in terms of things which are better known--if only by way of comparison and contrast.

Ah. I suppose you are right.

I'm not sure that it's the correct way, but it seems to be the only viable way.

We're simply deciding whether to include something in the set of existent things, or in the set of non-existent things. If we place something in the set of existent things, that thing will have the property of having been recognized in reality. Conversely, if something is placed into the set of non-existent things, that thing will have the property of not having been recognized in reality.

And without determining how something is recognised within reality, this seems somewhat arbitrary.
Essentially I feel that ascribing the property of existence based on these assumptions and without even fully knowing what existence is isn't entirely convincing to me.

[EduChris]

...I am already highly sceptical of most arguments...

You are? What makes you think so?

...As long as you are entirely aware of what is entailed in the assumptions you are making, you are as justified in accepting the argument as I am in rejecting it if not more so...

I'm okay with anyone who rejects this argument, as long as they admit that their criteria would entail rejecting pretty much any argument at all.

...To do otherwise is to end up like me, so you're probably making a good move...

I think you've helped me strengthen my argument (or at least my own understanding of the argument) so I'm glad you're you.

...And without determining how something is recognised within reality, this seems somewhat arbitrary. Essentially I feel that ascribing the property of existence based on these assumptions and without even fully knowing what existence is isn't entirely convincing to me.

Again, if we don't know what existence is, then we're not going to have much luck arguing anything at all. If the objections to the argument is reduced to this, then Godel's argument is as valid as any other argument. That might be good or bad, depending on one's confidence in rational argumentation--which gets us back to the human condition: we just don't have access to any ultimate truth. Any knowledge we think we have rests on faith and hope.

[LiamOS]

You are? What makes you think so?

I should've qualified that further. I'm sceptical of arguments for the existence of entities(Physical or otherwise) which cannot yet be tested, and I'm also sceptical of logic in cases outside the norm.

I'm okay with anyone who rejects this argument, as long as they admit that their criteria would entail rejecting pretty much any argument at all.

"Try telling that to Joey or Zzyzx."

I think you've helped me strengthen my argument, so I'm glad you're you.

Better me than you, I guess.

Again, if we don't know what existence is, then we're not going to have much luck arguing anything at all. If the objections to the argument is reduced to this, then Godel's argument is as valid as any other argument. That might be good or bad, depending on one's confidence in rational argumentation--which gets us back to the human condition: we just don't have access to any ultimate truth. Any knowledge we think we have rests on faith and hope.

I'm still of the opinion that the argument is necessarily assuming things about the universe and other universes which I cannot accept.
Neglecting the Cosmological Argument, this is probably one of the strongest arguments for the existence of a Deity(I define Deity loosely).

[JoeyKnothead]

From Post 224:

...My problem is that we haven't even defined what conceivability entails or means and that 'standard logic' hasn't even been shown to apply to our universe...

Try telling that to Joey or Zzyzx. As far as I'm concerned, if standard logic doesn't even apply to our universe, then any word salad makes as much sense as any other. If the objection to Godel's argument is reduced to this level, then either Godel's argument is valid or else all arguments are invalid.

Just in case I'm the Joey referred to here, I question the "logic" behind such as, "If I can imagine it, it exists".

I've yet to hear any valid argument that shows the mere act of imagining something exists means that something exists.

This is the gist of Godel's argument, and as I posted previously, it is high on the list of the goofiest notions in the history of mankind.

[Zzyzx]

.

...My problem is that we haven't even defined what conceivability entails or means and that 'standard logic' hasn't even been shown to apply to our universe...

Try telling that to Joey or Zzyzx.

Zzyzx takes no position regarding logic and the universe -- and makes no claim of great knowledge of the latter (only some knowledge of the biosphere of one of the planets and some knowledge of the solar system). Beyond that, I leave astronomy to the astronomers and logic to the "logicians" (or those who fancy themselves as such) with reservations where appropriate.

As far as I'm concerned, if standard logic doesn't even apply to our universe, then any word salad makes as much sense as any other.

I acquiesce to your command of word salad.

[EduChris]

...Also notice Axiom 2 says "either a property is positive, or its negation is". This is giving the God object rather a lot of properties - as the God object has all positive properties and any property it has is positive. For example, in the Christian belief system it would be necessary that "created horses in this universe" is a positive property but that "created unicorns in this universe" is not a positive property. Similarly "is believed to not exist by some people" would be a positive property...

Having re-read much of the material on Godel, I wanted to add something here even though it is obviously very late.

The theorem says, "If P is a property, then either P or its negation is positive, but not both." The salient point is the initial "IF". Not all things are properties as Godel is using the term. In other words, to string words together and say that these words represent a "property" in the Godelian sense is to commit the fallacy of equivocation.

With respect to "having created horses" or "not having created unicorns," neither of these represent properties in the Godelian sense, since the object of the verb "create" is overly specific, and therefore not independent of the accidental structure of the world (axiom 1). The attribute of "having created" is all we need for an attribute, and this attribute of creativity is indeed a "positive property" in the Godelian sense.

Another typical complaint (not raised by Zeeby, but rather by Abraxas early in this thread) is that "properties" of measurement don't seem to filter through the axioms. For example, it is claimed that having the measurements of HxWxD does not, at first glance, lend itself to either "positivity" or "negativity" in common parlance. Putative "properties" of specific measurements are not properties at all in the Godelian sense (they are not independent of the accidental structure of the world, per axiom 1). This raises the issue of whether the more general property of "being limited within the finite dimensions of space and time" is a property in the Godelian sense. It appears that "not being bound within the finite dimensions of space and time" is a positive Godelian property, which in turn would make its negation (being bound to some specific and quantified measurements in space and time) a negative Godelian property.

Therefore, I will add the positive Godelian property of "not being quantitatively bound within the finite dimensions of space and time" to the previous list of Godelian positive properties.