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?

[LiamOS]

Is positive just an inane label in this context or can these properties be considered objectively positive?

[EduChris]

Is positive just an inane label in this context or can these properties be considered objectively positive?

I think the "positive" aspect refers to the type of qualities without which numbers would not exist.

In other words, IF numbers exist, THEN the aforementioned positive properties must apply.

[LiamOS]

So positive is arbitrarily defined in this case?
I don't see how the properties necessary are necessarily positive.

[EduChris]

So positive is arbitrarily defined in this case?
I don't see how the properties necessary are necessarily positive.

I suppose you could substitute "abracadabra" for "positive," so long as your overall definition makes clear the salient point that in order for "existence" to be affirmed, the X-property must necessarily be present (where X is a stand-in for "positive" or "abracadabra" or any other term you wish to use).

[EduChris]

Numbers (plural) cannot exist unless we also have the "positive property" (or "X-property") of (distinctiveness vs. sameness/uniformity). In other words, any given number does not exist in separation, but rather exists (if it exists at all) only in terms of relationality. But in turn, there must be distinctiveness or distinguishability between the individual numbers, in order to have the plurality that is required for relationality.

So here is my newly proposed list of "positive Godelian properties":

1. Relationality vs. isolation
2. Contextuality vs. acontextuality
3. Logic vs. absurdity
4. Significance vs. inanity
5. Order vs. chaos
6. Distinctiveness vs. sameness

Any objections so far?

[LiamOS]

I suppose you could substitute "abracadabra" for "positive," so long as your overall definition makes clear the salient point that in order for "existence" to be affirmed, the X-property must necessarily be present (where X is a stand-in for "positive" or "abracadabra" or any other term you wish to use).

Exactly. So what implications does this have for the Ontological Theorem?

[EduChris]

I suppose you could substitute "abracadabra" for "positive," so long as your overall definition makes clear the salient point that in order for "existence" to be affirmed, the X-property must necessarily be present (where X is a stand-in for "positive" or "abracadabra" or any other term you wish to use).

Exactly. So what implications does this have for the Ontological Theorem?

Well, Godel's theorem doesn't seek to prove that "X" exists. It only states that IF "X" exists--that is, IF you are prepared to say that you believe it is reasonable for "X" to exist (whatever "X" may be, and in our case we have been using "numbers" as our "X")--THEN certain properties or attributes must necessarily be present in order for "X" to exist. Within the context of Godel's theorem, those attributes which are necessary for the existence of "X" are labeled, "positive properties," and those "properties" which logically negate the "positive properties" are labeled, "negative properties."

So in our specific example, we are claiming that IF numbers exist (and it seems that they really do exist) THEN the "positive properties" (i.e., the "existence enabling" or "existence affirming" properties) of Relationality, Logic, Distinctiveness, Order, Significance, etc. must necessarily apply to numbers. If any of these "positive properties" were negated or denied, then you would no longer be justified in claiming that numbers exist.

Note also that neither numbers nor relationality, logic, distinctiveness, order, significance, etc. are physical, material "objects" that can be measured or examined under a microscope.

[Abraxas]

...explain the difference between 1 and -1, as your designation would require 1 x -1, which would be a circular definition, thus empty.

The attribute of "evenness" depends on something being disivible by two (without remainder). The numerical attribution of "negativeness" or "positiveness" depends on whether the number in question is less than or greater than zero.

Thus, -2 or + 4 or -8 or +2 are all divisible by 2 without remainder, so their evenness follows from that alone, rather than from their having some conglomerate status of "negativeness and divisibility by two." Similarly, -1 or +1 are attributed as less than or greater than 0, which alone determines their "negativeness" or "positiveness," independently of whether (or not) they happen to be divisible by 2.

Which is exactly my point. Two is positive and entails evenness but evenness is not inherently positive. This seems to pose a problem for his second axiom.

...Greater than or less than zero is not a property of a number? That makes certain sets rather hard to work with...

Here is where your equivocation comes in. The word "property" can have different meanings in different contexts. "Negative" or "positive" may be a "property" of a number in certain contexts, but Godel's axioms clearly show that these are not the sort of "properties" his theorem is concerned with.

Fair enough, but, as I said, we would then have to justify any property at all exists using his definition, something I consider to be extremely unlikely.

...How does that matter? Something can't have a negative length either, and yet size is an attribute.

All you're saying here is that length or size of physical objects, to the extent they can't be negated, are not properties within the context of Godel's theorem. So we are making some progress, it seems, in that we have thus far ruled out duration, length, size, etc. as falling within the parameters of Godel's "properties." We need to look elsewhere if we are to find the sort of "properties" that we need if we're going to get anywhere with Godel's theorem.

See above.

...Which drags us back to my objections to Axiom 1, both in defining it and then determining if any such can properties exist at all. In particular, if there is no moral aesthetic sense independent from the accidental structure of the world, there can be no such thing as positive properties as he defines them.

Well, you have said that numbers actually exist. But what is a number, any number, taken in isolation? The number "2" is only significant in relation to other numbers, such as "1" and "3." So I would say that if numbers exist (as you claim they do) then relationality must be a "positive" property in Godel's usage, where as "isolation" or "acontextuality" would be the negation of "relationality."

Perhaps we should try to come up with a list of "positive properties" (along with their negations) that meet the criteria of a Godelian "property." Once we have found examples of such "properties" that fit within the context that Godel has laid out, then we can work on a definition of those properties.

For example, (relationality vs. isolation) might be one set of "positive" vs. "negative" properties. Another example might be (logic vs. absurdity) given that numbers have no use or function or value apart from the logic of mathematics--that is, the meaning of 2 + 2 would be empty or inane unless this addition logically compels a result that can be nothing other than 4.

Perhaps we could flesh this out further by saying that (significance vs. inanity) is another set of positive vs. negative Godelian properties.

So far we have the following possibilities for sets of positive vs. negative "properties" within the framework of Godel's theorem:

1. Relationality vs. isolation
2. Contextuality vs. acontextuality
3. Logic vs. absurdity
4. Significance vs. inanity
5. Order vs. chaos

I submit that without the above positive properties, numbers could not "exist." Do you agree?

No. For numbers to have the range of usefulness we require for advanced systems using them, those things are all required, but the capacity to use the item is not inherent to the object itself. I don't think any of those things are necessary for numbers to exist, however, I would suppose they are necessary for mathematics, a form of categorical thinking and relation building to exist.

A number in and of itself is an instantiation of a value, it does not get is meaning from other numbers or from any given context or another, logic only comes into play in the relationship between objects, as does order. If a number is it's own meaning, I would grant significance, but only in the sense of "it is not nothing", however, this again borders dangerously close to Kant's existence not being predicate, which I think is a flaw of trying to define properties in terms of prerequisites for something to exist. Is something a property if it is fundamental to even the possibility of something existing, or would the asserted property being an object in and of itself independent of that which exists only because of it? That is to say, if X (significance) must exist before Y (numbers) could even possibly exist, can X rightly be said to be a property of Y?

I would also then question whether this approach could ever leave Godlikeness as a positive property if we can only define such properties as that for which things are necessary to exist.

[EduChris]

...Two is positive and entails evenness but evenness is not inherently positive. This seems to pose a problem for his second axiom.

It poses no problem for two reasons. Firstly, the number "two," all by itself, does not entail "evenness." "Divisibility by two" entails evenness. Secondly, the positive/negative status of a number is not even a "property" within the context of Godel's theorem.

...as I said, we would then have to justify any property at all exists using his definition, something I consider to be extremely unlikely.

You're getting the cart before the horse here. If numbers exist, then certain "positive properties" (that is, "existence affirming" properties) are required. If one can show that any of those "positive properties" do not exist, then you have shown that numbers do not exist. The real question is, are we prepared to assert that anything at all exists? If so, then that "something" must have certain "positive" properties, or "existence-affirming" properties.

...For numbers to have the range of usefulness we require for advanced systems using them, those things are all required, but the capacity to use the item is not inherent to the object itself. I don't think any of those things are necessary for numbers to exist, however, I would suppose they are necessary for mathematics, a form of categorical thinking and relation building to exist.

If numbers have no inherent capacity for use, then in what sense is it meaningful to say that they exist? Numbers are not physical objects that can be examined under a microscope. If they have no inherent capacity for use, how is that different from saying that numbers are irrelevant? How is that different from saying that humans make up numbers arbitrarily without any correlation to "how things really are"?

...A number in and of itself is an instantiation of a value, it does not get is meaning from other numbers or from any given context or another...

Certainly we can say that 2 is not 3, and 3 is not 1. If we cannot define 2 as "exactly that number which lies between 1 and 3," then how do we define two in a matter that distinguishes it from 1 or 3?

...logic only comes into play in the relationship between objects, as does order...

So are you admitting that "logic, relationality, and order" actually exist?

...If a number is it's own meaning, I would grant significance, but only in the sense of "it is not nothing"...

If a number were its own meaning, apart from its relationality with other numbers and apart from its distinctiveness compared to any other number, then we could say that "2 is not nothing" in exactly the same way that "4 is not nothing." In such a case, 2 would be no different from 4 (and of course we usually want to deny that 2 and 4 are equivalent).

...this again borders dangerously close to Kant's existence not being predicate, which I think is a flaw of trying to define properties in terms of prerequisites for something to exist. Is something a property if it is fundamental to even the possibility of something existing, or would the asserted property being an object in and of itself independent of that which exists only because of it? That is to say, if X (significance) must exist before Y (numbers) could even possibly exist, can X rightly be said to be a property of Y?

I think this gets back to the "cart before the horse" situation that we addressed previously. We do not say that "X" exists and therefore "Y" exists; rather, we say that IF "X" exists, THEN "Y" must necessarily apply. In the case of numbers, it might be possible to say that relationality can exist without numbers, but whether or not that is true, it is certainly the case that IF numbers exist, THEN Relationality must be applicable.

Beyond this, has Kant done anything to "prove" his own views? Is Kant using "existence" and "predicate" in a way that fits the context of what Godel is trying to do? Are you granting Kant a greater "expert status" than Godel? I don't know the answers to these questions, but I am throwing the questions out for consideration.

...I would also then question whether this approach could ever leave Godlikeness as a positive property if we can only define such properties as that for which things are necessary to exist.

This appears to be a different matter than the definition of "positive" properties or "existence affirming" properties in terms of Godel's theorem. Shall we bring this matter up again after we have settled upon a definition for Godelian "positive properties"?

[Abraxas]

...Two is positive and entails evenness but evenness is not inherently positive. This seems to pose a problem for his second axiom.

It poses no problem for two reasons. Firstly, the number "two," all by itself, does not entail "evenness." "Divisibility by two" entails evenness. Secondly, the positive/negative status of a number is not even a "property" within the context of Godel's theorem.

Which is why I said entails evenness, not is evenness. You cannot have two without evenness, two entails it.

...as I said, we would then have to justify any property at all exists using his definition, something I consider to be extremely unlikely.

You're getting the cart before the horse here. If numbers exist, then certain "positive properties" (that is, "existence affirming" properties) are required. If one can show that any of those "positive properties" do not exist, then you have shown that numbers do not exist. The real question is, are we prepared to assert that anything at all exists? If so, then that "something" must have certain "positive" properties, or "existence-affirming" properties.

On the contrary, given the highly restricted nature of the definition of properties, the only thing that seems to qualify is existence itself, which is ruled out ala Kant. Everything else, as far as we can determine, is an accident of the universe. I would also request you defend the idea that "existence-affirming" properties, as Godel is using it, are required for anything to exist, as opposed to the more common use of the word being sufficient to fit the bill.

...For numbers to have the range of usefulness we require for advanced systems using them, those things are all required, but the capacity to use the item is not inherent to the object itself. I don't think any of those things are necessary for numbers to exist, however, I would suppose they are necessary for mathematics, a form of categorical thinking and relation building to exist.

If numbers have no inherent capacity for use, then in what sense is it meaningful to say that they exist? Numbers are not physical objects that can be examined under a microscope. If they have no inherent capacity for use, how is that different from saying that numbers are irrelevant? How is that different from saying that humans make up numbers arbitrarily without any correlation to "how things really are"?

Simple, usefulness is a relationship that may or may not exist. For something to be useful, something must be able to use it. This is not a property of an object, but rather a descriptor of how other incidental things can relate to the object. The difference lies in your equivocation, numbers have all kinds of observed properties, they simply don't conform to "positive properties" or, as you keep using "existence-affirming properties" as you define them.

...A number in and of itself is an instantiation of a value, it does not get is meaning from other numbers or from any given context or another...

Certainly we can say that 2 is not 3, and 3 is not 1. If we cannot define 2 as "exactly that number which lies between 1 and 3," then how do we define two in a matter that distinguishes it from 1 or 3?

1.7 is 2 under your definition. Further, I'm not sure definition is really appropriate. I would be hard pressed to come up with a definition of the keyboard I am using. I would be hard pressed to come up with a definition of you or me, for that matter. Sure, I could define keyboards in general, listing the specific characteristics of it exhaustively, but is that the definition of this keyboard or just a complete description? This conversation can only end in a debate on what constitutes identity which is rather off topic, but two is the number that has the identity two. Two is the name of a specific object. You are treating it like a concept to be defined rather than an object to be described, which, in my view, is a category mistake.

...logic only comes into play in the relationship between objects, as does order...

So are you admitting that "logic, relationality, and order" actually exist?

They are forms of categorical thinking, yes, but, being systematic rather than instantiated, they are not properties or defining characteristics of individual items but rather descriptors of relationships between items as we perceive them, and, they seem to be accidental rather than necessary at that. They would not qualify as positive properties (or really, properties at all) as the term is being used.

...If a number is it's own meaning, I would grant significance, but only in the sense of "it is not nothing"...

If a number were its own meaning, apart from its relationality with other numbers and apart from its distinctiveness compared to any other number, then we could say that "2 is not nothing" in exactly the same way that "4 is not nothing." In such a case, 2 would be no different from 4 (and of course we usually want to deny that 2 and 4 are equivalent).

Two is that object which has the identity two. Four is that which has the identity four. There are a number of properties in the traditional sense that apply, but not in the "existence affirming" sense. Otherwise it just boils down to the identity of the object.

...this again borders dangerously close to Kant's existence not being predicate, which I think is a flaw of trying to define properties in terms of prerequisites for something to exist. Is something a property if it is fundamental to even the possibility of something existing, or would the asserted property being an object in and of itself independent of that which exists only because of it? That is to say, if X (significance) must exist before Y (numbers) could even possibly exist, can X rightly be said to be a property of Y?

I think this gets back to the "cart before the horse" situation that we addressed previously. We do not say that "X" exists and therefore "Y" exists; rather, we say that IF "X" exists, THEN "Y" must necessarily apply. In the case of numbers, it might be possible to say that relationality can exist without numbers, but whether or not that is true, it is certainly the case that IF numbers exist, THEN Relationality must be applicable.

You completely missed my point. My point was that, going from your example, it is debatable as to whether you could then go on to say relationality is a property of numbers or something independent, but applicable.

Beyond this, has Kant done anything to "prove" his own views? Is Kant using "existence" and "predicate" in a way that fits the context of what Godel is trying to do? Are you granting Kant a greater "expert status" than Godel? I don't know the answers to these questions, but I am throwing the questions out for consideration.

Yes, Kant specifically formulated the objection to ontological arguments and, in "Critique of Pure Reason" Kant defended it quite thoroughly, to the degree in logic and philosophy it is considered a settled issue. Plantinga, as I recall, attempted to work around that by claiming necessary existence was even though existence wasn't, however, I never found the argument particularly compelling. As far as who is the greater logician, I don't know, both were top notch, however, the strength of their arguments are a separate question from how skilled they are. On this topic, I consider Kant to have the stronger argument.

http://en.wikipedia.org/wiki/Ontologica ... _predicate

...I would also then question whether this approach could ever leave Godlikeness as a positive property if we can only define such properties as that for which things are necessary to exist.

This appears to be a different matter than the definition of "positive" properties or "existence affirming" properties in terms of Godel's theorem. Shall we bring this matter up again after we have settled upon a definition for Godelian "positive properties"?

If you like.