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?

[Meow Mix]

Axiom 3: If P1, P2, P3, ..., Pn are positive properties, then the property (P1 AND P2 AND P3 ... AND Pn) is positive as well.

I find no reason to accept this axiom without further argumentation. For instance, while I agree that things like "differentiality" and "relationality" are ostensibly positive (considering Godel hasn't really given us a strong definition for what "positive" means) there is no reason to assert that the subset {differentiality & relationality} is positive.

The other axioms seem to be okay because, for instance:

(Axiom 2) It does follow that if a property p is positive and it entails another property q that property q is also positive.

(Axiom 4) There is no problem with the definition that if p is a property that either it or its negation is positive, but not both.

However, it seems to go wrong with Axiom 3. (I'd also argue that it goes wrong with Axiom 5, but that's a separate issue).

What is "positiveness?" In the end it doesn't matter so much other than that it's definitely a property of a property. "Positiveness" isn't denoting an object but rather a property, in other words.

So, let's look at the structure of Axiom 3 in general. It looks something like this:

If P1, P2, P3, ..., Pn are properties with the property X, then the property (P1 & P2 & P3 ... & Pn) has the property X as well.

Is the general form of the axiom true? Can we plug in any "property of a property" as X and end up with a true statement, or something that's intuitive to accept? Certainly not.

Consider the property of properties "extrinsicity." An extrensic property is one which depends on the object it describe's relationship with other things. For instance, "length" is an extrinsic property because it depends on the spatiotemporal extension of the object in question, which can be situational (especially in relativistic physics). That is to say, the property of "length" itself has a property of being extrinsic.

So, is it fair to say that all "properties of properties" can form meta-sets which themselves share the "property of a property" in question? Let's refer back to the general form of Axiom 3 above and plug in "extrinsic:"

Is the following axiom intuitively true, or does it even make sense?

If P1, P2, P3, ..., Pn are extrinsic properties, then the property (P1 & P2 & P3 ... & Pn) is extrinsic as well.

The answer is that it isn't at all intuitively true, nor does it seem to make sense. So, why do we get to declare that it makes sense for a meta-set of positive properties to be positive in itself? I hardly see a way to produce a clear answer, and thus I hardly see any reason for us to accept Axiom 3 whatsoever without further argumentation. A collection of properties with property X does not necessarily form a meta-set which also has property X.

[Zeeby]

...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.

That's fine. I've been doing some reading myself while reading the Arguments against God thread.

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.

This goes against everything I've read so far on the subject - the Wikipedia article gives examples of properties as "P(s) = s is pink" or "P(x) = x is taller than 2 meters". The more detailed article at http://sas.uwaterloo.ca/~cgsmall/ontology2.html states

Just as a predicate or property provides a truth-functional assignment to individuals (i.e., Rx, where "x=Santa Claus" and R="wears a red suit"), so the positivity operator Pos provides a truth-functional assignment to properties themselves.

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.

Axiom 1 defines positive properties (or rather, claims that they are definable). It does not limit what a general property can be.

Would it be an issue if the God object had non-positive properties? Must a being that exists non-contingently also act non-contingently?


Meow Mix has already raised some of the issues I have with the axioms. For a more concrete example of the difficulties Axiom 3 faces, consider Arrow's impossibility theorem. The 3 properties presented in the article are aesthetic and independent of the accidental political parties of the world. Yet it is impossible for all of them to be satisfied (and so the collection is not a positive voting system property).

Obviously God is not a voting system, but hopefully the analogy will explain my doubts.

[EduChris]

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.

This goes against everything I've read so far on the subject - the Wikipedia article gives examples of properties as "P(s) = s is pink" or "P(x) = x is taller than 2 meters". The more detailed article at http://sas.uwaterloo.ca/~cgsmall/ontology2.html states

Just as a predicate or property provides a truth-functional assignment to individuals (i.e., Rx, where "x=Santa Claus" and R="wears a red suit"), so the positivity operator Pos provides a truth-functional assignment to properties themselves.

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.

Axiom 1...does not limit what a general property can be... [emphasis added]

As I stated previously, you are assuming that Godel is claiming that all putative "properties" conform to his axioms. But in fact, that is not what Godel is claiming at all. Godel is using his axioms to define the ways in which "positive properties" differ from other properties which are "accidental" rather than "positive."

...it seems to go wrong with Axiom 3...let's look at the structure of Axiom 3 in general...Is the general form of the axiom true? Can we plug in any "property of a property" as X and end up with a true statement, or something that's intuitive to accept? Certainly not...

This appears to be the same objection that Zeeby and Abraxas have raised previously--and the answer is the same regardless of how the objection is stated. Godel is obviously not trying to say that his axioms are true for anything at all that might be considered a "property." He is using his axioms to filter out "properties" that correspond to the accidental structures of the world.

Accidental "properties" are simply variations on a more general theme. Godel is interested in the general themes rather than all of the endless variations on a particular theme. For example, any attribute that has to do with measurement (space, time, color spectrum) is simply a variation on the general theme of being bound or unbound in some quantifiable way within finite spatio-temporal dimensions. Similarly, notions of "having created horses, or not having created unicorns" are simply variations on the general themes of creativity and volition.

At this point, I am not interested in defending any interpretation of Godel that assumes he is trying to come up with axioms which hold for anything at all that might be construed as a "property." I am interested in defending the interpretation of Godel wherein his axioms serve as a "filter" to weed out endless variations on general themes.

Can we agree to limit our discussion to the interpretation of Godel which I wish to defend, rather than some other interpretation which I do not wish to defend? Given my interpretation of Godel, what objections might you raise?

[Zeeby]

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.

This goes against everything I've read so far on the subject - the Wikipedia article gives examples of properties as "P(s) = s is pink" or "P(x) = x is taller than 2 meters". The more detailed article at http://sas.uwaterloo.ca/~cgsmall/ontology2.html states

Just as a predicate or property provides a truth-functional assignment to individuals (i.e., Rx, where "x=Santa Claus" and R="wears a red suit"), so the positivity operator Pos provides a truth-functional assignment to properties themselves.

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.

Axiom 1...does not limit what a general property can be... [emphasis added]

As I stated previously, you are assuming that Godel is claiming that all putative "properties" conform to his axioms. But in fact, that is not what Godel is claiming at all. Godel is using his axioms to define the ways in which "positive properties" differ from other properties which are "accidental" rather than "positive."

...it seems to go wrong with Axiom 3...let's look at the structure of Axiom 3 in general...Is the general form of the axiom true? Can we plug in any "property of a property" as X and end up with a true statement, or something that's intuitive to accept? Certainly not...

This appears to be the same objection that Zeeby and Abraxas have raised previously--and the answer is the same regardless of how the objection is stated. Godel is obviously not trying to say that his axioms are true for anything at all that might be considered a "property." He is using his axioms to filter out "properties" that correspond to the accidental structures of the world.

Accidental "properties" are simply variations on a more general theme. Godel is interested in the general themes rather than all of the endless variations on a particular theme. For example, any attribute that has to do with measurement (space, time, color spectrum) is simply a variation on the general theme of being bound or unbound in some quantifiable way within finite spatio-temporal dimensions. Similarly, notions of "having created horses, or not having created unicorns" are simply variations on the general themes of creativity and volition.

At this point, I am not interested in defending any interpretation of Godel that assumes he is trying to come up with axioms which hold for anything at all that might be construed as a "property." I am interested in defending the interpretation of Godel wherein his axioms serve as a "filter" to weed out endless variations on general themes.

Can we agree to limit our discussion to the interpretation of Godel which I wish to defend, rather than some other interpretation which I do not wish to defend? Given my interpretation of Godel, what objections might you raise?

I want to clarify that I have understood your interpretation correctly: only some properties may be substituted in the axioms, and out of these, only some are positive?

[EduChris]

...I want to clarify that I have understood your interpretation correctly: only some properties may be substituted in the axioms, and out of these, only some are positive?

If some ostensible "property" is proposed, we apply the axiom filter. If (and only if) the putative property is compatible with the axioms, then it truly is a property in the Godelian sense. And if this property is compatible with the "superpositives" then it is "positive" in the Godelian sense; otherwise, it is "negative" in the Godelian sense.

I also would say that the majority of putative "properties" are reducible to more general themes, and Godel is strictly interested in the general themes rather than the endless variations on those general themes.

[ScotS]

Axiom 3: If P1, P2, P3, ..., Pn are positive properties, then the property (P1 AND P2 AND P3 ... AND Pn) is positive as well.

I find no reason to accept this axiom without further argumentation. For instance, while I agree that things like "differentiality" and "relationality" are ostensibly positive (considering Godel hasn't really given us a strong definition for what "positive" means) there is no reason to assert that the subset {differentiality & relationality} is positive.

The other axioms seem to be okay because, for instance:

(Axiom 2) It does follow that if a property p is positive and it entails another property q that property q is also positive.

(Axiom 4) There is no problem with the definition that if p is a property that either it or its negation is positive, but not both.

However, it seems to go wrong with Axiom 3. (I'd also argue that it goes wrong with Axiom 5, but that's a separate issue).

Meow Mix,

If you accept Axiom 1, {P(p) x[p(x) q(x)]} P(q), (Wiki has this as axiom 2, which is wrong.), then what Wiki calls Axiom 3 (which is actually a combination of Def 1 and Axiom 3), logically follows.

{P(p) x[p(x) G(x)]} P(G)

[EduChris]

...If you accept Axiom 1, {P(p) x[p(x) q(x)]} P(q), (Wiki has this as axiom 2, which is wrong.), then what Wiki calls Axiom 3 (which is actually a combination of Def 1 and Axiom 3), logically follows.

{P(p) x[p(x) G(x)]} P(G)

This link from ScotS may be of general help for obtaining a better grasp of Godel's argument

[Zeeby]

Meow Mix,

If you accept Axiom 1, {P(p) x[p(x) q(x)]} P(q), (Wiki has this as axiom 2, which is wrong.), then what Wiki calls Axiom 3 (which is actually a combination of Def 1 and Axiom 3), logically follows.

{P(p) x[p(x) G(x)]} P(G)

Could you prove this? I can't find a way to combine properties solely using axiom 1 - indeed it seems that axiom 1 can only be used to find weaker positive properties (e.g. from Wikipedia "P(x) = 'x is taller than 2 meters' entails the property Q(x) = 'x is taller than 1 meter'"), whereas "Axiom 3" (as you pointed out, equivalent to Def 1 and the modal Axiom 3) finds stronger positive properties.

-------------

As a general note regarding the Wikipedia article, it is a little frustrating that the modal argument is presented differently from the 'Axioms' section of the article. I am generally trying to refer to the modal argument because mathematical language has a precise meaning. It may also be worth noting that the modal argument has been fixed slightly since the OP was made (so bogus lines such as "Th. 3. G(q) -> G ess x" have been changed to "Th. 3. G(x) -> G ess x", and so on), which may improve understanding.

[Meow Mix]

This appears to be the same objection that Zeeby and Abraxas have raised previously--and the answer is the same regardless of how the objection is stated. Godel is obviously not trying to say that his axioms are true for anything at all that might be considered a "property." He is using his axioms to filter out "properties" that correspond to the accidental structures of the world.

Accidental "properties" are simply variations on a more general theme. Godel is interested in the general themes rather than all of the endless variations on a particular theme. For example, any attribute that has to do with measurement (space, time, color spectrum) is simply a variation on the general theme of being bound or unbound in some quantifiable way within finite spatio-temporal dimensions. Similarly, notions of "having created horses, or not having created unicorns" are simply variations on the general themes of creativity and volition.

At this point, I am not interested in defending any interpretation of Godel that assumes he is trying to come up with axioms which hold for anything at all that might be construed as a "property." I am interested in defending the interpretation of Godel wherein his axioms serve as a "filter" to weed out endless variations on general themes.

Can we agree to limit our discussion to the interpretation of Godel which I wish to defend, rather than some other interpretation which I do not wish to defend? Given my interpretation of Godel, what objections might you raise?

Well, if we limit our discussion to your interpretation of Godel -- which, without further argumentation is utterly arbitrary -- then we can simply reject Axiom 3 and declare the argument as you interpret it has failed to convince us and has no force.

Your interpretation makes an assumption that's ultimately dubious -- why should we assume that a set containing elements with some metaproperty would itself have the same metaproperty without further argument? The answer is that we shouldn't, and so Godel's proof fails unless it's preaching to the choire.

I get that it's just a filter, but the same argument that I've been making in the other thread -- that the choice of filter can be arbitrary and that said arbitrariness undermines any objectivity the argument is supposed to have -- has still gone unanswered.

[Meow Mix]

Meow Mix,

If you accept Axiom 1, {P(p) x[p(x) q(x)]} P(q), (Wiki has this as axiom 2, which is wrong.), then what Wiki calls Axiom 3 (which is actually a combination of Def 1 and Axiom 3), logically follows.

{P(p) x[p(x) G(x)]} P(G)

No, I don't see how it logically follows. It does follow that if some property p entails a property q that if p has some metaproperty that q would have that metaproperty; but it does not follow that a set of properties sharing the same metaproperty would itself have the same metaproperty.

Thus, indeed if p is positive and p entails q we can assert that q is positive. However, given a set S = {p, q, r} where p/q/r are positive we can't necessarily make the statement that S is positive without further evidence or argumentation.

If you disagree, then please explicitely show how this would be the case. It isn't there in the modal logic from what I can see.