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?

[ScotS]

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.

Ack, I think you are right. I probably shouldn't be posting while busy at work. Correct application of G to Axiom 1 would be:

{P(G) ∧□∀x[G(x) → p(x)]} →P(p)

which doesn't tell us anything useful.

-------------
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.

It's a lot frustrating that wiki presents it differently and I agree that the actual argument is more precise than wiki shows (although I admit I am still muddling through it trying to figure out what difference the precision makes.)

[ScotS]

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.

See my previous reply to Zeeby.

However, I think the key to Axiom 3 is in Godel's definition of positive, as opposed to just any metaproperty. "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)." It seems unreasonable to me that a property that is defined by the conjunction of only other properties that contain no privation would itself contain privation.

[EduChris]

...why should we assume that a set containing elements with some metaproperty would itself have the same metaproperty without further argument?...

Godel's axiom appears reasonable to me and to others here on this thread--and certainly to Godel, who must be considered an "expert witness" on the subject. Given this, perhaps the burden is on you to provide some counterexample that clearly shows why the axiom should be rejected. Your "extrinsicity" example involved measurement (length) which is clearly not a "positive Godelian property." Can you provide an example of a set of Godelian positives such that the conjunction of the members of the set turns out to be anything other than a Godelian positive? I don't believe that you can (but if you could, then I would almost certainly have to concede this debate).

...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.

Godel's filter is far from arbitrary. He intends to obtain a certain set of properties which are consistent with one another and which do not depend on the accidental structure of the world. From there, he demonstrates by modal logic that one and only one entity manifests all of the properties in that set, which includes necessary existence in all possible worlds. Why anyone should consider the filter "arbitrary" is beyond me, especially since I have gone beyond Godel and tied the properties to the "superpositives" or "incorrigibles" that pertain in all possible worlds.

Even if my selection of properties (necessary existence, differentiation, relationality, et. al.) were simply picked at random (and they weren't) it wouldn't matter in the least. As long as that set of properties passed Godel's filter, then modal logic would tell us that there exists an entity which has all of the properties in the set.

[Meow Mix]

See my previous reply to Zeeby.

However, I think the key to Axiom 3 is in Godel's definition of positive, as opposed to just any metaproperty. "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)." It seems unreasonable to me that a property that is defined by the conjunction of only other properties that contain no privation would itself contain privation.

Interesting. If so, we can define it as the set that contains no properties with privation. Isn't this in itself privation?

On a more whimsical note, does the set of all sets which do not contain privation contain itself?

[EduChris]

...the set that contains no properties with privation. Isn't this in itself privation?...

Is my net worth diminished if I have no debts? Probably not, since we are talking here only about a specific property that is measured in monetary units.

But experientially am I deprived if I never have first-hand knowledge of what it means to pay back a loan? In essense, I believe this is the rationale behind the incarnation in Christian theology: God assumes human nature so that he can have first-hand experience of the human condition.

[Meow Mix]

Is my net worth diminished if I have no debts? Probably not, since we are talking here only about a specific property that is measured in monetary units.

But experientially am I deprived if I never have first-hand knowledge of what it means to pay back a loan? In essense, I believe this is the rationale behind the incarnation in Christian theology: God assumes human nature so that he can have first-hand experience of the human condition.

I'm not sure what you're getting at with your analogy. Privation in the philosophical sense is just the lack of something. If Godel's "positive" is a metaproperty meaning without privation, then there is no problem entailed with that on the first step. However if you try to compose a set containing no properties without that metaproperty then the set itself doesn't have the metaproperty: so Axiom 3 would be false. The set of all sets that don't have privation would not itself lack privation (i.e., the set of all sets which are positive would not itself be positive).

Thus Godel's Ontological Proof goes wrong with Axiom 3 and fails.

[Meow Mix]

Godel's filter is far from arbitrary. He intends to obtain a certain set of properties which are consistent with one another and which do not depend on the accidental structure of the world. From there, he demonstrates by modal logic that one and only one entity manifests all of the properties in that set, which includes necessary existence in all possible worlds. Why anyone should consider the filter "arbitrary" is beyond me, especially since I have gone beyond Godel and tied the properties to the "superpositives" or "incorrigibles" that pertain in all possible worlds.

Even if my selection of properties (necessary existence, differentiation, relationality, et. al.) were simply picked at random (and they weren't) it wouldn't matter in the least. As long as that set of properties passed Godel's filter, then modal logic would tell us that there exists an entity which has all of the properties in the set.

You're misunderstanding why it's arbitrary. The point I've been trying to make is that just because you can filter properties based on a metaproperty, it doesn't mean that you can slap all those properties together in the same set as if that set represents anything.

Go back to my Gojira analogy. Say I want to filter blue stuff from non-blue stuff non-arbitrarily. In this case it's simple: I just look at stuff and see if it's blue or not. I end up with blue jeans, a gatorade bottle (again!), scissor handles, a thumb tack, etc.

Ok, how nice -- I've non-arbitrarily filtered objects that appear blue to me from not. I can pat myself on the back.

The arbitrary part comes where I decide to slap some or all of them together in a set and treat that set as if it's something that exists; as if these objects have some explicit relation to one another just because they share the same property or metaproperty: can I slap together a set {blue jeans, gatorade, scissors, thumb tack} and call that set Gojira and then exclaim that Gojira exists and ask for someone to prove that Gojira doesn't exist? It would be silly of me to do that -- what reason do I have for throwing together things into a set just because they pass through some filter based on some property or metaproperty and what business do I have declaring the set itself is non-arbitrary?

The answer is that it's completely arbitrary.

[EduChris]

...if you try to compose a set containing no properties without that metaproperty then the set itself doesn't have the metaproperty: so Axiom 3 would be false...Thus Godel's Ontological Proof goes wrong with Axiom 3 and fails.

I don't think you can say that the lack of privation is itself a privation (double negatives and all). If I make a complete list of my assets (deliberately omitting all of my liabilities) there is no sense in which you could say that the resulting set is marked by liability. It would only exhibit privation if it failed to provide a complete list of all of my assets (not that I have many of them, of course).

[EduChris]

...what reason do I have for throwing together things into a set just because they pass through some filter based on some property or metaproperty and what business do I have declaring the set itself is non-arbitrary?...

Godel has demonstrated that a set of properties that filter through his axioms will necessarily be exhibited in all possible worlds by a single entity. Your set of blue things, by contrast, would not filter through the axioms, and therefore there is no modal logic (that I know of) which demonstrates the necessary existence of an entity comprised of all blue things.

[Meow Mix]

I don't think you can say that the lack of privation is itself a privation (double negatives and all). If I make a complete list of my assets (deliberately omitting all of my liabilities) there is no sense in which you could say that the resulting set is marked by liability. It would only exhibit privation if it failed to provide a complete list of all of my assets (not that I have many of them, of course).

I think it's too ill-defined for us to be able to tell either way.

Godel has demonstrated that a set of properties that filter through his axioms will necessarily be exhibited in all possible worlds by a single entity. Your set of blue things, by contrast, would not filter through the axioms, and therefore there is no modal logic (that I know of) which demonstrates the necessary existence of an entity comprised of all blue things.

Fine, let's say that we use the same filter for a different metaproperty; one which will be in accordance with similar axioms: intrinsicity.

So let's say:
Axiom 1: It is possible to single out INTRINSIC properties from among all properties.

We then assume that the following three conditions hold for all INTRINSIC properties (which can be summarized by saying "the INTRINSIC properties form a principal ultrafilter"):

Axiom 2: If P is INTRINSIC and P entails Q, then Q is INTRINSIC.
Axiom 3: If P1, P2, P3, ..., Pn are INTRINSIC properties, then the property (P1 AND P2 AND P3 ... AND Pn) is INTRINSIC as well.

Finally, we assume:

Axiom 4: Necessary existence is an INTRINSIC property (Pos(NE)).

Now just plug it into a similar argument to Godel's and you'll come up with the "answer" that sets containing elements with the metaproperty will also have the metaproperty and that includes "necessary existence" and so we can "conclude" that some infinity of sets with this metaproperty exist.

It's absurd. And so is Godel's.