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:
WikiPedia wrote: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, ..., Pn are positive properties, then the property (P1 AND P2 AND P3 ... AND Pn) 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?
AkiThePirate wrote:...by what means are they determined to be positive within the criteria you have chose?...
I believe that only Rationality (as opposed to irrationality) is compatible with the super-positive "Differentiation." The assertion that "X is not Y" cannot be maintained in the absence of rationality (or reason or logic or whatever term you wish to use).
Irrationality would permit "X is not Y" and "X is Y" to be equally acceptable, at the same time and in the same sense. Since irrationality is incompatible with Differentiation, irationality must be "negative," and its counterpart, Rationality, must be positive.
AkiThePirate wrote:I don't think the possibility of an ideas conception has any bearing on reality, and as such I fail to see why an argument would posit such.
We are not talking about "just any idea" or "just any conception." We are talking about "positive properties" (in the Godelian sense).
In other words, the notion that "a unicorn exists" has no necessary validity (because "unicornness" is not a positive property); but in order for anything at all to "exist," the positive property of "existence" is necessary.
This is why we're spending so much time trying to get a handle on Godelian "positive properties." I think we've made a great deal of progress so far.
For existence, existence is a necessary property, but I'm still not seeing how it is a positive property. If I'm not mistaken, the argument holds that it is.
AkiThePirate wrote:For existence, existence is a necessary property, but I'm still not seeing how it is a positive property. If I'm not mistaken, the argument holds that it is.
Our discussion has considered this in some depth already, and Abraxas has (tentatively, for purpose of discussion) allowed Existence, Differentiation, and Relationality as comprising a trinity of "super-positive properties" that hold for all conceivable universes, given our initial assumption that "a universe such as ours is conceivable."
Although Abraxas' knowledge with philosophical concepts quite outranks mine, I do seem to follow most of his points.
From the three properties you've given, I still can't see how a positive property can be defined and determined as it seems entirely subjective in nature, as Abraxas raised in the post to which you linked.
I am still thinking this all over but besides the history of forceful argument criticizing the ontological argument in general and those in particular to Godel there are other problems I have with this line of reasoning provided by EduChris.
The problems are many.
If God were the ground of all being then everything that exists would be a emergent property of Gods nature including all positive or negative properties and defining God as all positive properties has no meaning except as an assumed property of God that is all things positive. Why exclude negative propertied from God the ground of all being or existence? Seem rather arbitrary except by desire to say God is of a certain nature already assumed, which is all positive properties. It is arbitrary, circular and question begging besides unfounded.
We see both positive and negative properties exist just as much as the universe so unless you presume some kind of dualism there is no justification for excluding negative properties, except maybe desire.
If we are talking about moral properties we fail to take into account the problems wit morality. It looks like you are just jumping from claim to claim without ever showing anything is true as correspondences to the actual universe which includes negative moral qualities.
I dont think it is God that makes morality as much as God would recognize morality or valuing social relationships and bond that include sympathy, affection and the desire to good which is relative except maybe in the extreme.
There is more then exist and not exist as there is imagination.
It is the confusion of language that the argument is taking advantage.
That sounds backwards.
Something can exist in my imagination and not exist in reality and I can imagine the impossible which can be expressed as existing, at least in our minds.
What are God like qualities except for those we insist God already has defined as positive to humans and assumed by believers?
Why should God be all powerful except for our desire for power? Why cant God be limited in power yet unlike us Gods power would never fail God?
Why can all knowing mean know all contingent knowledge?
If God is the ground of all being then why be redundant and demand God is everywhere too? It seems God would already be there as the very ground of all being. If God is outside of time and space then how is God the ground of All being or even related to the universe?.
Why should god-like properties be positive or better yet why would positive properties be god-like except by definition b and because for god knows why, you demand be such that you think God should be?
Not only do you want to impose positive qualities upon God, because of dogma or such, but you insist God be defined by those very same properties. It is like saying for God to be God, God must be God. I can hardly think anyone can really conceive of the one universe we are apt to assume exists but how does one conceive of God which by definition should be inconceivable unless we are going to commit some form of idolatry and insist our concepts of God are really God himself. It would seem even he most arrogant of atheists and agnostic would be humbled by any concept of God, or the universe being what we should insist it should be without it being obvious and so necessary we could not help but take notice.
If we are gong to have three positive properties, why stop there unless you just favor for dogmatic reason you need three, why not have an infinite or even really billions of positive properties? Surly you can imagine a God having more then three positive properties? I might wonder how much you could conceive but I am sure we can imagine almost anything, but not everything.
God even if necessary should also be .more then just necessary (given just contingencies in a universe we have already assumed) showing how it is relevant to unnecessary or contingent things.
AkiThePirate wrote:Although Abraxas' knowledge with philosophical concepts quite outranks mine, I do seem to follow most of his points.
From the three properties you've given, I still can't see how a positive property can be defined and determined as it seems entirely subjective in nature, as Abraxas raised in the post to which you linked.
It seems it would also be subjective to nature as well as in nature. A positive property seems to be a value dependent upon a limited but accumulating number of relationships and some yet to be defined satisfactions.
I grant that we all desire the good for good's sake even if it is relative, conditional and contingent. But I tend to think it is a mistake to mistake essence for values we make of events, relationships, outcomes or whatever.
Evil and good are something that is done not something that is.
