Meow Mix wrote:EduChris wrote: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.
EduChris wrote: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.
With all due respect to you, if defeating Godel's argument was this easy, then I'm sure much better logicians and philosophers than any of us would have done so decades ago.
All you have really done here is conjectured a similar, but different argument using another metaproperty. I'm not entirely sure you could substitute an Intrisic operator in place of Godel's Positivity operator and still have a valid argument. Whether or not your argument is valid doesn't really matter as it does not address Godel's. His argument has to be addressed "on its own turf", so to speak.
Getting back to Axiom 3, Godel's argument does not state that just any metaproperty would be preserved under aggregation, just positivity, however this does not follow from the rest of the argument. C Small gives some indication on his website that it could be possible to prove this, but doing so for a potentially unlimited number of properties would be extremely tedious. He also had this to say which I found interesting (
full link here):
It might well be said that Gödel’s ontological argument stands or falls on the coherence and the interpretation of his concept of the positive. So it is of particular importance that we examine Gödel’s intentions in this matter, as well as other possible interpretations. To do so, we shall consider Gödel’s axioms of positivity
and privation in light of two semantic systems, which I shall call Leibnizian semantics and Plotinian semantics. The two semantic interpretations are roughly analogous to the One and the Many as described in Plato's Parmenides, the debate being resolved in favour of the God as conjoined complexity in Leibnizian semantics, and undifferentiated unity in Plotinian semantics. Both interpretations are ontological, rather than moral-aesthetic.
In Leibniz' metaphysics, the collection of all attributes is constructed by combining
together simple attributes using the rules of conjunction, disjunction or negation. Simple attributes are always positive, although positive attributes need not be simple. When simple attributes are “combined� using conjunction, positivity is preserved. Gödel would agree with this, and would add that when attributes are “relaxed� by disjunction, positivity is preserved whenever at least one of the attributes is simple. In this system, all attributes, including those which are not positive can be written as a Boolean combination of simple positive ones. Privation enters the picture because a Boolean combination can include the negation of the simple attributes.
Given what seems to be Godel's underlying philosophy regarding properties, and absent effective argumentation (that specifically addresses positivity) to the contrary, I see no reason not to accept Axiom 3 at face value at this time.
That said, I am probably not any more convinced by his argument than you are. I feel like there is a fatal flaw in it somewhere, but I am having a hard time pinpointing it. The logic itself seems to be valid, as you would expect from one of the best logicians of the 20th century.
Here is where I would look for a problem:
1) Are there any positive properties that contradict each other when exemplified together?
2) Does Godel's argument lead to a state of affairs where everything is necessary?
3) While maybe not explicit, is there any implied circularity between Godel's conclusion that the godlike property is exemplified in all possible worlds and the other parts of the argument? (This one I am giving much thought to, myself.)
