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]

In that I'm not educated at all in philosophy or theology, where I ask a question here I am genuinely interested.
I'm not being rhetorical as I'm sure you've grown accustomed to(But hopefully not from me).

You are far too hung up on a particular slant on "positive." In Godel's usage, the term should be "non-arbitrary."

I fail to see how such a property proves the existence of a supreme being in the context of this or any other ontological argument.

Could you outline how such properties show the existence of such a being?

That objection is handled by my move to superpositives, which are entailed for all conceivable universes.

Being honest, I can't actually agree with you that differentiation is a property in our universe.
Quantum theory will mess up your philosophy.

Kant was objecting to earlier ontological formulations, and Godel's Proof avoids Kant's objection.

I was of the opinion that it necessitated that existence was a property of this supreme being. If not, I fail to see how it's an argument at all.

[Grumpy]

EduChris

Put up or shut up. I challenge you to come up with one property which is:

1) consistent with my super-positive properties of Existence, Differentiation, Relationality, and Information

2) able to filter through Axioms 2 through 4

3) either counterintuitive or damaging to my case or not already implicit within the superpositives or within my other "positive" (i.e., non-arbitrary) properties of Consciousness, Volition, Creativity, and Love.

If you can do that, it will either strengthen my argument or undermine it. I welcome any advance in either direction.

You are kidding, right? You want me to argue using your convoluted "logic"? I have a better idea. How about you(since it is you pushing the proof)show how something we imagine MUST be real, and why everything we imagine is not real. I imagine an invisible pink Unicorn. It is the most beautiful pink, it is totally invisible. No greater thing than that could possibly be imagined. Why, if existence is the greater vis-a-vis non-existence, it MUST exist. Therefore we are neck deep in invisible pink Unicorns, in fact , invisible pink Unicorns are all that exist. Since the property of invisible pinkness is, by definition the greatest Unicorn attribute and the greatest entails that which is the ultimate good, and the ultimate good requires existence(as compared to non-existence)then everything is pink but invisibly so, even the blues and greens are, in reality, invisibly pink......

The above is non-sense, of course, but then so is Anselm's argument(well, not the argument itself, but the claim that the argument has anything to do with reality), and any similar "proof" based on it(Godel included). That the proof is valid and logically consistent means nothing if the Axioms are not based on reality. As Gould put it...

"The final proofs of logic and mathematics flow deductively from stated premises and achieve certainty only because they are not about the empirical world.

(Yes, real scientists do have a poor opinion of such sophistry. Math and logic are TOOLS, not sciences. Without connection to reality the results cannot be about reality.).

So, no, I have better things to do with my time than to waste it arguing logic that tells us absolutely nothing about reality, absolutely nothing to an extent that is greater than it is possible to imagine, it's greatness is enhanced by it's existence so therefore it's uselessness MUST exist as a necessary property of the Universe.

Grumpy

[Grumpy]

To put it in terms most will understand I present the computer. It is internally hardwired or programed to follow strict rules of logic, it's use of math and logic is flawless(as Godel's Theorem is). But there is an old truism that illustrates what the problem with Godel(and all the other ontological arguments) is...

GIGO. Garbage In=Garbage Out.

Anslem's argument is...

"Thus even the fool is convinced that something than which nothing greater can be conceived is in the understanding, since when he hears this, he understands it; and whatever is understood is in the understanding. And certainly that than which a greater cannot be conceived cannot be in the understanding alone. For if it is even in the understanding alone, it can be conceived to exist in reality also, which is greater. Thus if that than which a greater cannot be conceived is in the understanding alone, then that than which a greater cannot be conceived is itself that than which a greater can be conceived. But surely this cannot be. Thus without doubt something than which a greater cannot be conceived exists, both in the understanding and in reality."

"that something than which nothing greater can be conceived(should be read as "is not possible")."

Greater by what standard? It is here where theists insert their favorite god's properties(a priori). These are assumptions about what constitutes greatness. As I tried to point out in my last post(badly, I fear)is that ANYTHING can be entered at this point and what comes out of the argument will be "proof" that that anything actually MUST exist.

"is in the understanding"

It "exists" in the mind.

"And certainly that than which a greater cannot be conceived cannot be in the understanding alone. For if it is even in the understanding alone, it can be conceived to exist in reality also, which is greater. "

Actually existing is "greater" than just existing in the mind. Therefore...

"that than which a greater cannot be conceived " exists in reality.

So, if you can conceive and understand ANYTHING that is the "greatest" such thing that could possibly exist in the mind, and since it is greater to exist in reality than to just exist in the mind, then that ANYTHING must exist in reality.

You have just logically "proven" that what you can conceive and understand actually exists.

Let's apply this to something we KNOW exists...

I can understand that the Bugatti Veyron is the car "than which nothing greater can be conceived". It would be much greater if it were mine and sitting in my driveway right now with the keys in the ignition(I know, it doesn't have a key, how great is that?). I just looked out the window and, sadly, there is no 1001 horsepower, 250+mph car there. So the axiom that says that what exists in the mind would be even greater if it exists may be valid in logic, but it is garbage in reality.

Insistence that I put this in modal logic terms(ala Godel)is a transparent attempt to bury this point under a steaming pile of symbology. Godel's argument is based on Anselm's and Anselm's is much more understandable by those not educated in modal logic symbology and usage. And understanding the problems with this logic is the goal, not baffling everyone with BS. I could even stipulate that Godel's use of modal logic is without flaw and it still would not change the fact that if you assume a priori that god exists you will accept the proof as valid, if you don't, you won't. Again, what you put in is what will come out. If you assume that what you can conceive and understand actually exists that is what your "proof" will "prove"(that it exists). Unfortunately, your argument has nothing to do with reality.

Grumpy

[JoeyKnothead]

From Post 213:

...
Greater by what standard? It is here where theists insert their favorite god's properties(a priori). These are assumptions about what constitutes greatness. As I tried to point out in my last post(badly, I fear)is that ANYTHING can be entered at this point and what comes out of the argument will be "proof" that that anything actually MUST exist.
...

Exactly. To say that merely imagining something means that something exists is right up there with the goofiest notions of all time.

It is wisful thinking, no more.

[EduChris]

...I fail to see how such a property proves the existence of a supreme being in the context of this or any other ontological argument...

Perhaps "Supreme Being" contains baggage which needn't concern us at the moment. Instead, the main idea is that there is metaphysical or supraphysical framework within which universes--our universe, of course, and perhaps any other conceivable universes--exist. Godel's proof seeks to determine what we can know about this metaphysical "framework for reality."

...Could you outline how such properties show the existence of such a being?...

Ultimately, we are not discussing "a being," or "a thing" like all other "things," but rather a "framework for existence," a framework which can be seen as entailing those properties (of any and all conceivable universes) which we can identify as "non-arbitrary." If various non-arbitrary properties can be identified in various conceivable universes, then it is also possible to conceive of some universe in which all non-arbitrary properties can be identified (note that if it would be impossible for all of those properties to exist in the same universe, then one or more of our proposed properties is not "positive" or "non-arbitrary" at all; that is, we have made a mistake and must go back and correct it by eliminating the arbitrary properties which have somehow slipped past us). The metaphysical "framework" (henceforth referred to as "metaframe") must necessarily encapsulate or entail at least all of these non-arbitrary properties, and in this way we can obtain at least some partial information about our "metaframe."

...Being honest, I can't actually agree with you that differentiation is a property in our universe...Quantum theory will mess up your philosophy...

Are you saying that there is no such thing as distinct fundamental particles?

At any rate, differentiation is entailed by the fact that we have multiple conceivable universes. The set of conceivable universes, all by itself, means that each universe is itself distinct, unique, and related by virtue of its membership in the set of universes which can be conceived.

...I was of the opinion that it necessitated that existence was a property of this supreme being. If not, I fail to see how it's an argument at all...

Kant's objection that "Existence is not a predicate" is circumvented by Godel's use of set theory. He implicitly proposed a set of "all existent things." Anything which is placed into that set entails the property of "existence." Moreover, Godel further (implicitly) stipulated a subset of "all non-arbitrarily existent things." This subset does not include unicorns or hippos at all, since unicorn-ness or hippo-ness fail the axiomatic filter. The other "conceivable universes" do not act so much as license for unbridled imagination as instead a means for helping us to distinguish arbitrary from non-arbitrary properties.

To me, the real trick is to decide what belongs in the set of "existent things," as opposed to what belongs in the set of "non-existent things." I think that my superpositives, in conjuction with the filter of Axioms 2 through 4, work well to weed out arbitrariness. But what things actually are "existent"? That is the biggest question.

For example, we would place people, trees, rocks, water, oxygen, hydrogen, fire, fundamental particles, etc. into the set of "existent things." We would probably also place dinosaurs and dodo birds into the set as well, on the basis of their past existence. But what about things such as numbers? Rational numbers, irrational numbers, imaginary numbers, and so on. What about such things as truth, love, creativity, volition? Do these things belong in the set of existent things, or in the set of non-existent things? To me, this is the most difficult aspect of Godel's Ontological Proof.

In the end I have to adopt a pragmatic approach and say that numbers exist, as do other commonly used abstract or metaphysical concepts. These belong in the set of existent things, rather than in the set of non-existent things. From there, we eliminate the "arbitrarily existent things" from the set of "all existent things," so that we end up with the set of "all non-arbitrarily existent things." And this set of all non-arbitrary things provide us with some core of information about the metaframe which entails them.

To my thinking, the superpositives of Existence, Differentiation, Relationality, and Information--along with the "positives" of Consciousness, Volition, Creativity, and Love--are the only things that I can see existing non-arbitrarily. Therefore, these properties constitute the core of information that we can detect (on our own) about our metaframe.

[LiamOS]

Perhaps "Supreme Being" contains baggage which needn't concern us at the moment. Instead, the main idea is that there is metaphysical or supraphysical framework within which universes--our universe, of course, and perhaps any other conceivable universes--exist. Godel's proof seeks to determine what we can know about this metaphysical "framework for reality."

But why must it be metaphysical?

Ultimately, we are not discussing "a being," or "a thing" like all other "things," but rather a "framework for existence," a framework which can be seen as entailing those properties (of any and all conceivable universes) which we can identify as "non-arbitrary." If various non-arbitrary properties can be identified in various conceivable universes, then it is also possible to conceive of some universe in which all non-arbitrary properties can be identified (note that if it would be impossible for all of those properties to exist in the same universe, then one or more of our proposed properties is not "positive" or "non-arbitrary" at all; that is, we have made a mistake and must go back and correct it by eliminating the arbitrary properties which have somehow slipped past us). The metaphysical "framework" (henceforth referred to as "metaframe") must necessarily encapsulate or entail at least all of these non-arbitrary properties, and in this way we can obtain at least some partial information about our "metaframe."

I don't see any reason to assume that it is either metaphysical or pertinent to theological matters.

Are you saying that there is no such thing as distinct fundamental particles?

How are you defining distinct?

At any rate, differentiation is entailed by the fact that we have multiple conceivable universes. The set of conceivable universes, all by itself, means that each universe is itself distinct, unique, and related by virtue of its membership in the set of universes which can be conceived.

Inductive logic and I don't get along. You are assuming that what is applicable to 'standard' matters is applicable not only to universes but to everything in this universe.
I'm not saying you're wrong, but I can't see any reason to make that leap.
Also, should we not have a working definition of what conception actually entails?

[...]For example, we would place people, trees, rocks, water, oxygen, hydrogen, fire, fundamental particles, etc. into the set of "existent things." We would probably also place dinosaurs and dodo birds into the set as well, on the basis of their past existence. But what about things such as numbers? Rational numbers, irrational numbers, imaginary numbers, and so on. What about such things as truth, love, creativity, volition? Do these things belong in the set of existent things, or in the set of non-existent things? To me, this is the most difficult aspect of Godel's Ontological Proof.[...]

As much as I hate to ask, I must ask what exactly 'existence' is and is defined as.

[Goat]

In that I'm not educated at all in philosophy or theology, where I ask a question here I am genuinely interested.
I'm not being rhetorical as I'm sure you've grown accustomed to(But hopefully not from me).

You are far too hung up on a particular slant on "positive." In Godel's usage, the term should be "non-arbitrary."

I fail to see how such a property proves the existence of a supreme being in the context of this or any other ontological argument.

Could you outline how such properties show the existence of such a being?

That objection is handled by my move to superpositives, which are entailed for all conceivable universes.

Being honest, I can't actually agree with you that differentiation is a property in our universe.
Quantum theory will mess up your philosophy.

Kant was objecting to earlier ontological formulations, and Godel's Proof avoids Kant's objection.

I was of the opinion that it necessitated that existence was a property of this supreme being. If not, I fail to see how it's an argument at all.

Kant would not like axiom number 5, according to some analysis of Godel's proof. There also is the problem that many terms are not defined. For example, Godel does not define what is 'positive' is , or 'God like' is, .. even though he equates them.

From http://plato.stanford.edu/entries/ontol ... #GodOntArg

Some philosophers have denied the acceptability of the underlying modal logic. And some philosophers have rejected generous conceptions of properties in favour of sparse conceptions according to which only some predicates express properties. But suppose that we adopt neither of these avenues of potential criticism of the proof. What else might we say against it?

One important point to note is that no definition of the notion of positive property is supplied with the proof. At most, the various axioms which involve this concept can be taken to provide a partial implicit definition. If we suppose that the positive properties form a set, then the axioms provide us with the following information about this set:

1. If a property belongs to the set, then its negation does not belong to the set.
2. The set is closed under entailment.
3. The property of having as essential properties just those properties which are in the set is itself a member of the set.
4. The set has exactly the same members in all possible worlds.
5. The property of necessary existence is in the set.
6. If a property is in the set, then the property of having that property necessarily is also in the set.

On Gdel's theoretical assumptions, we can show that any set which conforms to (1)"(6) is such that the property of having as essential properties just those properties which are in that set is exemplified. Gdel wants us to conclude that there is just one intuitive, theologically interesting set of properties which is such that the property of having as essential properties just the properties in that set is exemplified. But, on the one hand, what reason do we have to think that there is any theologically interesting set of properties which conforms to the Gdelian specification? And, on the other hand, what reason do we have to deny that, if there is one set of theologically interesting set of properties which conforms to the Gdelian specification, then there are many theologically threatening sets of properties which also conform to that specification?

In particular, there is some reason to think that the Gdelian ontological argument goes through just as well"or just as badly"with respect to other sets of properties (and in ways which are damaging to the original argument). Suppose that there is some set of independent properties {I, G1, G2, } which can be used to generate the set of positive properties by closure under entailment and necessitation. (Independence means: no one of the properties in the set is entailed by all the rest. Necessitation means: if P is in the set, then so is necessarily having P. I is the property of having as essential properties just those properties which are in the set. G1, G2, are further properties, of which we require at least two.) Consider any proper subset of the set {G1, G2, }"{H1, H2, }, say"and define a new generating set {I*, H1, H2, }, which I* is the property of having as essential properties just those properties which are in the newly generated set. A proof parallel to that offered by Gdel establishes that there is a being which has as essential properties just those properties in this new set. If there are as few as 7 independent properties in the original generating set, then we shall be able to establish the existence of 720 distinctGod-like creatures by the kind of argument which Gdel offers. (The creatures are distinct because each has a different set of essential properties.)

Even if the above considerations are sufficient to cast doubt on the credentials of Gdel's proof, they do not pinpoint where the proof goes wrong. If we accept that the role of Axioms 1, 2, 4, and 6 is really just to constrain the notion of positive property in the right way"or, in other words, if we suppose that Axioms 1, 2, 4, and 6 are analytic truths about positive properties"then there is good reason for opponents of the proof to be sceptical about Axioms 3 and 5. Kant would not have been happy with Axiom 5; and there is at least some reason to think that whether the property of being God-like is positive ought to depend upon whether or not there is a God-like being.

Now, one thing .. what a 'positive property' ?? It's not really defined in the proof, and any attempt to 'define' it I have seen sort of relies on subjective, metaphyiscal or subjective terms... and they don't make sense to me.

Next, what is 'Godlike'?? and why should we accept any of definitions, and accept that 'Godlike' actually exists?

It seems that the argument boils down 'If this undefined emotional term' has this unsupportable and undefined properties, then if this undefined emotional term actually exists, it is logical for it to exist.

This shows nothing at all.

[EduChris]

...why must it be metaphysical?...

We don't have to use the word "metaphysical." We simply need to acknowledge that the set of all conceivable universes is not less than, and is in all likelihood more than, our own physical universe.

...I don't see any reason to assume that it is either metaphysical or pertinent to theological matters...

Drop "metaphysical," if you like, per my previous point. Whether or not it is theologically pertinent depends on what non-arbitrary properties we find, and how well these may or may not correspond to any given theological tradition.

...How are you defining distinct?...

There is some X and some Y such that X is not Y. Of course I'd be willing to accept some help here, if you have a better definition.

...Inductive logic and I don't get along. You are assuming that what is applicable to 'standard' matters is applicable not only to universes but to everything in this universe...

I'm not sure how it could make sense to assume otherwise. I suppose we could assume all kinds of things about the metaframe, but I don't know any way to talk about "that which is lesser known" except in terms of "that which is better known."

...should we not have a working definition of what conception actually entails?...

This is why I said that the introduction of "conceivable universes" serves more as an aid to help us distinguish between arbitrary and non-arbitrary properties. We can't very well conceive of any other universes except some variation of our own. These variations might be quite similar to our universe, or quite dissimilar in terms of extreme limits. We can conceive of unicorn-like animals because they have comparisons and contrasts to animals which we already know.

...As much as I hate to ask, I must ask what exactly 'existence' is and is defined as.

That is sort of a catch, isn't it? After all, suppose we postulate the set of all existent entities, and also the set of all non-existent entities? What is the basis by which we place some "thing" or "concept" or "idea" or "principle" into one set rather than the other? This gets back to the whole, "Do numbers exist?" question. I don't know any way around this except to use a pragmatic approach. Perhaps all we can say is that if [insert previously stipulated superpositives and positives here] exist, then the metaframe entails these properties, and as such these properties constitute the base core of knowledge that we can obtain (by our own effort) about the metaframe.

[LiamOS]

We don't have to use the word "metaphysical." We simply need to acknowledge that the set of all conceivable universes is not less than, and is in all likelihood more than, our own physical universe.
Drop "metaphysical," if you like, per my previous point. Whether or not it is theologically pertinent depends on what non-arbitrary properties we find, and how well these may or may not correspond to any given theological tradition.
[...]
This is why I said that the introduction of "conceivable universes" serves more as an aid to help us distinguish between arbitrary and non-arbitrary properties. We can't very well conceive of any other universes except some variation of our own. These variations might be quite similar to our universe, or quite dissimilar in terms of extreme limits. We can conceive of unicorn-like animals because they have comparisons and contrasts to animals which we already know.

Again, I still don't see how conception at all helps this matter. The something can be conceived should have absolutely no bearing on anything as best I can tell.
I get that this isn't central to the argument inherently, but the mechanics of ascribing and determining properties still somewhat hangs on it.

There is some X and some Y such that X is not Y. Of course I'd be willing to accept some help here, if you have a better definition.

Then I'd have a very hard time saying that fundamental particles are distinct.

I'm not sure how it could make sense to assume otherwise. I suppose we could assume all kinds of things about the metaframe, but I don't know any way to talk about "that which is lesser known" except in terms of "that which is better known."

Sorry, I'm not following you here.

That is sort of a catch, isn't it? After all, suppose we postulate the set of all existent entities, and also the set of all non-existent entities? What is the basis by which we place some "thing" or "concept" or "idea" or "principle" into one set rather than the other? This gets back to the whole, "Do numbers exist?" question. I don't know any way around this except to use a pragmatic approach. Perhaps all we can say is that if [insert superpositives and positives here] exist, then the metaframe entails these properties, and as such these properties constitute the base core of knowledge that we can obtain (by our own effort) about the metaframe.

Even forgetting the numbers for a moment, does the computer you're using exist?
By what criteria was its existence or lack of established?

[EduChris]

...I still don't see how conception at all helps this matter. The something can be conceived should have absolutely no bearing on anything as best I can tell...

Suppose we conceive of a universe consisting entirely of liquid water. Suppose we then conceive of another universe consisting entirely of solid gold. We cannot then conceive of a universe consisting entirely of liquid water and entirely of solid gold (although of course we can have a universe which consists of some part liquid water and a remainder of solid gold). This tells us either that "entirely consisting of liquid water" is not a positive (non-arbitrary) property or that "entirely consisting of solid gold" is not a positive, or perhaps neither are positive (non-arbitrary) properties. The conceptualization helps us to see which properties cannot be "positive" in the Godelian sense.

...I'd have a very hard time saying that fundamental particles are distinct...

Again, I've never seen any particles at all, so I can't disagree. But can you say whether there any physical forces (e.g., gravity, attraction, repulsion, etc) or any concepts or any persons which are distinct?

...Sorry, I'm not following you here...

What aren't you following?

...does the computer you're using exist? By what criteria was its existence or lack of established?

The computer seems to compel certain familiar electrical impulses in my brain, and it seems to produce similar effects in other people--to the extent that I can believe that other people exist independently of my own mind.

But isn't this the same problem we all face, regardless of our worldview? This sort of question isn't any more damaging to Godel's theorem than it is to any matter involving human cognition and epistemology. We simply have to face the uncertainty of all human knowing, and move forward as best we can with the tools we have.