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?

[Zzyzx]

.

In the meanwhile, perhaps you could discuss the matter with ZZyzx and Cathar--maybe they have some ideas on how the level of discourse here might be raised--which, after all, is our shared common goal.

The level of discourse would be raised considerably if people would DEBATE rather than making excuses, and respond openly and honestly -- without preaching, pontificating, obfuscating or resorting to sophism and solipsism, AND without delusion of superiority for themselves or their beliefs.

[EduChris]

...as a former math teacher, I think I can help here. The "two-ness" you are referring to is the attribute we call absolute value. (2) and (-2) do have the same absolute value, but they are indeed two separate and distinct numbers.

Yes, that helps me to better see the equivocation between the two separate meanings of the word "positive" in Abraxas' original claim. Essentially, Abraxas has used these double meanings to piggyback two independent properties and treat them as though they were a single property.

The number +2 does not imply positiveness. It is the + that implies positiveness, not the absolute value of 2. Similarly, it is the absolute value of 2 (rather than the +) that implies evenness.

The problem in Abraxas' logic could be rephrased as follows: "gray elephants" entail X; therefore, "gray" entails X (where X represents some property, say, "largeness"). Obviously here we have two independent properties: "gray" is one property, and "elephantness" is another property. Since not all elephants are gray, and not all gray things are elephants, neither of these two properties entails the other.

..."Incivility" is generally defined here as remarks that reference the individual personally or to personal attributes. For instance, it's obviously unacceptable to say that a person is "stupid"; but it's almost as bad to say that an argument is "stupid" because that clearly refers to the person who uses it. One is expected to explain WHY an argument is "stupid" without actually applying that word...I am posting this as a member and not as a moderator, which means anyone is free to argue...

Does incivility include the following:

1) continously repeated "loose canon" accusations of "logical fallacies" where the accuser gives no evidence of having even tried to understand the accused person's argument?

2) continual application of double standards, wherein one party is constantly demanded to "prove" every single assumption, even though there is no "proof" to be had either for the assumption in question OR for any of the competing/alternate assumptions?

3) stalking behavior, wherein one party makes a general nuissance of himself by cluttering up every post in every thread with cut-and-paste demands for "proof," while at the same time insisting that such "proof" must conform to the accuser's own self-serving definition?

4) flame-baiting?

Depending on how you answer these questions, I may be willing to honor Abraxas' request that I remove the individuals from my Ignore List and give the moderators a chance to deal with the individuals directly. Obviously I can always put the individuals back on the Ignore List if the resulting moderating task becomes too burdensome.

[Zzyzx]

.

1) continously repeated "loose canon" accusations of "logical fallacies" where the accuser gives no evidence of having even tried to understand the accused person's argument?

Yes, one of the Theists is famous for emphasis on trying to show logical fallacies (with particular emphasis on validity). Some answer his ploy but most overlook the ploy and challenge his assertions.

Anyone is free to point out logical fallacies in opposition arguments. Is that a problem somehow?

Notice that Holy Huddle is a safe environment for those who do not wish to have their god assumptions and assertions challenged by Atheists.

2) continual application of double standards, wherein one party is constantly demanded to "prove" every single assumption, even though there is no proof to be had either for the assumption in question OR for any of the competing/alternate assumptions?

EVERY party is REQUIRED by Forum Rules and Guidelines to substantiate their claims. Those who make claims that they cannot substantiate (except by quoting non-authoritative sources) are inclined to complain.

Alternative / competing assumptions are NOT required in reasoned debate or by Forum Rules and Guidelines. It IS valid to challenge CLAIMS made by any person and to ask for reason to accept what they say as truth. Because I think so or Because my favorite book says so or Because leading academic theologians say so are NOT valid reasons (except perhaps in Holy Huddle).

3) stalking behavior, wherein one party makes a general nuissance of himself by cluttering up every post in every thread with cut-and-paste demands for "proof," while at the same time insisting that such "proof" must conform to the accuser's own self-serving definition?

Any member is free to respond to any post. Some members choose to respond to many posts, some to few. Thats life in the big city folks. If you cant stand the heat, get out of the kitchen (as Harry Truman famously said).

Makes a general nuisance of himself may translate to Makes points that I cannot rebut or counter and Asks questions that I cannot (or will not) answer because to do so honestly would destroy my argument.

4) Flame-baiting?

Kindly identify exactly what you intend by the term flame bating and cite examples. You ARE aware that actual instances of flame bait are prohibited by Forum Rules and should be reported, arent you?

Depending on how you answer these questions, I may be willing to remove the individuals from my Ignore List and give the moderators a chance to deal with the individuals directly.

It might be wise to leave people on an ignore list when one cannot deal effectively and convincingly with the points they raise or the questions they ask.

Doing so leaves them free reign to expose defects in ones arguments for readers to consider without possibility of rebuttal. However, shooting sitting ducks isnt much of a challenge and doesnt make for interesting debate.

Obviously I can always put the individuals back on the Ignore List if the resulting moderating task becomes too burdensome.

Moderators have been on duty all along " and evidently do not regard as uncivil the actions that upset some people. Surely, those offenses have been reported, havent they? Perhaps it would be prudent to reexamine ones own definitions and supposed grievances.

[Abraxas]

...Last I checked, numbers were considered actually existing abstract objects.

Does that mean that numbers (real and imaginary) are actually existing immaterial objects? In other words, is there more than just "physical matter" in the universe? Are there some objects which actually exist apart from anyone's ability to examine them under a microscope, so to speak?

That is one of the great unresolved questions of philosophy, is it not, whether abstract objects actually exist or are merely constructs of categorical thinking? None the less, it is extremely common, as I said, to treat numbers as being actually existing abstract objects that do exist, though not as a physical substance of the universe.

http://en.wikipedia.org/wiki/Mathematical_object
http://en.wikipedia.org/wiki/Number

...Further, you are making the mistake of equating two with negative two. They are two independent numbers that happen to partially share a symbolic designator but they are separate numbers...

I'm still not satisfied that I've correctly understood you. To me, there is such a thing as "two-ness" that is independent of the CR'ness or the DB'ness. For example, suppose the boss goes to the bookkeeper and says, here's a two-dollar check, please enter it into the books. The bookkeeper can understand the "two-ness" of the check, but until she figures out whether the check is written out of the company's bank account (CR'ness) or instead to be deposited into the company's bank account (DB'ness) she won't be able to enter it into the books. The "two-ness" is a separate and distinct property from the negative or positive effect it will have on the company's bank account.

As CNorman said, that is the absolute value of each, but there is no "two-ness". Two and negative two are independent numbers. Further, I have already shifted the example to four and negative eight to get away from this, an example I note you did not reply to.

...how about stars? Luminosity is a positive property of stars, as is longer life span. Adding brightness to a star inherently involves shortening its lifespan, thus brighter entails shorter life span. Lengthening the lifespan entails dimming the star...

I'm no expert on stars, but nevertheless I feel that you aren't understanding the concept of "positive properties" the same way that I am. Luminosity (or light) is a property in and of itself, independent of the duration of that luminosity/light. The total light output is dependent not on duration, but rather on the total amount of fuel (hydrogen, or whatever) that can produce the light. Duration is not a property, but rather a measurement of the rate of change (same with the battery example). So again, your example doesn't seem plausible to me, though it might seem that way to others.

How is lifespan not a property of an object? If size is a property of an object, duration is just length in a time dimension. The more fuel you add, the faster the star burns. That is why hypergiants have the shortest lifespan of all stars, just a few million years. Conversely, red dwarf stars have life spans running into the tens of trillions of years, but emit very little light by star standards. Luminosity is a property of stars, as is lifespan, both are determined by a third property, mass. Mass goes up, lifespan goes down and luminosity goes up, mass goes down, lifespan goes up and luminosity goes down. The two properties are intrinsically linked, you cannot change one without changing the other.

I think you would have to defend the idea that duration is not a property, especially when you have already stated it is a measurable quantity of an object. Last I checked, (forty seconds ago on wiki) the traditional definition of property was a characteristic but non-essential qualities of which duration would certainly qualify.

...not one of the people you have indicated is on it or should be on it or is close to it, in my opinion, has acted uncivilly...

Obviously one's evaluation and experience of "incivility" is at least partially a subjective matter, and that is ultimately why the list is necessary, in my opinion. But I will continue to keep your opinions in mind, and perhaps a better solution will emerge in time. In the meanwhile, perhaps you could discuss the matter with ZZyzx and Cathar--maybe they have some ideas on how the level of discourse here might be raised--which, after all, is our shared common goal.

They are free to weigh in as they like, however, neither of them in my experience has been anything but a positive influence on the community and each has been recognized by both theists and nontheists many times as such.

Yes, that helps me to better see the equivocation between the two separate meanings of the word "positive" in Abraxas' original claim. Essentially, Abraxas has used these double meaning to piggyback two independent properties and treat them as though they were a single property.

The number +2 does not imply positiveness. It is the + that implies positiveness, not the absolute value of 2. Similarly, it is the absolute value of 2 (rather than the +) that implies evenness.

You just got wrong everything he said.

Two, the number, does imply positiveness in a numerical sense, negative two, the number, implies negativeness in a numerical sense. You are equivocating two, the number, with two, the absolute value. The plus sign only implies positiveness symbolically, however, the actual numbers two and negative two are positive and negative respectively regardless of how they are notated, that is simply a characteristic of the actual number.

The problem in Abraxas' logic could be rephrased as follows: "gray elephants" entail X; therefore, "gray" entails X (where X represents some property, say, "largeness"). Obviously here we have two independent properties: "gray" is one property, and "elephantness" is another property. Since not all elephants are gray, and not all gray things are elephants, neither of these two properties entails the other.

This is a complete misrepresentation of my argument. You are again acting like the absolute value is what is at the core and that numbers are not independent of each other. You are the only one who has used absolute value as part of your argument and this is the very reason I switched to four and negative eight to avoid confusion.

You are treating numbers like they are an absolute value with a positive or negative designator tacked on instead of treating the numbers as independent objects, which is incorrect.

[JoeyKnothead]

From Post 63:

I feel like I have special insight here, and I'm proud about it and I'm gon' let let y'all know...

I've studied (in my amateur way) Zyzzxeses posts. I'm aware this man seldom concerns himself with what an individual believes, but what that individual states as truth. Notice that Zyzzx, if you really examine his methods and all, will ensure the READER is aware of problems in some folks' arguments. To not "fight back" when I know he's targeting a larger audience, and especially when I know I'm right, and can prove it, is to go against every fiber of my being.

This is why, because I generally just try to steal the good stuff from the good debaters here, is why I fear using an ignore feature. It ain't just to make sure he ain't mean-mouthing my kinfolk. If I can show I speak truth, why would I ignore one who asks me to show I speak it?

I challenge everybody to understand that if you can get your claims by Zyzzx (and SO MANY others) on this site, you've said something. If you can't get him to sign off on it, you better hope to whatever god you worship that the READER never finds out you couldn't.

[Zzyzx]

.

Finally, as a general request to EduChris, I would ask that you have the "Ignores the Uncivil" user group removed...If there are certain individuals you do not wish to debate with, speak to, or read, by all means, don't, but the continual advertisement of such only serves to, ironically, increase the level of incivility and lower the level of discourse...it would be best if the group were dissolved (or at the very least never mentioned again).

I am not happy that the group needs to be used at all--I had hoped it would serve merely as a deterrent. Unfortunately, given the situation, I feel it is the lesser of evils (though I remain open to suggestions and would certainly remove any individuals from the list if/when there was some indication that doing so would raise, rather than lower, the level of discourse).

Evidently, members do not agree that the group needs to be used (or to exist). It has exact ONE member besides the originator.

Perhaps the existence of the group says more about the originator than about any "need" in the Forum??? What is regarded as "uncivil" by one member must NOT be regarded as such by Moderators or there would be action taken. Actual incivility is not tolerated (and should be reported by all of us).

A charge of "incivility" and use of "ignore" may be applied by some people in lieu of attempting to rebut points raised or answering questions asked rather than any actual incivility by others.

[EduChris]

...there is no "two-ness". Two and negative two are independent numbers. Further, I have already shifted the example to four and negative eight to get away from this...You are treating numbers like they are an absolute value with a positive or negative designator tacked on instead of treating the numbers as independent objects...

I maintain that -2 is simply a different way of writing +2 x -1; there is complete equality in either case, but writing it out the longer way makes it impossible to accidentally combine the two attributes, as you seem to have done (and this applies with your "positive-four" and "negative-eight" example as well).

But I guess the point is moot anyway, since the "negative" or "positive" of a number is not a "property" at all in terms of Godel's theorem (more on this below).

...I think you would have to defend the idea that duration is not a property, especially when you have already stated it is a measurable quantity of an object.

Can time go backward? How does one negate a duration after it has already been measured?

...Last I checked...the traditional definition of property was...

Here is the biggest problem that I see, and if I'm right it pretty much invalidates your entire argument. It involves the one thing that both Aki and I have already determined to be the main sticking point: namely, Axiom 1. We haven't come up with an adequate definiton of "positive property." It is clear that we can't just whip out our dictionary and use the common definition of "property," since Godel goes out of his way to list at least some of the requirements for a "property" within the context of his theorem. Let's look at Axiom 4 again:

Axiom 4: IF P is a property, then either P or its negation is positive, but not both.

In other words, according to Godel's axiom, if P and its negation are both positive, then P is not a "property" at all within the context of the theorem. Godel's "properties" are not garden variety "properties," but something else entirely.

...suppose you flip a coin. You have heads. Heads entails not tails. Thus not tails is a positive. You flip again, and you have tails, which entails not heads, which, from this must both be positive. Heads and not heads, tails and not tails, all seem to be positive per axiom 2 and yet axiom 4 says the negation cannot be positive if the property is....in any existing binary system, the negation of one state is identical (per the law of identity) to the positive state of the other. If a switch is either up or down, if it is not up, it is not just not up, it is down. Not up and down will necessarily share all properties with each other. As such, axiom 4 is demonstrably false.

The problem is not that the Axiom is false; rather, the problem is that you are claiming something to be a "property" that is most definitely not a "property" within the context of Godel's theorem (see above).

The more I think about it, the more I agree with Aki that we all need to work on coming up with a working definition for "positive property" if we're ever going to be able to comment on the validity of the theorem. I'm not sure if we can succeed, but that does seem to be where the crux of the whole argument lies.

[Abraxas]

...there is no "two-ness". Two and negative two are independent numbers. Further, I have already shifted the example to four and negative eight to get away from this...You are treating numbers like they are an absolute value with a positive or negative designator tacked on instead of treating the numbers as independent objects...

I maintain that -2 is simply a different way of writing +2 x -1; there is complete equality in either case, but writing it out the longer way makes it impossible to accidentally combine the two attributes, as you seem to have done (and this applies with your "positive-four" and "negative-eight" example as well).

Mathematically, you are wrong. If you feel otherwise, explain the difference between 1 and -1, as your designation would require 1 x -1, which would be a circular definition, thus empty.

But I guess the point is moot anyway, since the "negative" or "positive" of a number is not a "property" at all in terms of Godel's theorem (more on this below).

Greater than or less than zero is not a property of a number? That makes certain sets rather hard to work with...

...I think you would have to defend the idea that duration is not a property, especially when you have already stated it is a measurable quantity of an object.

Can time go backward? How does one negate a duration after it has already been measured?

How does that matter? Something can't have a negative length either, and yet size is an attribute.

...Last I checked...the traditional definition of property was...

Here is the biggest problem that I see, and if I'm right it pretty much invalidates your entire argument. It involves the one thing that both Aki and I have already determined to be the main sticking point: namely, Axiom 1. We haven't come up with an adequate definiton of "positive property." It is clear that we can't just whip out our dictionary and use the common definition of "property," since Godel goes out of his way to list at least some of the requirements for a "property" within the context of his theorem. Let's look at Axiom 4 again:

Axiom 4: IF P is a property, then either P or its negation is positive, but not both.

In other words, according to Godel's axiom, if P and its negation are both positive, then P is not a "property" at all within the context of the theorem. Godel's "properties" are not garden variety "properties," but something else entirely.

...suppose you flip a coin. You have heads. Heads entails not tails. Thus not tails is a positive. You flip again, and you have tails, which entails not heads, which, from this must both be positive. Heads and not heads, tails and not tails, all seem to be positive per axiom 2 and yet axiom 4 says the negation cannot be positive if the property is....in any existing binary system, the negation of one state is identical (per the law of identity) to the positive state of the other. If a switch is either up or down, if it is not up, it is not just not up, it is down. Not up and down will necessarily share all properties with each other. As such, axiom 4 is demonstrably false.

The problem is not that the Axiom is false; rather, the problem is that you are claiming something to be a "property" that is most definitely not a "property" within the context of Godel's theorem (see above).

The more I think about it, the more I agree with Aki that we all need to work on coming up with a working definition for "positive property" if we're ever going to be able to comment on the validity of the theorem. I'm not sure if we can succeed, but that does seem to be where the crux of the whole argument lies.

Which drags us back to my objections to Axiom 1, both in defining it and then determining if any such can properties exist at all. In particular, if there is no moral aesthetic sense independent from the accidental structure of the world, there can be no such thing as positive properties as he defines them.

[EduChris]

deleted

[EduChris]

...explain the difference between 1 and -1, as your designation would require 1 x -1, which would be a circular definition, thus empty.

The attribute of "evenness" depends on something being disivible by two (without remainder). The numerical attribution of "negativeness" or "positiveness" depends on whether the number in question is less than or greater than zero.

Thus, -2 or + 4 or -8 or +2 are all divisible by 2 without remainder, so their evenness follows from that alone, rather than from their having some conglomerate status of "negativeness and divisibility by two." Similarly, -1 or +1 are attributed as less than or greater than 0, which alone determines their "negativeness" or "positiveness," independently of whether (or not) they happen to be divisible by 2.

...Greater than or less than zero is not a property of a number? That makes certain sets rather hard to work with...

Here is where your equivocation comes in. The word "property" can have different meanings in different contexts. "Negative" or "positive" may be a "property" of a number in certain contexts, but Godel's axioms clearly show that these are not the sort of "properties" his theorem is concerned with.

...How does that matter? Something can't have a negative length either, and yet size is an attribute.

All you're saying here is that length or size of physical objects, to the extent they can't be negated, are not properties within the context of Godel's theorem. So we are making some progress, it seems, in that we have thus far ruled out duration, length, size, etc. as falling within the parameters of Godel's "properties." We need to look elsewhere if we are to find the sort of "properties" that we need if we're going to get anywhere with Godel's theorem.

...Which drags us back to my objections to Axiom 1, both in defining it and then determining if any such can properties exist at all. In particular, if there is no moral aesthetic sense independent from the accidental structure of the world, there can be no such thing as positive properties as he defines them.

Well, you have said that numbers actually exist. But what is a number, any number, taken in isolation? The number "2" is only significant in relation to other numbers, such as "1" and "3." So I would say that if numbers exist (as you claim they do) then relationality must be a "positive" property in Godel's usage, where as "isolation" or "acontextuality" would be the negation of "relationality."

Perhaps we should try to come up with a list of "positive properties" (along with their negations) that meet the criteria of a Godelian "property." Once we have found examples of such "properties" that fit within the context that Godel has laid out, then we can work on a definition of those properties.

For example, (relationality vs. isolation) might be one set of "positive" vs. "negative" properties. Another example might be (logic vs. absurdity) given that numbers have no use or function or value apart from the logic of mathematics--that is, the meaning of 2 + 2 would be empty or inane unless this addition logically compels a result that can be nothing other than 4.

Perhaps we could flesh this out further by saying that (significance vs. inanity) is another set of positive vs. negative Godelian properties.

So far we have the following possibilities for sets of positive vs. negative "properties" within the framework of Godel's theorem:

1. Relationality vs. isolation
2. Contextuality vs. acontextuality
3. Logic vs. absurdity
4. Significance vs. inanity
5. Order vs. chaos

I submit that without the above positive properties, numbers could not "exist." Do you agree?