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?

[EduChris]

...If by that you mean you will clarify at a later stage, that's acceptable. Simply saying that most people know is not appropriate or acceptable to define something on which an entire theorem relies while believing the theorem to be valid.

Actually I mean I'm tired and I'm going to take the weekend off from this forum.

My undergraduate field was mathematics, and one of the few things I remember from my studies was that whenever we needed to we could just stick a variable X or Y or something into the formula, and then go on working with the formula with the understanding that X or Y or whatever represented something. In other words, I don't have to define what the square value of -1 might be; all I have to do is give it a label, i, and then do my thing.

For now and the foreseeable future, I am content to say that Godel's theorem is valid to the extent that "meaningfulness" is reasonable. I think it is reasonable, and so the theorem is valid for me. For those who don't find "meaningfulness" reasonable as a concept, the theorem won't make much of an impact on them. I'm okay with that right now, unless someone can convince me otherwise.

[LiamOS]

My undergraduate field was mathematics, and one of the few things I remember from my studies was that whenever we needed to we could just stick a variable X or Y or something into the formula, and then go on working with the formula with the understanding that X or Y or whatever represented something. In other words, I don't have to define what the square value of -1 might be; all I have to do is give it a label, i, and then do my thing.

This is true, but it's meaningless until the variable or function is defined.

And have a wild weekend.

[Board]

And for what it's worth, I think God is pleased that you find joy in such activity. That's why he created you and gave you the capacities for creative action and pleasure.

While I disagree this, I do thank you for the good discussion.

This all reminds me of my early 90s teenage angst.

Suffer - Smashing Pumpkins. Good song...

What takes meaning is cleaning the meaning of who you are

[JoeyKnothead]

From Post 37:

...The proof can summarized as: IF it is possible for a rational omniscient being to exist THEN necessarily a rational omniscient being exists...

I don't believe that summary really gets at the heart of Godel's theorem. A physicist summarized the theorem for me as, "If anything at all exists, something must necessarily--i.e., non-contingently--exist."

Oh please. This is nothing more than an attempt to siddle "god exists" onto "anything exists".

Can anything exist? I'll give ya that'n. How does that mean "It (God) does exist"?

Here is another link and a highlighted section from the paper. To the extent that the author is an "expert witness" himself, one should not be too quick to claim "fallacy" when presented with Godel's argument:

Nor should one conclude that just because this'n here declares something true that it is true.

If one's claims can't be shown to be true those claims can't be shown to be true.

Best I can tell, the referenced article does not addresss my central argument here, that just because one can concieve a notion does not mean that notion is accurate. To put it in perspective, the possibility of something occurring, existing, etc. does little or nothing to show that something occurs, exists, etc.

"It could well be" is a poor replacement for "Well there it is right there".

The ontological argument attempts...

And there's the problem right there. It attempts to, but doesn't verify what claims it makes.

[Abraxas]

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?

For brevity, I will assume so. Ontological arguments in general tend to fall prey to equivocation fallacies and at the moment I lack the time to check this one over.

-Are all the axioms of the theorem valid?

I would go so far as to say none of them are.

1. This one has a number of unjustified build it assumptions, such that there is a moral aesthetic independent of the accidental structure of the world and that positive properties exist at all as opposed to merely being a language or mathematical construct.

2. Absolutely untrue. If a value of 2 requires positiveness (it does) and 2 entails evenness, then evenness must be positive? What of -2 that entails evenness as well? Much stronger justification would be needed to demonstrate that all things that result from something positive, or that are required for something positive, are in fact themselves positive. It is entirely possible for them to be value neutral or even negative in and of themselves.

Another example, one that directly contradicts axiom 4, 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.

3. I would challenge that having properties is in fact a property in and of itself and would further challenge that possession of properties raises the value of the possessor, as opposed to the properties themselves possessing all the value and the collection in which they fall merely existing as an example of categorical thinking with no concrete basis.

4. I touched on this one already, but this one is inconsistent with axiom 2. Beyond that, however, I would state that 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.

5. I am going to defer to Kant on this one, existence is not predicate. Existing is not a property, it is a prerequisite to have properties. Any assertion of different kinds of existence (contingent, necessary) would require further backing to demonstrate an actual existence.

-Can the argument hold without the axioms being valid, if they are not necessarily so?

No. The conclusion might be true, but the argument itself does not hold.

Approaching this from a different angle, I would argue a being such that Godel describes requires inconsistent standards for possible worlds. In order for his argument to work all possible worlds must be open to the possibility of a supreme being when I would argue this is not so. I can imagine a world about the size of a basketball the same consistency as solid brick that exists independent of all other worlds and incapable of being interacted with in any form. So such supreme being is possible here because the world is by definition immutable and without the capacity to change it, such a being would be severely deficient. I would in turn argue the opposite of Godel, that if a possible world exists where Godlikeness is not possible, God cannot exist in any world under his criterion.

Finally, as a general request to EduChris, I would ask that you have the "Ignores the Uncivil" user group removed. I do not wish to return the thread to personal discussion, however, more than anything the group seems to have become a bludgeon with which one can hammer any individual who disagrees on style instead of the substance of their post. Indeed, I have seen it used as such when the bludgeoner does not seem to hold the civil high ground from the bludgeonee. 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. I have noticed a growing trend in recent threads for this user group to be brought up and then promptly devolve the thread into several pages of personal discussion, accusations, and ad homs and think for the good of the board and a general sense of order, it would be best if the group were dissolved (or at the very least never mentioned again).

[EduChris]

...2. Absolutely untrue. If a value of 2 requires positiveness (it does) and 2 entails evenness, then evenness must be positive? What of -2 that entails evenness as well?

I had intended to take the weekend off from the forum, but I couldn't resist responding here.

I am confused regarding your use of positive and negative numbers in their relation to Godel's use of positive and negative properties. The words are the same, but the associated meanings (in their respective contexts) might not be. For example, suppose we drop the "negative" and "positive" with respect to the number 2; let's use CR ("credit" in accounting terminology) to refer to the "negative" numbers, and let's use DB ("debit" in accounting terminology) to refer to "positive" numbers. In this case, a value of 2 would entail "evenness," but not DB'ness (or CR'ness). In other words, 2 would entail "evenness," but "evenness" would not entail either DB'ness or CR'ness--and of course, neither would 2 entail DB'ness nor CR'ness. It seems to me that your objection doesn't hold up. Am I wrong here? Do you have another example that demonstrates your objection more clearly? (on another line of thought, does the number 2 actually "exist"? does the number -2 actually "exist"? What about 2i ? do numbers, imaginary or otherwise, "exist" anywhere other than the noosphere of human thought?)

...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).

[Abraxas]

...2. Absolutely untrue. If a value of 2 requires positiveness (it does) and 2 entails evenness, then evenness must be positive? What of -2 that entails evenness as well?

I had intended to take the weekend off from the forum, but I couldn't resist responding here.

I am confused regarding your use of positive and negative numbers in their relation to Godel's use of positive and negative properties. The words are the same, but the associated meanings (in their respective contexts) might not be. For example, suppose we drop the "negative" and "positive" with respect to the number 2; let's use CR ("credit" in accounting terminology) to refer to the "negative" numbers, and let's use DB ("debit" in accounting terminology) to refer to "positive" numbers. In this case, a value of 2 would entail "evenness," but not DB'ness (or CR'ness). In other words, 2 would entail "evenness," but "evenness" would not entail either DB'ness or CR'ness--and of course, neither would 2 entail DB'ness nor CR'ness. It seems to me that your objection doesn't hold up. Am I wrong here? Do you have another example that demonstrates your objection more clearly? (on another line of thought, does the number 2 actually "exist"? does the number -2 actually "exist"? What about 2i ? do numbers, imaginary or otherwise, "exist" anywhere other than the noosphere of human thought?)

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

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 could have just as easily selected four and negative eight. The abstract object four entails the property of evenness as does the abstract object negative eight. We cannot "drop" the negative because negative is not a modifier but rather part of the name of the number, that would be like dropping sev from seven.

But, if you don't like numbers for this, 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. If axiom 2 were correct, it increasing luminosity, entailing a shorter lifespan would conclude shortening a star's lifespan is positive and vice versa. This should not be so according to this axiom, indeed the entire concept of "trade-off" seems to require what positive properties entail not necessarily being positive. Another example that comes to mind is processing power vs. battery life in a laptop.

...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).

Firstly, let it be said it is not "the list" but "your list", and in all honesty, 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. Goat and Zzyzx are forceful debaters, certainly assertive, occasionally aggressive, but those qualities are not uncivil. You even threatened Aki with it because he indicated he believed your argument was fallacious. That you keep referencing this list and threatening people with it is probably the single most inflammatory aspect of this board right now, judging by the number of brush fires started in the last few days alone. If you want to increase board civility, set the standard with your behavior, report uncivil posts to the moderators, even stop reading replying to people you think bring down the discourse if you must, but continuing to draw attention to your list of people you don't like and threatening people outright merely for going after your argument is just going to have the opposite effect of what you intend.

[Zzyzx]

.

Firstly, let it be said it is not "the list" but "your list", and in all honesty, 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. Goat and Zzyzx are forceful debaters, certainly assertive, occasionally aggressive, but those qualities are not uncivil.

The group has exactly ONE member other than the originator, so evidently there is little agreement with EduChris about the need.

Perhaps the group could be renamed Ignore those whose arguments I cannot handle (by calling them uncivil).

[EduChris]

...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?

...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.

...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.

...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.

[cnorman18]

...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?

...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.

I mainly want to comment on the incivility issue (below), but 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.

...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.

...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.

"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.

Objections to another member's ARGUMENTS can be abrasive and repeated multiple times, but as long as they address the argument and not the arguer, they are acceptable. Obviously obscene or vituperative language is unacceptable too; but vigorous debate is sometimes emphatic and passionate, and that is acceptable as long as it doesn't cross those lines.

In any case, publicly accusing another member of incivility is not appropriate, and is almost always itself an uncivil comment. I repeat; the proper procedure is to REPORT that perception and let a moderator handle it. WE are here to handle those disputes; when one tries to do it oneself, directly, one is contributing to the problem, not solving it. Accusations of that kind, we know from long experience, INEVITABLY turn a thread into an unpleasant and off-topic battleground with an atmosphere of bickering and personal remarks.

I hope that helps. For the record, I am posting this as a member and not as a moderator, which means anyone is free to argue, but my intent is only to clarify the concerns of the moderating team.