A modern symbological assessment of the ontological argument for the existence of God

Newton meter, thank you. I understand you much better now. Thank you for going out of your way to make your point more clear.

My problem is with your first axiom being derived from Tisthammer’s first axiom. I think you’ve taken the inverse, rather than the contrapositive. The same truth value might not apply. The contrapositive of “if God exists, he is necessary” would be “if God is not possible, then He does not exist”. You’ve said, “if God does not exist, then he is not possible”.

I also think you’re beating around the bush too much with your concern about popping into and out of worlds. There is no argument in this case about whether what is possibly necessary is true. This isn’t a more general metaphysical treatment, but an ontological one. Therefore, all that is addressed is whether what is possibly necessary is real.

I can’t disagree with you that we make positive statements about God. However, we make them informally. And it’s well known that such statements cannot be made formally. So it isn’t relevant to introduce them into an ontological argument, any more than it would be relevant to inject into a scientific paper on Mandelbrot sets how beautiful they are. Interesting, maybe, but beside the point.

So, where are we now?

Just a sec, Newton. You say that you agree with Apos. But surely you do not agree that defining God as having no positive descriptions translates to an admission that existence is indistinguishable from nonexistence, do you?

I wasn’t clear about where I was getting that from. It’s not the inverse or the contrapositive. All these modal quantifiers floating around get hairy.

The inverse of Tisthammer’s first axiom is ~p->~p, or ~p-><>~p: if God does not exist then it is possible that God does not exist. I doubt either one of us will have any problem with that. (Showing my work: ~p = ~~<>~p = <>~p)

My claim is stronger: ~p->~<>p: if God does not exist, then God is not possible. This claim comes from insisting that Becker’s postulate (modal status is necessary) should apply to the sentence ~p. I suggest that if you apply Becker’s postulate to p (as Tisthammer insists), you should also apply it to ~p. Clearly, this is not generally valid so something about God must permit it. I’m hoping that the reasons that would cause one to reject my “first axiom” would also cause one to question Tisthammer’s “first axiom”, but the case is not compelling.

Examining the contrapositive of Tisthammer’s first axiom is interesting. It is ~p->~p, or <>~p->~p: if God possibly doesn’t exist, then God doesn’t exist. If that seems strange, then I think you also ought to question Tisthammer’s axiom.

In the actual world, still having difficulty convincing everyone that God’s existence is a guaranteed consequence of what we mean by “God”. :wink:

Oh dear. I do not agree with that. Your ability to lead me around by the nose depending on the way you phrase the question convinces me that I really don’t understand this positive/negative mess.

Sorry for the double post, I couldn’t access the site. If someone can remove the second one, that’d be cool.

—To say that no positive descriptive term can be applied to God is merely to say that He is not, for example, a dodecahedron.—

I’m starting to see the same problem that ethicallynot had with you. You did not address my argument, and indeed focused on a point which, while true, has nothing to do with my criticism. Yes, negative definitions define what a thing is not. I know that well. And indeed that was the basis for my criticism. You can’t play the game of negative definitions forever. If you want to discuss an entity in any meaningful sense, at some point you’re going to have to commit yourself to some positive description. If you can’t do this, I submit that you’ve still yet to escape the criticism that you’re not discussing anything at all, but merely listing a lack of characteristics, devoid of any sense of WHAT lacks them.

—To say that no positive descriptive term can be applied to God is merely to say that He is not, for example, a dodecahedron.—

As I said, if this is all you are prepared to suggest, then I have to point out that non-existence lacks the characteristics of a dodecahedron too (as it lacks all characteristics). How is this entity in the proof in any way different from something which does not exist?

Existence is not, itself, a perfection. It is a necessary condition required for an entity to be able to be perfect in some or any respects.

Some things only lend themselves to negative descriptions. God is not the only example. Infinity is usually defined in terms of sets. And you can tell whether a set is infinite if you can throw away one element and still have a set of the same size.

You lost me here, as I’m once again unsure of where you’re coming from. I thought, from your earlier clarification, that you were arguing Tisthammer’s inverse to be axiomatically true. So here, I’m wondering what you mean: are you saying that every rule that applies to p ought to apply to ~p, or that there is something about God that permits you to apply it when you ought not?


What game? Every standard form categorical proposition has an affirmative or negative quality. A mediate assertion is no more or less valid than an immediate one.

It’s nice to stretch these long-unused mental muscles!
I believe erislover has identified the real problem. Tisthammer himself alludes to it and then, oddly, dismisses it.

**
No, it isn’t nuts, merely absurd – precisely the point of the exercise.

Tisthammer hasn’t proved the existence of “God”. He has simply defined God as “existence.” There is an enormous logical difference in saying “God is perfection” and in saying that which is as perfect as possible is God. In any set (apart from null) there is always one member of the set which is “the most X” where X is some objectively definable characteristic. If the set is the universe, by definition the member that is “most X” is the “most possible X.”

I’d also point out that Tisthammer’s definition of God runs afoul of his own refutation of “the greatest possible island” problem. He claims that this is meaningless because it is subjective. So is his definition of God, really. By defining several characterstics of God and then asserting that that being which is as omniscient “as possible,” as omnipotent “as possible,” and, therefore, as perfect “as possible,” he overlooks that you cannot maximize over more than one variable. In other words, assume that that being which is as omnipotent “as possible” is a different being than that which is as omniscient “as possible.” Which of these is the “most perfect?”

—In either case, once we agree to engage in modal discourse, it becomes meaningful to talk about whether an object exists, and to talk about properties of non-existent objects.—

Sure: but that still doesn’t justify talking about existence as a special property of things. I can meaingfully talk about Santa Claus without saying that it exists: because I am applying characteristics to a conception in my mind, which exists, not to a Santa Claus which does not necessarily exist. Indeed, my “conceptual” Santa Claus is really going to end up just as a definition that I would then have to try to apply to all existing entities to find out if my definition fits anything that exists or not.

—Lib: But surely you do not agree that defining God as having no positive descriptions translates to an admission that existence is indistinguishable from nonexistence, do you?—

I, for one, do agree with myself. However, I didn’t claim that god IS indistinguisable from non-existence: I just inquired as to HOW it could be distinguished. I do feel that swearing off positive descriptions is essentially swearing off any commitment to discuss anything at all. Perhaps you ARE discussing something, but how can we know what it is, so that we can evaluate it’s definition? Are you really saying that this argument successfully pulls out a conclusion of necessary existence for an entity that so far doesn’t even have a single defined characteristic?

Not even close. I’ll try one last time on this non-issue. A negative definition IS a definition. We are not leaving God with undefined characterstics. We define God negatively as the intrinsic maximum of existence. That is why His existence, if possible, is necessary. And vice-versa.


The being different from both of them that is both omnipotent and omniscient.

**
Lib,
I think there are a couple of problems with this response,[list=1]
[li]You have just proven polytheism.[/li]If there is necessarily a being who is “maximally omniscient” and “maximally” ominpotent, there is also necessarily a being who meets these criteria and has the largest shoe size (or the smallest) or some other criterium unrelated to the defined attributes of “God.” These beings would equally be “God” as per whatever definition you are using.

[li]You assume that the “properties” of God are orthogonal.[/li]Your response assumes that all the properties of God are independent. We are not talking about an omnipotent God, we are are talking about a God who is “as omnipotent as possible.” However, what if the “rules” were to require a trade-off between omnipotence and omniscience? In other words, what if it is possible to be “more” omniscient if you are not as omnipotent “as possible?”

In this case, you would have to make a subjective choice between maximizing omnipotence and omniscience. [/list=1]

I think the reductio ad absurdum bit works just fine with all these “proofs”, from Anselm onwards.
The problem is that Anselm etc. say "We define God as ‘That which there can be no greater’ " without demonstrating that God is anything like what they define. (Actually IMO it’s a partial description, not a definition, but let that go.)
Fine. I “define” The President of the United States as “The wisest, most beloved, and tallest American citizen.”
David Robinson is an American citizen, and is seven feet tall.
Therefore, George Dubya is at least seven feet one inches tall.
Q.E.D.

  1. Premise.
  2. Premise.
  3. I don’t believe that was ever formally defined, but I assume that its definition includes this statment.
  4. True.
  5. Becker’s postulate should be stated as a premise, as no effort is made to prove it. And even if it were were one of the premises, this would not follow. The postulate only refers to p and <>p, not ~p.
  6. True, given the preceeding.
  7. Follows from modal modus tollens, but modal modus tollens was never justified.
  8. True, given the proceeding.
  9. True, given the proceeding.

Analysis: this argument presents the conclusion as being proven based on two premises. However, not only is there no reason to believe the original premises, several other premises were snuck in during the middle of the proof without any justification. And even with additional premises, the proof still doesn’t work. Furthermore, an obvious argumentum ad absurdum makes it hard to accept the argument.

Conclusion: the sole result of this is to show that a set of premises (which weren’t even clearly articulated) might result in a ridiculous conclusion. The obvious reaction, at least to me, is to suspect that there is something wrong with the premises, rather than accept the conclusion.

— I’ll try one last time on this non-issue.—

An apt name for it! For non-existence, everything is a non-issue. I suppose that’s the appeal of such proofs.

—A negative definition IS a definition.—

But not a litterally meaningful one for describing an actual entity. Negative definitions tell you what something is not, not what it is. For instance, rocks do no have god beliefs. They don’t have any beliefs, actually. Neither does non-existence. But if you are going to meaningfully describe an entity like a “rock,” at some point you’ve got to bite the bullet and specify what it is that you’re talking about.

—We are not leaving God with undefined characterstics. We define God negatively as the intrinsic maximum of existence.—

Oogly woogly who? What is a “maximum of existence”? Existence without limit? I wasn’t aware that “existence” was a concept that permitted graduations.

—That is why His existence, if possible, is necessary. And vice-versa.—

Let’s pretend that this proof works:

Onto: I have just proven the necessary existence of god!
Joe: Wow, ya sure have: so what is it that you’ve proven? What can we now say exists?
Onto: …
Joe: …
Onto: I can tell you what it ISN’T though.
Joe: …
Onto: It’s not an ardvark!
Joe: …
Onto: And no limits on its powers!
Joe: Which powers?
Onto: We cannot describe them, because they are limitless.
Joe: Look, I’d love to believe in this god of yours, but I still have no idea what I’d be believing in. Your proof seems valid, but for the life of me I can’t see what it’s a proof OF. This entity seems perfectly synonymous with no entity at all.

I sort of like this formulation, in the Biblical idiom:
And Jesus said unto them, “And whom do you say that I am?”
They replied, “You are the eschatological manifestation of the ground of our being, the ontological foundation of the context of our very selfhood revealed.”
And Jesus replied, “What?”

Egad, it occurs to me that you don’t know what we mean by positive and negative descriptions. Positive definitions are descriptions of immediate observations. Negative definitions are descriptions of mediate observations. Tisthammer et al are not saying what God is not. They’re describing what God is in terms that they cannot know directly, but can only conceive. That is what is meant by positive and negative descriptions.

Infinity is no different. You have never observed a set that remains the same size when an element is removed. It is something that you must conceptualize. And some of this childish ridicule at the God definition sounds like someone making fun of the idea that both the set of integers and the set of even integers are the same size. “What the heck kind of oogly woogly is that?

Infinity does not exist in the context of a number system where you can treat it like a number. There is no counting to reach it, and it is the successor of nothing (and thereby not applicable to the Induction Axiom). But it does exist in a context of topological space, where you can use it to describe what certain sequences of numbers converge to.

This is the sense in which God is described. He is the maximum of all convergences. No serious treatment of the ontological argument, even by its most famous detractors, wastes time and intellectual resources by quibling over whether the definition is adequate. They might offer a different definition, but you won’t see them pretending not to know what this one means.

You shouldn’t either.

No.

You can’t use half the definition, nor modify it with arbitrary qualifications. I mean, you can, but then you’re not dealing with this proof. A being that is all knowing without being all powerful is not God. Not for this proof. Nor can you use positive descriptors like shoe size. (A shoe cannot be larger than the universe.) Nor can you use descriptors that are ontologically dualistic. You may not, for example, assign to Him both the maximum power and the maximum impotence, nor the maximum goodness and the maximum evil.

Op. cit.

I’m still having trouble with this definition of “positive” and “negative”. What is an “immediate observation”? If it means “not mediated” - how can any observation not be mediated? All my observations are mediated by my senses.

Then how do you choose which to assign?

Lib said: “You have never observed a set that remains the same size when an element is removed. It is something that you must conceptualize.”
Sure I have. Draw a circle. Go all the way around it once, or a million times. Now count the number of times you have to go around it until you get to the end.
The “set” is the distance to the end of the circle. An element is once around. Remove one revolution, or a million, and the outcome remains infinite.

Lib, I’ll try to make my objections explicit. I have at least three objections, and I’ve been playing mix and match in successive posts. I’ll present the obvious non-modal objection first, followed by my modal objection as clearly as I can state it.

Non-modal objection

Tisthammer’s first assumption is G->G. Things have already gone terribly wrong. It’s like watching a chess game where the first move is a4.

I would ask that you accept <>~G, that it is possible God does not exist. If one is not willing to admit (in their proof of God’s existence) that it’s possible he doesn’t exist, then they are clearly engaging in question-begging. Now, in all systems of modal logic, <>~G = ~G. If it is possible God does not exist, then it is not necessary that he exists. Now, using G->G, ~G, and plain old propositional modus tollens, we immediately conclude ~G. God does not exist.

The only way G->G does not lead immediately to God’s non-existence is if you insist that ~G is false. That is, by the excluded middle, you insist that G is true. So G->G compels you to accept the axiom “God necessarily exists” if you don’t want to be able to (trivially) prove that he doesn’t.

Clearly, the argument is still valid. The question-begging makes is suspicious, and unlikely to convince a non-believer.

Modal objection

There is a way out of the conundrum above. You can recognize that when Tisthammer claims G->G and when I claim <>~G, we are using two completely different notions of accessible possible world. Of course, this is exactly Tisthammer’s modal flaw: he fluidly varies his notion of accessible possible world to suit his needs.

The modal sentence G->G doesn’t mean “God has the greatest possible existence”. It’s really just a bit of syntax that doesn’t mean anything without a semantics. To give a semantics, we present a set of possible worlds, and an accessibility relation on possible worlds. Then, G->G means that if God exists in a world, He exists in all worlds accessible to that world. The semantics is important, because it determines the valid modal axioms.

Tisthammer’s flaw is that he never specifies the sort of accessible possible worlds he is talking about. There are several possibilities, all resulting in different valid modal axioms (or axiom schemas). Modal logic can be used to talk about what is necessary, what is known, what is believed, what ought to be, what hold locally, what holds now and in the future, what is provable in Peano arithmetic, what is true after the computer program terminates, and many others. Since Tisthammer doesn’t specify a system of modal logic, we’ll have to try to infer it.

First, he has G->G. If God exists in a world, then he exists in all worlds accessible to that world. Though this is purported to be a statement about what we mean by God’s existence, it also determines the kind of accessibility relation we can have on possible worlds. Assume two types of worlds, those with God and those without (either set can be empty). If we ever find ourself in a God-world, the only accessible worlds are other God-worlds. Once we get to a God-world, there is never an accessible non-God-world.

Next, he has <>G. God is possible in all worlds. That is, whether we’re in a God-world or a non-God-world, there is always an accessible God-world. Together, these two imply that if we’re in a God-world, we can only ever “get to” other God-worlds. If we’re in a non-God-world, we can always get to at least one God-world. This asymmetry bothers me a bit.

It’s not damning to Tisthammer’s argument, though. It is quite possible that the nature of God induces this asymmetry. God is always possible (“innocent until proven guilty”, according to Tisthammer), and his existence spreads out to all accessible possible worlds (perfect existence). The conclusion is that we must have an asymmetric accessibility relation.

Now, in step 5. of his proof, Tisthammer uses Becker’s postulate (that modal status is necessary), applied to ~G (which is <>~G). This is what I called a “dirty trick” before. We use accessibility to go from here (whether there is a God or not) to a world with God, then back to here; concluding God is here. Becker’s postulate can only be applied to <>A if the accessibility relation is symmetric. All this after carefully convincing us that the accessibility relation was asymmetric (otherwise we would not have blithely accepted <>G based on Tisthammer’s weak “innocent until proven guilty” argument).

This is my objection. His two assumptions seem to require an asymmetric accessibility relation, his use of Becker’s postulate requires a symmetric accessibility relation. Without Becker, no proof. The only way to insist on his axioms and insist on a symmetric transition relation is if he requires all possible worlds to contain God. That is, as in my first objection, the only way he can make the proof work is if he insists on the assumption that God exists.

Again, it’s obviously a valid proof, but Tisthammer fails to make clear that his proof only works if all possible worlds already contain God. So, if God exists then God exists. QED.

—Egad, it occurs to me that you don’t know what we mean by positive and negative descriptions. Positive definitions are descriptions of immediate observations. Negative definitions are descriptions of mediate observations. Tisthammer et al are not saying what God is not.—

Aren’t they? Then where do they say what it is? And none of these definitions can come from any theoretical observation without at least one theoretical immediate observation. How would you know WHAT you are looking at to determine that it was not part of a particular set?

—They’re describing what God is in terms that they cannot know directly, but can only conceive.—

By concieving only of what it is not!

—That is what is meant by positive and negative descriptions.—

Every description used so far has been a case of describing what this entity god lacks. But an entire host of lacks does not add up to an entity (unless this is a “process of elimination” out of a finite number of known possible entities). A negative description is certainly useful for distinguishing elements of a set from everything else, but it alone CANNOT define a specific entity.

I’m simply making the point that it is trivially easy to prove anything you wish when you do away with the need to ever specify what it is that you are proving.

—Infinity is no different. You have never observed a set that remains the same size when an element is removed. It is something that you must conceptualize.—

But I can’t conceptualize it, can you? I can only mouth the words that make up its definition, and then try to analyze whatever logical content I can unpack from that definition. But that exercise alone is simply not enough guarantee that we are discussing anything at all.

—And some of this childish ridicule at the God definition sounds like someone making fun of the idea that both the set of integers and the set of even integers are the same size.—

You can address my arguments if you please, or not if they are beneath you, but no editorializing or accusations please. I apologize for the over the top oogly comment, but the gist of it is that “define negatively as the intrinsic maximum of existence” does not seem to have any meaningful content that I can analyze, especially containing, as it does, this strange term “maximum of existence.” How can “existence” have a “maximum”? What does this claim mean?

—But it[infinity] does exist in a context of topological space, where you can use it to describe what certain sequences of numbers converge to.—

As I noted, it exists because a particular definition is given, and portions of that definition are unpackable so as to draw workable logical points. But that simply does not guarantee that we are having a meaningful discussion. One can pack anything they wish into a definition for use later on, even the stipulation that none of its elements contradict.

—No serious treatment of the ontological argument, even by its most famous detractors, wastes time and intellectual resources by quibling over whether the definition is adequate.—

I suppose you are an arbiter of what a “serious” treatment is? To even give the argument a serious treatment one would first have to agree that it is a meaningful argument. But many do not: they can’t give it any sort of treatment at all. So obviously if you go looking for serious treatments of it, you aren’t going to find dectrators of my sort.