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

Great Debates
121

I was under the distinct illusion that some illustrious thinkers have questioned them (especially A). Aren’t the axioms:

A. G -> G (If God exists, he necessarily exists)

B. ~ ~G (God is not impossible)
Accepting the truth of the axioms depends extensively on the definition supplied for God, no? Indeed if we define God as the “greatest possible being”, we must accept both A and B. But this definition fails to enlighten me personally about much at all. G becomes nothing else than that which by necessity is. So:

That which by necessity is, is by necessity.

Great. It’s no more useful to me than the argument “if 2 is 2, 2 is 2”.

:confused:

Libertarian, I’d be happy if you could demonstrate the practical utility of this self-evident argument. It escapes me…

122

“”"""""""""“Conclusion: After correcting premise 2, the proof proves the existence only of the concept of god as the greatest possible perfection.”"""""""""""""

Woohoo !! Now I just have to figure out how to ensure that the concept of god never leaves my attention !!

(sorry, I couldn’t reisist =)

-Justhink

123

“”""""""""“Libertarian, I’d be happy if you could demonstrate the practical utility of this self-evident argument. It escapes me…”"""""""""""

Did what’s his name get any money from proposing it?

-Justhink

124

Eris

Because of the definition. The greatest possible form of existence is existence in all possible worlds. Existence in all possible worlds is called necessary existence.

Night

Oh, be nice.

That’s not a premise; it’s a definition.

Gah. That’s not even close to what it says.

Isn’t that what I’ve been explaining in some detail?

Ethic

How I look at it from a broader metaphysical perspective might not be meaningful to you (and actually is off-topic, but that’s okay because the proof itself has been established as valid). Here’s what it means to me, and if it means something different to you (or nothing at all), then I don’t have a problem with that. But you asked.

I know that I am not a perfect being. And there is only one world in which I know that I exist. That world is soaked through and through with corruption and cruelty. It is my fervent hope that the Perfect Being, in His perfect compassion, will see fit to use His omnipotence to bring me into a world that is just and fair, where life is more than just consumption and defecation until the cells rot. And because He loves perfectly, He will do this if I will allow it.

That is my own private, personal opinion, and there is nothing about it to debate, although you may, if you wish, tell me how stupid you think it is. But whatever you bring at me, I will hold onto it because I have glimpsed Him, and I know by immediate knowledge that He is there.

Thus endeth the off-topic musings. :wink:

Who? What?

125

all right.

Ah, but it is what it says, by implication. The whole reason this proof is so seductive to you is that it tries to hide this very implication. To prove this point:

So because it is the greatest possible perfection, it must have the greatest possible existence, necessary existence. Sounds reasonable… except WHY does it have to have existence at all? The only explanation is that existing in real life is more perfect than existing only as a concept in your head. But is this true? No. As I have said (and you have ignored), a concept is perfect by definition, and contains all possible perfection and no flaws. Something that exists in reality can never be more perfect than a concept, it can only be more flawed because now we can study it.

Oh! So you have known all along that you were only proving that there is such a thing as a concept of god. I was under the impression that you were attempting to prove the existence in reality of the god that we agree only exists in your mind.

126

:smiley:

It doesn’t. That’s why all the other steps were required to get to the conclusion, G.

I know nothing about your impresssions. But what was proved is that God, as defined, exists.

127

Hmm, I’m still having trouble with the necessary existence, or that “most perfect form of existence” is to exist in all possible worlds. I think that, were this carried to its logical conclusion, there would be a contradiction in many of the worlds (if not infinitely many of them) from a perfect being in all respects existing.

Apart from that, I still don’t see that a being that exists is more perfect than a being that doesn’t.

128

There seem to be two threads on this, but I think everyone involved is reading both, so I’m posting in the more appropriate one.

Lib, are you saying the argument for the first postulate something like:
[ul]
[li]Suppose an perfect being exists in this world.[/li][li]If he exists in this world only he isn’t as great as if he existed in all other worlds.[/li][li]So as he is perfect he must exist in all possible worlds?[/li][/ul]

I think I’d better make it clear that I was assuming that since we discussing the existance of God possible worlds would be ones potentially without God, that is we assume in any given world beings of various attributes exist, and if any of them are perfect, that is God (for that world), separate to any other possible worlds.

129

Shade (an Eris)

Yeah, there are two threads. As I see it, this one ought to deal with more technical aspects of the proof, and the other is a more informal discussion, although the nature of discussion itself causes some overlap. I think the other thread is where I gave the formal definition of perfection because of a comment that was made there. But really, only Newton and Truth have engaged the technical issues, so it really doesn’t matter. I’ll discuss wherever you want to.

With respect to the first postulate, “If God exists, then His existence is necessary,” there is no argument made. It is posited as axiomatic because it follows from the definition. In modal logic (and other forms of higher logic) there are two basic modes of existence: possible and necessary. Possible existence is defined as that which exists in at least one possible world. Necessary existence is defined as that which exists in all possible worlds. That’s why it was discovered by Hartshorne that modern modal logic applies itself so well to ontology, where existence can be easily understood in these terms.

Note that the first axiom is not saying that God exists, but merely that IF He does, then His existence is necessary. Aside from the technical aspects, this is intuitively easy to understand if you consider the greatness of a being Who exists in all possible worlds as compared to one who doesn’t, so long as you’re defining greatness with respect to their symmetry (another technical term that you can think of as meaning invariant under other changes).

We don’t begin by supposing that a perfect being exists in this world because we don’t know whether one does or not. Rather, we are dealing with a being whose existence, by definition, includes this world if He exists at all. It is very important to understand that the first axiom does not declare God’s existence. It is an implication, not a declaration.

130

But Lib, the more and more I think about this the more dissatisfied I am about those axioms in the first place. Consider what we’ve said here (not like you haven’t for the past three pages! I know…):

If the perfect being exists, it exists in all worlds; and, it is not impossible that a perfect being exists in at least one world.

When it is stated like that, do we really need all the formal logic anyway? The logic operations are more or less substitution and deduction. 14 = x-6 is similar: we state the problem and apply deductive rules in order to restate the axioms in a clearer sense.

But still, the first one does bother me greatly. The “all worlds” idea does seem to be intuitively at odds with “perfect”, or to use Anselm’s phrase, “which no greater can be conceived.” That seems to be a pretty hefty definition.

Secondly, the notion of “existing in all possible worlds” being somehow more perfect than “existing in one possible world” troubles me greatly, too. The most perfect being in world A might not have many qualities available to it that World B could offer. In world A, for example, there is no concept of identity. This is a possible world, you should grant. What does “existence” even mean there? Or a world where knowledge as we conceive of it is impossible (these two worlds overlap in some regard).

As I understand it, all possible worlds means all symbolic systems which may be defined. But can we not define a world where all qualities are finite but unbounded, that is, rings? Where b follows a follows c follows b etc? In such a possible world no concept of perfection exists, and so no such thing as a perfect being can exist, because no qualities have inherent maximums, and so we’re in quite a predicament. Either this world is impossible by the result of our G conclusion, or the axiom itself is flawed that if a perfect being exists it exists necessarily.

131

Not which may be defined, but which may be consistent, and even then consistency seems to be a view of a system in itself. I see no a priori reason to assume that all possible worlds must even be consistent.

132

Well, Lib, you seem to be coming very close to admitting the flaw in your argument, but then you keep pretending nobody pointed it out…

Lib says:

You are saying that because god is defined as perfect, he must exist in all worlds if he exists at all. This is NOT true. Existence does not increase perfection. In fact, concepts are completely perfect, and there is NO way a god that exists could be more perfect than your concept of him. You have already admitted this!

I said:

Lib said:

I don’t see what is left to argue about. By your own admission existence does not increase perfection, and therefore the first postulate is false. Without this postulate there is NO proof. Please ask questions if you need this explained in simpler terms.

133

Eris

The problem is that the “exists”, which I emboldened in your quote above, is followed by that nasty “If”, also emboldened. Note that the assertion that God does indeed exist is not made until the conclusion. It is simply “G”.

I don’t know why “hefty” would be a disqualifier. It doesn’t seem unreasonable that a definition of God might be substantive. But perfect is formally defined as all X, such that [~P(X) --> P(~X)], where P is any arbitrary type <<0>> positive substantive claim.

That’s exactly why only a priori synthetic descriptions of God are made. All we know about Him in the identityless World A is that He knows all there is to no about that world and that He can do anything in that world that can be done.

All possible worlds means all worlds that might possibly exist. The only excluded worlds are nonexistent ones. In the Jabberwocky world, if it exists, God exists in it and knows all there is to know about it, and can do everything the world allows. Consistency is certainly no prerequisite for God’s existence because there does exist a world (this one, for example) where consistency allows His existence to be proven in all others, including ones that are not consistent. We aren’t talking about what critters of those worlds might or might not be able to do, but merely whether God exists in them.

134

In your dreams. At least Newton and Truth were making valid modal arguments. You’ve yet to offer one at all.

Nor accurate. If He exists, then He exists in all possible worlds.

I don’t even know what that means.

What nonsense. Do you have a perfect concept of an electron field collapsing and producing a photon? Do you have a perfect concept of eternity and the notion that something can be simultaneously not yet begun, ongoing, and already finished? You seem to have concepts confused with their implications.

My question is when will your argument begin?

The first postulate does not even address any “increase” of any kind.

What is your problem? Rather than hearing you ramble any further with jingoistic red herrings and non sequiturs, what I’d like to see is your modal tableau.

135

Define god as the greatest possible perfection.

  1. If god (on the definition above) is only a concept, he is necessarily only a concept.
  2. god is not impossible.

Note that the first axiom is not saying that god is only a concept, but merely that IF he is, then his existence as only a concept is necessary. Aside from the technical aspects, this is intuitively easy to understand if you consider the greatness of a being who exists only as a concept in all possible worlds as compared to one who doesn’t, so long as you’re defining greatness with respect to their symmetry (another technical term that you can think of as meaning invariant under other changes).

136

I apologize for acting as though you needed things explained simpler. Perhaps my last post is what you were looking for?

137

Lib, what about Godel’s Incompleteness Theorems? Certainly a perfect logic system, as we are describing “perfect”, has clearly implicit ceilings. This proof should demand perfect logic systems to exist in all possible worlds, too. Yet this is something we’ve already demonstrated cannot exist.

138

A tableau is a backward chaining proof method that seeks to demonstrate that no counter-model for a proposition exists, or to find one if it does. Tableaux start by asserting all the assumptions of an argument and the negation of the conclusion. Why the negation of the conclusion? The aim is to discover whether there is a non-contradictory way that the assumptions can be true while the conclusion is false. If so, then the inference is not valid (and the counter-model can be exhibited). If not, the inference must be valid.

There are many different systems and notations of tableaux. It is sometimes hard to render without diagrams, but I’ll try using a set-based notation. The system I will use is similar to those found in the literature; and it can be proven sound and complete with respect to the (modal) logic. The argument below is somewhat technical, so feel free to trust me and jump to my conclusion :).

Example: Not Tisthammer’s Argument

At each step, we’ll have a set of propositions that are true at a particular world. I will write A:i, where i is an integer identifying a world to mean “A is true at world i”. The initial set contains the assumptions and the negation of the conclusion, all true at some “starting world” (world 0).


1. {G->[]G:0, <>G:0, ~G:0}

I am using a flawed version of Tisthammer’s proof (it best illustrates the tableau method). The initial assumption includes G->G, where -> is material implication (A->B is equivalent to ~A OR B).

The tableau method identifies subsets of this set that can be removed and replaced with other sets of propositions. An atomic proposition or its negation cannot be further reduced, and will remain in the set. Thus, there is nothing to be done about ~G:0.

The rule for possibility is that <>A:i can be removed, and replaced with a pair of propositions {iRj, A:j}. We assume a new world j that is accessible to i via the relation R, and where A is true. The world j “witnesses” the possibility of A at i. Making this change:


2. {G->[]G:0, 0R1, G:1, ~G:0}

Notice that <>G:0 has been removed. The rule for an implication in a world is that the implication is true if either the antecedent is false at the given world or the consequent is true in the given world. This requires that the tableau branches into two possibilities, which must both be explored. The success of either branch leads to a counter-model that makes the inference invalid.


3a. {0R1, G:1, ~G:0}
3b. {[]G:0, 0R1, G:1, ~G:0}

Notice that 3a. actually includes two instances of ~G:0 (one from the initial set, and one from the negation of the antecedent of G->:0). We remove duplicates. First, let’s explore 3b. The rule for necessity is that a proposition A:i is not removed, but propositions A:j are added for every world j such that iRj is already known. A:i is left behind to remind us that we must add A:j everytime we encounter a new j such that iRj. This allows modal tableau to be possibly infinite.


4b. {[]G:0, 0R1, G:1, ~G:0}

results from adding G:1 to the set, due to G:0 and 0R1. Note that G:1 was already included. At this point, there is no rule that we can apply that will add anything new to the set, and there is not yet a contradiction. This means that the inference is not generally true (this set demonstrates a way that the assumptions can be true and the conclusion can be false). However, if we require R to be reflexive, we must add iRi for every known world i.


5b. {[]G:0, 0R1, G:1, ~G:0, 0R0, 1R1}
6b. {[]G:0, 0R1, G:1, ~G:0, 0R0, 1R1, G:0} X

At the last step (6b), we added G:0 from G:0 and 0R0. The X indicates that this branch of the tableau is closed, since it contains a contradiction (~G:0 and G:0) at a world. That means that (along this branch), there is no non-contradictory way that the assumptions can be true and the conclusion false.

Returning ot 3a:


3a. {0R1, G:1, ~G:0}

there is no rule left to apply, and no contradiction. Thus the inference is not valid, and this set provides a counter model: two worlds 0 and 1, with 0R1 and ~G in world 0 and G in world 1.

Variation: Still not Tisthammer’s Argument

What if we replaced G->G with G<->G? The initial set is:


1. {G<->[]G:0, <>G:0, ~G:0}

as before, we eliminate <>G:0


2. {G<->[]G:0, 0R1, G:1, ~G:0}

The rule for A<->B is that either both A and B are true, or they are both false. Thus, the tableau branches again:


3a. {G:0, []G:0, 0R1, G:1, ~G:0} X
3b. {~[]G:0, 0R1, G:1, ~G:0}

Notice that the duplicate ~G:0 was removed from 3b. 3a contains a contradiction. We continue with 3b. All negative modalities are rewritten (~A is <>~A, and ~<>A is ~A):


4b. {<>~G:0, 0R1, G:1, ~G:0}

And the rule for <> is used:


5b. {0R2, ~G:2, 0R1, G:1, ~G:0}

With no rule left be applied and no contradiction. The inference is not valid, and the counter-model can be read from the set 5b.

Tisthammer’s Argument

Finally, let’s consider Tisthammer’s argument:


1. {[](G->[]G):0, <>G:0, ~G:0}
2. {[](G->[]G):0, 0R1, G:1, ~G:0}

we immediately eliminate <>G:0 as before. Next, we use the rule to make G->G at every world known accessible from world 0:


3. {[](G->[]G):0, 0R1, G:1, ~G:0, G->[]G:1}

We break G->G:1 into the two possibilities:


4a. {[](G->[]G):0, 0R1, G:1, ~G:0, ~G:1} X
4b. {[](G->[]G):0, 0R1, G:1, ~G:0, []G:1}

4a contains a contradiction. 4b. does not, and there is no rule that can be applied to add anything new to the set. If we could get G:0 or ~G:1, we would have a contradiction in 4b. There is no immediate way to add ~G:1; but we could derive G:0 from G:1 if 1R0. Let’s add that:


5b. {[](G->[]G):0, 0R1, G:1, ~G:0, []G:1, 1R0}
6b. {[](G->[]G):0, 0R1, G:1, ~G:0, []G:1, 1R0, G:0} X

The way to get 1R0 from set 4b is to insist that the accessibility relation R is symmetric. Since all branches of the tableau are closed (contain a contradiction), there is no way that the premises of the argument could be true and the conclusion could be false in a modal logic with a symmetric accessibility relation.

Conclusion

Thus, the Tisthammer argument is valid (remember that his first axiom is a strict (or necessary) implication). To reject the argument, one must reject at least one of , <>G, and that the accessibility relation on possible worlds is symmetric. I’ve argued in this thread (and I still maintain), that adoption of any two of these nearly compels rejection of the third. I personally find it easiest to reject one of either <>G or symmetry, since I am inclined to grant that Libertarian has made a good case for his .

139

erislover, I think you’ve caught on to something intriguing.

There is no reason to assume that. Non-normal modal logics allow possible worlds where logic does not hold. They insist that the actual world is normal (logic holds). I don’t see any reason that the argument is invalid in a non-normal logic (equipped with a symmetric accessibility relation).

I’m not sure incompleteness comes into play. We’re not considering a true statement that cannot be proven, but a proven statement that might not be true. You would have to argue that the modal logic is unsound.

I think that is a promising avenue: is there any reason to imagine that the modal logic used in the argument is sound with respect to reality? Is there a reason to imagine that the actual world is non-contradictory? Those two questions seem to be at least as hard as “does God exist?”.

140

Perfect logical system: one that is both complete and consistent and powerful enough to express a notion about God. Godel’s theorems state this is impossible (unless God is less complicated than Peano’s Axioms!).

We’ve already determined that specific properties need not hold to god if they have no inherent maximum. Thus anything which can be described as having a set of properties which contain an absolute maximum is demonstrated to exist.

The properties of the “perfect logic system” are stated above. Since we’ve already decided that God can be described logically by attempting to use Tishammer’s proof, we should state that describing the most perfect being is an inherent maximum. What else is there? Also, a complete system. There’s nothing more we can add to completeness to make it more than complete. Also, absolute consistency. If there’s no contradiction, then there’s no contradiction. We can’t have less than no contradiction.

I. If a perfect logical system (as described above) exists, it necessarily exists.
II. It is not impossible that a perfect logical system exists.

Now, (II) directly contradicts GITs inasmuch as we’ve defined “perfect logic system”. So now what? It seems to me that we’ve justified (I) and (II) as much as Tishammer has justified his (I) and (II) about God… in fact, I feel I’ve done him one better since we’re evern assuming his definition is possible in the first place (something that also doesn’t sit well with me, but for other reasons that I will mention if this tactic fails).

If (II) directly contradicts GITs and we accept that GITs are, in fact, correct, then any proof which allows a formulation contrary to GITs are themselves suspect of GIT’s inconsistency, and anything they prove should remain considered unproven.

If (II) directly contradicts GITs and we reject, then GITs… well, now what?