The Ontological Argument

[Brian Bosse]

I realize there are many Christians who think the traditional arguments for the existence of God are not sound. I would like to explore this claim with the ontological argument. I will present a modified argument based on the formal modal argument presented in Hartshorne’s The Logic of Perfection, Open Court, 1962, p. 51.

A Formal Presentation of the Ontological Argument

p = there exists a God such that He is a perfect being.

Modal Operators

□p = p is necessarily true. That is to say, that in all possible worlds p is true.
◊p = p is possibly true. That is to say, there exists a possible world where p is true.
p = p is actually true. That is to say, in the actual world, p is true.

It should be noted that we can define both □ and ◊ in terms of each other. For instance:

Rule N: □p ↔ ¬◊¬p.
Rule P: ◊p ↔ ¬□¬p.

Logical Rules

A. Disjunctive Syllogism: [(a V b) Λ ¬b] → a
B. Modus Ponens: [(a → b) Λ a] → b
C. Modal Modus Tollens: [□(a → b) Λ □¬b] → □¬a
D. Substitution: [(a V b) Λ (b → c)] → (a V c)
E. Becker’s Postulate: □a → □□a; ◊a → □◊a (Modal status is always necessary.)
F. Excluded Middle: a V ¬a
G. Modal Axiom: □a → a

Formal Proof

1. □(p → □p) (Anselm: It is necessarily the case that if there exists a perfect God, then this God necessarily exists.)
2. ¬□¬p (Anselm: It is not necessarily the case that there does not exist a perfect God. This is the same as saying: It is possible for there to exist a perfect God, i.e. ◊p by Rule P.)
3. □p → p (Rule G)
4. □p V ¬□p (Rule F)
5. ¬□p → □¬□p (Rule E)
6. □p V □¬□p (Rule D – 4 and 5)
7. □¬□p → □¬p (Rule C – 1)
8. □p V □¬p (Rule D – 6 and 7)
9. □p (Rule A – 8 and 2)
10. p (Rule B – 3 and 9)

If this proof is not sound, where and why does it break down?

Sincerely,

Brian

 

[Brian Bosse]

*Bump*

 

[Vytautas]

I think the problem is that (1) or (2) are false because the argument is valid and not sound.

 

[staythecourse]

What is "perfect" as defined in the argument. Could perfection include complete "non-beingness" is for example?

 

[Brian Bosse]

Hello Gentlemen!

I finally got a taker.

I think the problem is that (1) or (2) are false because the argument is valid and not sound.

Yes, I agree that the argument is valid. It may not be sound. If (1) or (2) are false, then it is not sound. Let's begin with (1). I would argue that it is true by definition. Think of it this way: if the Christian God exists, then He exists necessarily. This is true by definition. This leaves us with (2), but before I comment on this I will let you respond.

What is "perfect" as defined in the argument. Could perfection include complete "non-beingness" is for example?

I don't think the predicate of perfection would indicate non-being. In fact, if it did include this, then it would be incoherent to speak of any being as perfect. As far as a definition, I would say that perfection is the state of being flawless. Of course, you could ask me what flawless means. But I don't think this line of thinking is going to be helpful. Let me go back to my claim regarding (1). If the Christian God exists, then He exists necessarily. This is no different than saying that if the Christian God exists, He exists as a Trinity, or whatever. It is true by definition. Now, I know this is a little different than what Anselm argued, but who cares?

Sincerely,

Brian

 

[Vytautas]

Yes, I agree that the argument is valid. It may not be sound. If (1) or (2) are false, then it is not sound. Let's begin with (1). I would argue that it is true by definition. Think of it this way: if the Christian God exists, then He exists necessarily. This is true by definition. This leaves us with (2), but before I comment on this I will let you respond.

If one can say that (1) is true by definition, then could not one also easily say that (2) is true by definition as well? If God exists, then he exists necessarily, because he does not require another being to explain his existence because he is the greatest being. Something is possible only because there is another being to explain or ground the cause of that being. So to say that it is possible that a perfect God exists would require another being to ground the possibility that a perfect God exists.

 

[Cheshire Cat]

Here is my two cents, even though I haven’t studied modal logic at all, and I barely know regular logic.

If one can say that (1) is true by definition, then could not one also easily say that (2) is true by definition as well?

No, I don’t think so. Proposition 2 says ‘It is not necessarily the case that there does not exist a perfect God’. Meaning I think that it is logically possible for this perfect God to exist. This doesn’t have the same definitional attribute as proposition #1.

Something is possible only because there is another being to explain or ground the cause of that being. So to say that it is possible that a perfect God exists would require another being to ground the possibility that a perfect God exists.

I think you might be equivocating on the word ‘possible’. It seems to me at least that possible is being used as logically possible in the argument, while you are using it in a different sense.

 

[Brian Bosse]

Hello Gentlemen,

Great comments.

If one can say that (1) is true by definition, then could not one also easily say that (2) is true by definition as well?

Vytautas, I agree with Caleb's comments. (2) needs to be established. It is not clear that the God Christianity asserts exists can actually exist. If this God is somehow logically unable to exist, then the proof fails. So, the person putting forth this argument needs to argue for (2); whereas, the person putting forth the argument needs only to show that (1) is true within the Christian worldview.

Something is possible only because there is another being to explain or ground the cause of that being. So to say that it is possible that a perfect God exists would require another being to ground the possibility that a perfect God exists.

Caleb points out that there may be an equivocation on the modal term 'possible'. Certainly, within modal logic, 'possible' has a particular meaning. Here is a definition:

Possible: X is possible if and only if X is true in at least one possible world.

Right away one can see there is a problem here. We used 'possible' in our definition when we used the term 'possible world'. Here is a way to define 'possible world'...

Possible World: X is a possible world if and only if all events in X logically cohere.

Now, this definition is little more helpful leaving us with this definition...

Possible: X is possible if and only if X is true in any world where all events logically cohere.

An event logically coheres if it is logically possible (again, that nasty problem of using possible) and if it is not logically precluded by other events. Even though this still may be problematic, I think for our purposes it suffices. With all of that said, Vytautas's argument is an applicable argument against (2). He is essentially saying that for X to be possible, then X must be grounded in some other being. Since there is no other being to ground God, then God is not possible. If this can be established, then I would grant that this refutes the proof. The difficult part will be to justify the assertion that all beings require a ground outside of themselves. This is mere assertion, and as such is inauthentic. Since it is mere assertion and consequently inauthentic, then the possibility of God's existence still exists! That is to say, the person putting forth the objection still has not yet removed the possibility of God's existence. (Note: I may now be equivocating on 'possibility' here.)

Now, this is all fine and dandy, but the apologist (or whomever) still needs to justify (2). I will let you all comment before I put forth an argument for (2). Thanks for all of the great comments.

Sincerely,

Brian

 

[B.J.]

Brian,
Even if it is granted that the argument is valid and sound, so what?

A perfect God? Can the predicate "perfect" stand alone in a complete discription of this God? I say no! Is this God "perfectly evil",or "perfectly good?" Is God perfectly female, or male? Maybe It is none of these. I think whomever uses the argument is being ambigious with the term perfection. What do you think? I know you defined it as flawless above, but can it be that flawless is another ambigious term? What does that mean? For the time being I charge your argument with being ambigous with terms. That is to say, you are applying arbitrary, meaningless predication to a concept known as God.

 

[JohnV]

B.J.:

I believe you have it backwards. It is more that the concept and ideal of perfection does not exist without God. It is exactly the opposite that is the case: perfection is worse than ambiguous, it is meaningless if there is no God. We're no longer talking about whether or not God exists, but we're talking about the world as we know it and how we know it. Can it be that one thing is no better, of no more value, or of no greater importance than another? If we erase perfection (which is what we eventually do when we deny God's existence) then the mere fact that things are, without value or qualification, is all that's left. And even that disappears altogether, if we continue to follow through consistently. The point is that perfection loses its meaning if God does not exist; the proposition that God does not exist loses its meaning, and is no more or less true than the proposition that God exists. The point is: you lose a lot more than a proposition that God exists when you deny that God exists. Just the fact that we have such a notion of perfection is evidence enough that He does indeed exist.

Words can be wrangled, but sooner or later the chickens come home to roost.

 

[Vytautas]

For 7), could you have □~□p → □~p (Rule D and G)?

Vytautas, I agree with Caleb's comments. (2) needs to be established. It is not clear that the God Christianity asserts exists can actually exist. If this God is somehow logically unable to exist, then the proof fails. So, the person putting forth this argument needs to argue for (2); whereas, the person putting forth the argument needs only to show that (1) is true within the Christian worldview.

In order to show that it is possible for there to exist a perfect God, it must be shown that a perfect God is not self-referentially incoherent which is done by explaining what God is by using a coherent definition.

 

[Vytautas]

I think whomever uses the argument is being ambigious with the term perfection. What do you think? I know you defined it as flawless above, but can it be that flawless is another ambigious term? What does that mean? For the time being I charge your argument with being ambigous with terms. That is to say, you are applying arbitrary, meaningless predication to a concept known as God.

The argument does not depend on God being perfect but on God being a necessary and possible being given propositions one and two.

 

[PresReformed]

Isn't the question of whether the ontological argument is a valid argument or not moot? Even if it is a valid argument it cannot prove the existence of the trinitarian God of the Bible. It could just as easily be used as a proof for the Jewish god or the Muslim god couldn't it?

 

[Brian Bosse]

Hello Everyone,

For the time being I charge your argument with being ambigous with terms.

Fair enough. I grant that the concept of perfection is vague, but when one pushes any predicate far enough it ends up being ungrounded. One can always adjust the argument by further defining this perfect God. I have spoken of the God of Christianity. That is to say, “It is necessarily the case that if the God of Christianity exists, then He exists necessarily.”

It is more that the concept and ideal of perfection does not exist without God.

I agree with this. However, I think you may have missed B.J.’s point. He is looking at the given argument and raising objections to it. His point was that even if the proof is sound it has not proven the God of Christianity. What you have done JohnV is to present another argument.

For 7), could you have □~□p → □~p (Rule D and G)?

I don’t think this follows. What previous steps were you applying rules D and G to?

In order to show that it is possible for there to exist a perfect God, it must be shown that a perfect God is not self-referentially incoherent which is done by explaining what God is by using a coherent definition.

I think you are right on track here. The way to show that something is impossible is to demonstrate that it does not logically cohere. If some item is logically coherent, then it is possible. For instance, green Martians may not be actual, but in terms of possible world theory, they are possible. The existence of green Martians is not a logical incoherence. Therefore, they are possible.

The argument does not depend on God being perfect but on God being a necessary and possible being given propositions one and two.

This is exactly the point!

Isn't the question of whether the ontological argument is a valid argument or not moot? Even if it is a valid argument it cannot prove the existence of the trinitarian God of the Bible. It could just as easily be used as a proof for the Jewish god or the Muslim god couldn't it?

Three thoughts. Firstly, I think there is some sense in which the Trinitarian God of the Bible is the Jewish God. Secondly, one could argue that the Jewish conception of God and that the Muslim God are not possible. This is where one could wed the ontological argument with another type of apologetic. One could argue that these religions cannot account for knowledge, and as such are not possible. Lastly, if one modifies the initial premise by making more explicit the God being spoken of (the Christian God) then the objection that this does not prove the Christian God goes away. It should be noted that in terms of this thread, this last objection is not an objection against the soundness of the proof.

Sincerely,

Brian

 

[JohnV]

It may indeed be true that someone may pose an argument for "the perfect island", or some object that is not existent necessarily but may yet be logically necessary, or for the impossibility of casting down innumerable possibilities besides the God of the Bible; all that may be true, but none of these address the argument in full. The point still remains that all refutations or counters to the ontological argument, when carried to their logical ends, eventually and inexorably lose all their terms of meaning, while the ontological argument remains internally coherent.

But that's not the argument itself; that is only an attribute of the argument.

In other words, the ontological argument is not merely a logical or internally coherent proposition. Though it is these things, that quite misses the point. It seems, Brian, that you have attempted to show its logic and internal coherence. It is these things that are on the block for discussion, as you said. You are proving some major attributes of the Ontological Argument. It is not a different argument than the one I posed, I believe, but more a different focus upon the same one.

 

[Vytautas]

I don’t think this follows. What previous steps were you applying rules D and G to?

7. □~□p → □~p (Rule G – 6)

 

[JohnV]

I'm not going to comment on missing the force of the objection, or on the more sophisticated Gaunilo-esque counters. I'm sticking by my assumption that they miss the force of the argument.

As for the two problems, the first is not a problem. The second is correct in that it does help me sleep at nights. But as to whether it's a mere parlour trick, I turn rather to the Belgic Confession, the Heidelberg Catechism, the Westminster Catechisms, and the Westminster Confession of Faith, and find that they all use the same approach. They see no excuse for anyone not to acknowledge and appreciate the introductory assertions of the Christian's faith in God, the God of the Bible, on the strengths of the witness of Him in the creation and its order and in the consciences of men. To assert it is to assume it has been proven. None of these Church documents waste time with "proofs" that they may satisfy the skeptic. The Ontological Argument for God satisfies these sufficiently, I will continue to hold. They go not only against the witness of God in creation, but against their own consciences. And I will continue to hold that there are no proofs that will satisfy those who will not believe, even when faced with the overwhelming proofs that we find in nature, save for the irresistible work of the Spirit to work in the hearts of men.

What has been done up until Brian's proposed theorum is what has been suggested also by Plantinga, namely that those who believe in the God of the Bible are at least no less warranted in their belief than those who will not believe in the God of the Bible are in their belief, based solely upon the philosophical necessities of the Ontological Argument. Though it may be said by those who do not see the force of the argument that they remain unconvinced by it, yet it remains just as true that those who believe in God are not without warrant for their belief. The very least that is accomplished by the Ontological Argument is that it puts into words the force of the argument supplied by the revelation of God through the works, order, and providence of the creation.

What Brian has attempted to do is to make a philosophically "tight" argument. It is not revelation itself, but is an outworking by Brian from revelation. Every attempt fails in some respect, I suppose, because we're working from the finite to the infinite, from the measurable to the immeasurable, from the changeable to the unchangeable; the former cannnot contain the latter. We could not hope to contain all that is necessary to provide for all the objections that are raised against the proofs that are clearly all around us. But it yet remains a philosophical pursuit that, if all these proofs are sufficient proofs indeed, as the Bible tells us they are, and as the Church has witnessed to us all these centuries, then a simple argument ought to be able to be formed according to those revealed truths, on the assumption that these truths have been authoritatively asserted (by God, not man.)

This, if I am not mistaken, is what Brian is trying to demonstrate. When we are charged with circularity, of having the conclusion defined into the premise, or such like objections, the charge is stated as if the original assertion came from Christians rather than from God Himself. The Ontological Argument attempts to take us from the temporal necessities to the eternal necessity, going through the revelation of God's attributes and deity in creation. It is God's revelation of Himself to us in the creation, even in the use of words and other symbols of defined meanings; it is not man's revelation to himself. This includes, then the possibility of philosphically defined terms. This is what Brian is trying to do, it seems to me.

At the very least it is an attempt to state in philosophical terms the clarity, sufficiency, and force of God's revelation of Himself in creation, putting it into a symbolic formula. As I said, I'm not up on these symbols that he uses. I prefer to study words and their meanings, as they are used in Scripture. I don't mean translation from Hebrew and Greek into English. I mean the use of simple terms with unequivocal meanings spoken with authority, witnessed by the Church as "clear and sufficient". Whether Brian has succeeded I will leave to others who have a well-practiced depth of understanding of the symbols. I don't claim to be a philosopher. The fact that it ought to be possible, no, that it is impossible to state the opposite, I think remains whether or not he has succeeded. We are at the very least at the position that belief in God is warranted and cannot be discredited, and therefore can be stated in a propositionally compelling form. So even if Brian has failed, the quest remains to "try, try again."

 

[Brian Bosse]

Hello Paul,

I am sorry I have been away so long. My replies are going to be sporadic for awhile.

That is, can one also prove the existence of a perfectly stupid being?

The argument I have presented is formally valid. I suppose one could argue againt the form, but the form I am using is standard. So, if we take your “perfectly stupid being” and apply it to the proof, then we are left with the following two propositions:

(1) It is necessarily the case that if a perfectly stupid being exists, then he exists necessarily.
(2) It is possible for a perfectly stupid being to exist.

These two propositions would need to be established for the proof to be sound. If they could be established, then it does prove the existence of a perfectly stupid being (assuming the Modal system is valid). However, I think it will be difficult for someone to establish the necessary premises.

Further, some would suggest that since (7) says that the existence of a perfect being is not impossible, one should back (7) up by refuting the myriad (or, at least the best) incoherency arguments put forth against the possibility of this being. Until then, some would say, they're justified in rejecting (7) as unsound.

This is a good observation, but rather it is step 2 that states this being is possible (not impossible). Here would be my response: To show that this being is not possible, one would need to establish that this being is logically incoherent. No one has ever been able to demonstrate this – leaving us with the possibility of this being’s existence. (Note: I italicized ‘possibility’ in the last sentence because I used it in a different sense than its technical sense in Modal Logic.) I grant this does not establish 2 in a deductive manner, but it does provide justification for 2, and it also means that someone is not rationally justified for rejecting 2. He has no good reason to do so. Saying that there is no positive proof is not good enough because we are speaking of mere possibility rather than actuality or necessity. To establish possibility is much easier. For instance, it is possible in the Modal Logic sense for there to exist green Martians even if this is not actually the case. Step 2 speaks of only the mere possibility of such a being.

Lastly, some like Michael Martin and Richard Purtill would argue that (1) is problematic.

I will interact with the quote you provided below. It does not surprise me that Martin would argue against (1) because that is the only basis he has to reject the proof. I understand that Martin grants that this being is logically possible (2). Maybe not? Nevertheless, here is my argument against this:

The necessary state of this being is established by definition. If the God of Christianity exists, then all truths that make up the Christian worldview are actual. One truth that makes up the Christian worldview is that God’s existence is necessary. So, if God exists, then He does so necessarily. Now, if this is incoherent, then it would refute step (1), and in fact it would refute the Christian worldview. Is this incoherent? I do not believe so. Here are my comments concerning the quote you provided:

It seems to be contrary to our idea of logical necessity that whether or not a statement is logically necessary should be determined by the existence or nonexistence of something. If by "logically necessary statement" we mean "theorem of a logical system" or "tautology" or "analytic statement," it seems quite clear that the existence or nonexistence of something is irrelevant to the question of whether or not a statement is a theorem, or a tautology, or is analytic. Even if our idea of logical necessity is claimed to be wider than any of these notions, it seems unlikely that any plausible account of logical necessity would allow it to be dependent on existence.

Paul, it seems to me that his argument is that logical necessity of something is independent of its existence. For him to make this point, it seems he is assuming that p → □p implies □p is dependent upon p. If not, then what is his objection?

Now, let’s say that A is a necessary precondition for B. That is to say, for B to exist, A is necessary. In this sense B is dependent upon A. How would this be represented logically? Martin answers A → B, an this is wrong. The way you show the dependence of B upon A is: B → A! If B is truly dependent upon A, then...

if anytime you have B, then you must have an A...

or

if B, then A...

or

B → A!

Martin, makes a very simple syntactical error in his interpretation of p → □p. p → □p does not say that □p is dependent upon p, but rather it says p is dependent upon □p! In Copi’s Introduction to Logic – Eleventh Edition, on page 318 you find this:

In general, “q is a necessary condition for p” ... (is) symbolized
as p → q.

This is taught in any freshman level logic class. p → □p says that p depends on the necessity of p and not the other way around as Martin would have it. Martin makes a beginner logic mistake in his syntactical interpretation of the implication and as such the objection fails.

Why think when we run into difficulties we should reject our basic presupposition? Don't most other people try to find a way to resolve the problems when they arise in their fields?

Yes, I agree. What do you think these guys would come back with based on my answers above?

Sincerely,

Brian