Wednesday, March 24, 2010

Modal ontological argument, revisited

Let's revisit the modal ontological arguments for God.  Though I have covered these arguments and variations at length, and have not changed my opinion about them, I didn't necessarily cover them in the clearest manner possible.  It's a tough balance, because for most people, symbolic logic is like math.  Math = scary!  Other readers understand the symbolic logic, but clearly have difficulties translating between the symbols and their underlying meaning.  My goal is to explain it so both groups understand what I'm saying.

Modal ontological argument, reviewed
Definition: If God exists, then God necessarily exists.
Premise: It is possible that God exists.
[Insert mess of logical reasoning here]
Conclusion: God exists.
I omitted the mess of logical reasoning because I don't want to scare away my readers. Unfortunately, this makes it difficult to convey the amount of respect the logical arguments deserve.  Richard Dawkins and other atheists often seem to think it's all a bunch of high-sounding gibberish.  And perhaps it is.  But that particular gibberish is absolutely solid, as solid as 2+2=4.

There are basically two reasonable objections to the argument.  Either we must object to the premise, or object to the definition.

Objection to the premise

To explain the problem with the premise, I must distinguish between two kinds of possibility.
Epistemological possibility: For all we know, God exists.
Modal possibility: Among the set of "possible worlds", there exists a world in which God exists.
The conclusion of the proof follows only from modal possibility, not from epistemological possibility.  However, the epistemological statement is the one that is intuitively true, while the modal statement could be true or false.  Modal possibility is intended to be a translation from epistemological possibility to logic, but the translation is not perfect.

In particular, the translation fails when we talk about the set of possible worlds itself.
Epistemological statement: For all we know, the universe is deterministic (ie there is only one possible world).
Modal translation: Among the set of "possible worlds", there exists a world in which it is true that there is only one possible world.
The epistemological statement is sensible, but if we naively copy it as a modal statement, it's a mess.  The first statement only says that the universe may be deterministic, while the second statement can be used to prove that the universe must be deterministic.  Clearly, we need to be careful with our translations when discussing determinism.

Similar to determinism, the existence of God also says something about the set of possible worlds.  If God exists, then s/he exists in all possible worlds.  If an object does not exist in all possible worlds, then we cannot call it God.  So we need to be careful with our translations when discussing God too.

Here is a much better translation to logic:
Epistemological statement: For all I know, God exists
Logical translation: God exists, or God does not exist.
The conclusion of the ontological argument does not follow from this proper translation.

Even proponents of the modal ontological argument must accept that there are problems with translation.  If we take the statement, "For all I know, God doesn't exist" and naively translate it to modal logic, then we would conclude that God doesn't exist.

Alternate premise: Logical consistency

Some ontological arguments have a cleverer premise to replace the old one.
Old Premise: It is possible that God exists.
New Premise: There is no contradiction in the existence of God.  In other words, God is "consistent."
The old premise logically follows from the new one.  If you're familiar with symbolic logic, I hope you already knew this.  If not, here is a simple proof.

Suppose that the old premise is false; it is not possible that God exists.  The statement "If P, then Q" is always true if P is false.  Likewise, the statement "If God exists, then Q" is always true, because God doesn't exist.  It's true even if Q is a contradictory statement (ie "God is blue and not blue").  Therefore, the new premise is false; the existence of God implies a contradiction.

If the old premise is false, then the new premise is false.  Equivalently, if the new premise is true, then the old premise is true.

But now we will run into another problem of translation.  I will distinguish between two kinds of consistency.
Self-consistency:  The object has no contradicting properties in its definition.
Logical consistency: The object implies no contradictions.
Proponents of the ontological argument often expect me to disprove the self-consistency of God.  Perhaps they expect me to argue that God's omnipotence contradicts its omniscience, or something like that.  But they fail to realize that I don't have to.  The proof requires logical consistency, not self-consistency.

Self-consistency is not sufficient to establish logical consistency.  For an object to be logically consistent, not only must its definition be properly formed, but the world must cooperate.  (More precisely, the set of "possible worlds" must cooperate.)  Suppose that the world does not cooperate, and the object does not exist.  If the object does not exist, then the existence of the object implies a contradiction.  Namely, it implies that the object both exists and does not exist.  I didn't even have to look at the definition of the object.

Of course, I don't know whether the world cooperates with the ontological proof or not.  The proponents have no idea either, but think they do.

Philosophers ought to teach themselves some mathematics. In geometry, there is a famous axiom called the Parallel postulate.  It is famous because many mathematicians thought they could prove it.  Modern mathematicians know that it is impossible to prove, because there is no contradiction in assuming it false.  Likewise, it is impossible to disprove.  The Parallel postulate is self-consistent.  The negation of the Parallel postulate is also self-consistent.  But in any given geometrical system, only one can be true.  Thus, only one can be logically consistent.

Objection to the Definition of God
Definition: If God exists, then God necessarily exists.
If we define a fork to be an object with a handle and prongs, then we can give the following statement: "If a fork exists, then it has prongs".  If a fork does not exist, we can't even talk about "it", much less ascribe it properties.  If an object does not have prongs, then it is not a fork.  That is the rationale behind the definition.

How can you disagree with a definition?  Can't we define any object we like?  If the definition makes no sense, can't we just say that the object doesn't exist?

I don't know about philosophy, but in mathematics, you can't just define any object you like.  Consider the set of all sets that do not include themselves (like the male barber who shaves all men who do not shave themselves).  Call this Russell's Set.  In "naive" set theory, you are allowed to define any set you like, including Russell's set.  But Russell's set leads to a paradox.  Therefore, "naive" set theory is inconsistent.

Naturally, mathematicians want a set theory that doesn't have paradoxes.  So they formulated Zermelo-Fraenkel set theory, which has specific rules dictating what sets you are allowed to define.  These rules do not allow us to define Russell's Set, and thus avoid its paradoxes.

Is modal logic more like "naive" set theory, or like Zermelo-Fraenkel set theory?  Can you define anything you like, or do you have to follow specific rules?  I suspect it depends on your choice of axioms.  It's a difficult question that I don't feel qualified to answer, which is why I prefer objections to the premise.

For what it's worth, Immanuel Kant's original objection to the ontological argument might fit in this category.  Kant argued that existence is not a property that you can include in the definition of an object.

Alternate definition: The greatest being

Many proponents of ontological arguments like to have it both ways.  On the one hand, we are allowed to define God.  On the other hand, we are not allowed to define "the unicorn which necessarily exists."

But to be fair, they're not exactly parallel.  In most ontological arguments, God is not defined as "the deity which necessarily exists".  Rather, God has a much more specific definition.
Definition: If God exists, then s/he is the greatest being conceivable.
Additional premise: We can conceive of a being as greater by conceiving it as necessarily existing.
I think that this new definition hurts the ontological argument.  For one thing, we have a whole new premise.  I don't have any particular problem with the premise, but it just seems so extraneous and unnecessary.  I refuse to argue with the additional premise, because it seems like a tactic to draw attention away from the real flaws of the ontological argument.  In my naive optimism, I expected this tactic from conspiracy theorists, not philosophers.

And the new definition does not help in the slightest.

Let's say that Russell is the name of "the male barber who shaves all men who do not shave themselves.  As I said before, in Zermelo-Fraenkel set theory, there are rules against defining Russell.  If we are not allowed to define Russell, then obviously, we are also not allowed to define Russell's wife!  Russell's wife may not have any self-referential paradoxes, but she requires the existence of Russell, who does have self-referential paradoxes.

Let's say we have a rule against defining necessarily existing beings.  Obviously, we are also not allowed to define the wives of any of those beings.  We are not allowed to make any definition which implies necessary existence.  If we accept the additional premise, then the definition of God implies necessary existence.  Therefore we are not allowed to construct the definition of God.

Concluding remarks

I've written over 1600 words now, so I think I'm done.  Dear readers, you are lucky that I'm on spring break (or unlucky, if you see it that way).

Multiple past experiences tell me that someone trained in philosophy will start lecturing me about implication, contingency, and other things.  You're welcome!  But a few notes: 1) I'm not an idiot when it comes to logic.  2) Philosophers who do not know how to communicate are useless.  3) I do not necessarily agree with the objections to the definition. 4) I view the ontological argument the same way I view a fun proof that 0=1.

82 comments:

The Barefoot Bum said...

Very well done.

Is modal logic more like "naive" set theory, or like Zermelo-Fraenkel set theory?

Everything, including modal logic and possible world semantics, is — I suspect — just another way of interpreting set theory.

...someone trained in philosophy will start lecturing me about implication, contingency, and other things. You're welcome! But a few notes: 1) I'm not an idiot...

Philosophy is just a fallback for people who couldn't make it through theology school.

Chuck said...

The premise "<>God" is actually not translated from "(x)[~Kx(god) & ~Kx~(god)]". The modality of God is something that can be *known*. Hence, we do not simply *believe* that God is possible. We *know* that God is possible.

What you are calling into question is whether or not we truly know what we claim to know.

Unfortunately, epistemic logic is not on your side on this one. According to one of its axioms, if we know that God is possible, then we know that we know God to be possible. And if we know that God is possible, then according to the knowledge of truth axiom, God is possible.

How do we know? If God can be conceived without contradiction, then God is possible. It does not need to be "intuitively true" in order for the modal ontological argument to work. It is derived using an actual metric (the law of non-contradiction).

To address your response to this, are you claiming that something can be logically consistent but not self-consistent? For example, are you claiming that a squared circle, though not self-consistent, may actually be logically consistent? That does not make much sense.

Anyways, in the context of Hartshorne's argument, there are no disparities between self-consistency and logical consistency because there is no real dichotomy between the "world" and objects which inhabit it. The world is simply the set of all real things. Something which is internally consistent does not require cooperation from something that has no real ontology beyond acting as a linguistic operator.

If an opponent is to become too fixated on the definition of "God", then we can simply employ sophistry and say that we can prove the existence of the greatest conceivable being and start off with the premise, "If the greatest conceivable being exists, then s/he cannot possibly not exist."

miller said...

To address your response to this, are you claiming that something can be logically consistent but not self-consistent?

No, you got it backwards. I said that something can be self-consistent but not logically consistent. You must not be reading very carefully to have missed that.

It's certainly the intention of modal logic to just be a linguistic operator. All things which are self-consistent are supposed to be possible within the formalism. But if you actually look at the rules, that cannot be the case, especially when we consider statements about the set of possible worlds itself.

Chuck said...

I know what you said, I just wanted to clarify on whether or not you believed that the reverse was true.

My point is that the dichotomy you are creating between self-consistency and logical consistency does not exist. It is true that there are many self-consistent ideas which do not exist in reality, but such ideas would be sustained by contingent states of affairs. For example, we now have a greater understanding of how organisms develop their physical attributes and we (more specifically, biologists) can thus talk about what would have to be true in order for a unicorn to exist, or for horses to develop horns on their heads.

But a being who is both eternal and immaterial would not be sustained by any contingent states of affairs and therefore the only heuristic that we need to go through is whether or not such a being can be conceived without contradiction.

miller said...

My point is that the dichotomy you are creating between self-consistency and logical consistency does not exist.
But it must! You have not offered any real arguments otherwise.

I would be interested to know whether you think the statement "God does not exist" is self-inconsistent, and what paradox you think it implies.

Chuck said...

(1) ~g AP
(2) ~g-->~<>g p
(3) ~<>g-->Cg p
(4) ~Cg p

Assume that God does not exist. Here is what you can infer from that along with the original 3 premises:

(5) ~<>g
(6) Cg
(7) Cg & ~Cg

If you presume that God does not exist, then you have to presume that God can be conceived without contradiction and that God also cannot be conceived without contradiction.

miller said...

Okay, but you took three extra premises. Premise (4) is exactly what we've been disputing. Can you prove that ~(4) implies a contradiction?

Chuck said...

All I can offer you are examples of contradictory ideas and show that such things are not similar to the idea of God.

Squared circles, unmarried bachelors, living dead people, good evil people, etc. are examples of contradictory things.

God is eternal. He is without beginning and without end, but not with beginning and without beginning.

God is omniscient. His knowledge is without limits. It is not both limited and unlimited.

God is morally perfect. He is not both good and evil.

God is omnipotent. He can do whatever is logically possible. He does possess both the ability and inability to do whatever is logically possible.

So, where are finding contradictions in the idea?

miller said...

Okay, since we're reduced to rhetorical arguments rather than logic, let me offer my argument that ~g is consistent too.

Let's say that one of the possible worlds is empty. Where are we finding contradictions in this idea? There's nothing within the world to contradict.

But I have only argued that an empty world is self-consistent. Likewise, you have only argued that God is self-consistent. Suppose, as you contend, that self-consistency is equivalent to logical consistency. Therefore, it is possible that nothing exists. It is possible that God exists. [Insert mess of logical reasoning here] Therefore God exists and does not exist.

We arrived at a contradiction; therefore, our proof has a false assumption. The false assumption was that self-consistency is equivalent to logical consistency.

Chuck said...

Aside from the fact that I've rejected possibilism as a coherent worldview, a empty world, which is just a fancy way of saying "a possible world with nothing in it" is a contradiction because if, in this possible world, nothing existed, then there could not even be a world, since a world is something.

Furthermore, in the context of this empty world, it would apply that there are no modalities of possibility, contingency, or necessity. Therefore, if you claim that such a world is logically consistent (since there are no real differences between logical and self-consistency), then you are claiming that it is logically consistent for the possible to be not possible. Unfortunately, axiom S5 disagrees with you.

Are you asking me to make the case for presuppositionalism? If so, I'll say it right here: There is no tenable worldview that you can hold which does not include some fully actualized being. You can complain about the variety of religions until the cows come home, but it does not change the fact that atheism is false.

miller said...

I also understand that an empty set is a contradiction, since the set itself is a something and an empty set cannot contain a something. Wait, what?

Chuck, I have a policy of offering my adversaries the last word around here. So shoot.

Chuck said...

And getting a "0" on your test is a contradiction because "0" is a number, so therefore, you did not really get a zero. Wait, what?

Of course we can use the word "nothing" if we treat it as an operative term instead of some hypothetical state of affairs in some possible world. For example, the set of dogs in my apartment is empty, or contains nothing. However, this is intelligible only if we understand what my apartment does contain, such that we can say it contains no dogs.

I'm talking about a completely empty existence, or absolute nothingness.

I guess you want to end this. Nice talking to you. Actually, most of your articles are a good read. You've just chosen the wrong worldview, but keep up the good work.

Metacrock said...

I like your stuff. Your conclusions are wrong but I applaud your fairness in the way you think about it and I appreciate the respect you show thinkers with whom you disagree.

This is a good article. It's wrong but it's well written and I understand your view point and I can respect it.

I'm going to answer it on my blog but not right away. I have some other things planned first but I'll try to get it up latter in the coming week.

I don't want my blog to be taken up with too much OA stuff, the readers don't like it. It gives me a head ache.

The Barefoot Bum said...

Oh Miller, you poor bastard, you've attracted Metacrock's attention. He and I go waaaaaaay back to the early days of IIDB.

I'm going to get some popcorn. This could turn out to be interesting.

The Barefoot Bum said...

Note that Metacrock has responded to your argument. But even making the most charitable allowances for his genuine and well-established learning disabilities, his response is the usual incoherent, sanctimonious babble.

miller said...

Metacrock has responded on his blog. I will leave a response on his blog later.

The Barefoot Bum said...

If you would be so kind, could you post a link to your response here? I don't follow his blog.

miller said...

The discussion has begun. Be aware that there is often a delay time between writing and publishing comments, since Metacrock needs to moderate them.

The Barefoot Bum said...

Good stuff, miller (your side, at least). Looks like pretty standard Metacrock to me: he's already going in circles. Let us know here if anything really interesting develops?

miller said...

I think my discussion with Metacrock has just ended. At least, I wrote my last comment. Metacrock said he would write another post about consistency by today, but it hasn't happened. If it does, I'll say so here.

Anyways, I think I'm bored by now. And he didn't ever add a link to this post! That was needlessly rude.

miller said...

A link for posterity: Plantinga has personally responded to me, though it seems he didn't invest enough time to really say anything coherent.

Charles said...

This is all a matter of stipulation. By "Possibly, God exists," I don't mean that I lack knowledge as it pertains to God's existence; rather, I mean that God is possible, in the same way that a twelve foot blue hot dog is possible. Now, an atheist is certainly within his or her epistemic rights to deny this, but the theist has equal warrant for accepting it.

At the very least, the argument goes to show that the existence of God is not a matter of empirical fact, but rather it's an issue of meaning. Either the atheist or the theist is talking nonsense.

Nueva Argentina said...

Dear Miller,

you tried to prove that the old premise 'God is possible' follows from the new premise 'God is consistent'.

I doubt very much this is so; anyway, your proof is wrong. You say:

[Suppose that the old premise is false; it is not possible that God exists. The statement "If P, then Q" is always true if P is false. Likewise, the statement "If God exists, then Q" is always true, because God doesn't exist. It's true even if Q is a contradictory statement (ie "God is blue and not blue"). Therefore, the new premise is false; the existence of God implies a contradiction.]

All you prove is that on the assumption that God is impossible, the assumption that God exists entails contradiction. But this triviality does not prove that God's being impossible entails God's being inconsistent. The inconsistency need not lie in the impossible God; rather it arises from the assumption of two contradictory statements.

Consider this spoof proof that if the moon (or whatever) does not exist, then it is inconsistent. Assume the moon does not exist; then the conditional 'if the moon exist, p and not p' is true. Assume now the moon exists; then p and not p; so, the existence of the moon implies contradiction and the moon is inconsistent.

As you claim to be trained in logic, you'll surely get it soon.

miller said...

Nueva Argentina,

"All you prove is that on the assumption that God is impossible, the assumption that God exists entails contradiction."

That's good, because that's exactly what I wanted to prove.

"But this triviality does not prove that God's being impossible entails God's being inconsistent."

It depends on what we mean by inconsistency. If by "God is inconsistent", we mean, "The existence of God entails a contradiction" (ie what I called logical inconsistency), then I have indeed proven it. And yes, under this definition, the moon's impossibility would imply the moon's inconsistency.

If by "God is inconsistent", we mean, "God has a contradiction within its own concept" (ie what I called self-inconsistency), then you are correct and I have not proven it.

The thing is, the modal ontological argument requires logical consistency. Why? Because the very proof you are disputing is something embedded within the the ontological argument itself. And as you just observed, the proof does not hold under the self-consistency definition.

miller said...

Nueva Argentina,
On second glance, I think I was misinterpreting you as being pro-ontological-argument, when I have no reason to think so. I apologize. But I hope I was able to clarify the issue.

The basis of the modal ontological argument is essential a bait and switch. We accept the premise because we're thinking of self-consistency (arguably the more intuitive definition), and then we accept the rest of the argument because we weren't paying attention and fail to see that it's really logical consistency which is required.

Nueva Argentina said...

Dear Miller,

think again of my example about the moon. It's not the moon's impossibility, but the sheer moon's nonexistence what is required in order to prove the moon's inconsistency with your technique!

--

If you were right, the modal ontological argument would really be stronger than it is.

That's so because it is easy to prove that Gödel's axioms governing 'positive property', (from which the possibility of God follows) are logically consistent: they have a model. Read 'positive property' as 'property of number 7', and, on the plausible assumption that number 7 has necessary existence, you have a model for the axioms (whether you take Dana Scott's version on wiki http://en.wikipedia.org/wiki/G%C3%B6del's_ontological_proof or Gödel's version reported by Sobel).

So, there is a logically possible world w1 where the axioms are true. At w1, God is (metaphysically) possible, hence logically possible; this implies there is a logically possible world w2 (accessible from w1 and perhaps identical with it) where God exists. But surely, the logically possible worlds are the same at all logically possible worlds, hence w2 is also accessible from the actual world (it is a logically possible world also at the actual world).

Now, if all logically possible worlds are also metaphysically possible worlds, w2 is a metaphysically possible world at which God exists. And you've proven the weakest part
of Gödel's argument: that God is possible. Now you only have to admit modal system S5 (or the somewhat weaker B) to prove that God exists.

But if there are logically possible worlds that are not metaphysically possible, w2 need not be metaphysically possible. I believe there are logically possible worlds that are not metaphysically possible; perhaps worlds where all laws of logic hold but things pop out of pure nothingness or the past can be changed or abstract objects are causally efficient or...

miller said...

Nueva Argentina,
Your replies are becoming increasingly unclear. Could you at least clarify what position you are trying to argue?

I argued that impossibility (ie non-existence in all metaphysically possible worlds) implies inconsistency (ie the object's existence entails a contradiction). Therefore consistency implies possibility. Is that what you object to?

"if all logically possible worlds are also metaphysically possible worlds"

I'm not sure if you think this is something I claimed. I never claimed that, and in fact I would object to it.

"And you've proven the weakest part of Gödel's argument: that God is possible."

I'm not sure if you're talking about me, or the rhetorical "you". I did not claim that God is possible, much less prove it. Consistency implies possibility. I did not claim consistency.

Nueva Argentina said...

Dear Miller,

you did claim that consistency (i.e. not entailing contradiction) implies possibility.

This is what I am objecting to.

One can prove that God can exist without logical contradiction because it is easy to provide a model for Gödel's axioms concerning 'positive property' (namely, read it as 'property of number 7'; please, check it).

These axioms entail God's possibility and they are consistent, because they have a model. Hence, it is consistent that God be possible. This could not be so if God were inconsistent because then God would also be impossible. So, God is consistent.

If consistency entails possibility, God is also possible.

I remark I don't believe consistency entails possibility. I think your proof to the effect is wrong.

I also remark that, in view of God's consistency, your position strengthens the modal ontological argument.

Larry, The Barefoot Bum said...

Again, we have to distinguish between domains of possibility.

By definition, all modal statements are necessarily true or necessarily false, i.e. true or false in all possible worlds (that are included in the specified domain of modality). "All possible worlds" is usually taken as the domain of worlds where simple, i.e. non-modal, statements can vary freely.

A simple statement is like:

(S) John F. Kennedy was shot by Lee Harvey Oswald.

A modal statement is like:

(M) There exists a possible world where John F. Kennedy was shot by Lee Harvey Oswald

By definition, since (presumably) neither S nor ~S implies any kind of contradiction, there are some possible worlds where S is true, and some where S is false. However, M is true in all possible worlds, including those worlds where S is false.

In order for M to be true in some possible worlds and false in other possible worlds, we have to define a meta-domain, perhaps sets of possible worlds.

Thus we might have:

M': There exists a set of possible worlds in which M is true.

Note that M' is still true in all sets of possible worlds, even the sets where M is false.

As I've argued before, the modal ontological argument makes a very straightforward category fallacy. There is some possible world where M is true violates the definition of possible world in a straightforward way. (Alternatively, it requires a definition of "possible world" that is internally contradictory.)

Larry, The Barefoot Bum said...

We can't just define "possible world" as a world where all the true statements are not logically contradictory. We run into exactly the same problem of Cantor's naive set theory. We have to insist on a strict North/Russell/Whitehead category of types, which means, per Godel, that any description of possible worlds is necessarily incomplete (i.e. there is no possible world where it is possible to describe all possible worlds).

Nueva Argentina said...

Larry,

you seem to be assuming that modal truths are necessary.

Whether this is so or not depends on the kind of accessibility relation you assume to hold between worlds. Roughly, we can say that your position is in agreement with system S5 or, equivalently, with those Kripke frames where the accessibility relation is an equality relation.

Since Gödel's reasoning is absolutely sound in S5 and relies precisely on the assumption that all modal truths are necessary, your position should not be in conflict with his argument.

Larry, The Barefoot Bum said...

you seem to be assuming that modal truths are necessary.

'Deed I am.

Whether this is so or not depends on the kind of accessibility relation you assume to hold between worlds.

Please explain in rigorous detail.

Roughly, we can say that your position is in agreement with system S5 or, equivalently, with those Kripke frames where the accessibility relation is an equality relation.

Yay. Please explain in rigorous detail.

Since Gödel's reasoning is absolutely sound in S5 and relies precisely on the assumption that all modal truths are necessary, your position should not be in conflict with his argument.

Please explain in rigorous detail.

miller said...

@Nueva Argentina

"These axioms entail God's possibility and they are consistent, because they have a model."

I contend that providing a model proves a certain kind of consistency, but not the kind of consistency which I speak of, and not the one used in the ontological argument. Consistency can be stated symbolically as follows:

~[]( G => (P ^ ~P) )

Where G is the statement "God exists", [] is the necessity operator, and P is any statement.

Your definition of "consistency" (if I understand correctly) is arguably the more sensible one, but it's very difficult to formalize in logical form, and thus is not the one used in the modal ontological arguments.

To summarize, we have the following argument:
1. There is a logical model in which the premises of Godel's argument are true.
2. The existence of a logical model implies consistency.
3. Consistency implies possibility.

I contend that the problem with the argument is step 2, and you contend the problem is in step 3. It really comes down to our definition of consistency.

"Since Gödel's reasoning is absolutely sound in S5 and relies precisely on the assumption that all modal truths are necessary, your position should not be in conflict with his argument."

I would not grant that Godel's reasoning is sound, I would grant that it is valid. I don't think we have any reason to believe the premises are true.

Nueva Argentina said...

Miller,

you have impeccably summarized our differences.

Let G be 'God exists'. By God's inconsistency you mean:

[](if G, then P ^ ¬P)

for some p.

Then any impossible proposition is inconsistent. For instance, if it is impossible to alter the past, then 'the past is altered' is inconsistent.

Then, by definition, your consistency and metaphysical possibility are the same. You need not prove that one implies the other, you'd better show they are the same by definition.

Yours is not the usual sense of consistency in formal logic.

I agree that Gödel's axioms concerning positive properties are not evident. I don't know if they are true, hence I meant to say that his argument is valid, not sound, as you point out.

The point that Gödel's argument can be used to show that G is formally consistent is not frequently made. Do you remember having read it elsewhere?

Best.

Nueva Argentina said...

Larry,

surely, you know of Kripke's semantics for modal logic. It's the semantics of frames. A frame is a triple where W is a set of worlds, R is an accessibility relations and v is a valuation function that assigns a truth value to each proposition at each world.

The usual interpretation of R is: Rw1w2 (w2 is accessible from w1) iff w2 is a possible world at w2.

In system S5 all modal truths are necessary. S5 is sound just in case R is reflexive, symmetric and transitive. You can find the proof of this in the technical litterature.

S5 is the system usually assumed in the modal ontological argument, though system B, which is slightly weaker, would also do. System B requires the actual world to be a possible world at all possible worlds, that is,

(w) Rw_0w

where w_0 is the actual world.

Hope this helps.

Larry, The Barefoot Bum said...

surely, you know of Kripke's semantics for modal logic.

Do not assume I know of or know anything in particular. I'm reasonably intelligent, but my knowledge is hardly encyclopedic.

More importantly, it is almost certain that my understanding of any given point differs in some subtle way from your own. I want to be crystal clear about precisely what you mean.

The usual interpretation of R is: Rw1w2 (w2 is accessible from w1) iff w2 is a possible world at w2.

I think you might have a typo here. But left undefined, at least for our readers, is precisely what you mean by "w2 is a possible world at w1".

Also, it might be nice to have a thesis, a simple statement about what you're trying to argue.

The contention is, as I recall (and I'm perhaps mistaken, since this is an old thread, I've discussed the MOA in several places, and I'm too lazy to check carefully) that the premise "it is possible that God exists" is not only controversial, indeed the essence of the controversy, but it is also not well-formed. On the former, the argument is just circular; on the latter it is actually invalid.

miller said...

@Nueva Argentina,
"Yours is not the usual sense of consistency in formal logic."
That may be true. I'm not formally trained in formal logic, but I'm familiar with how "consistency" is used in the context of the modal ontological argument.

I am curious what you think is the "usual" sense of consistency. I think you said it means that there exists a logical model, but is there a simple way to state this formally?

I suspect the formalization is nontrivial. Quote from Wikipedia on Godel's second incompleteness theorem:
"A technical subtlety in the second incompleteness theorem is how to express the consistency of T as a formula in the language of T. There are many ways to do this, and not all of them lead to the same result. In particular, different formalizations of the claim that T is consistent may be inequivalent in T, and some may even be provable."

Nueva Argentina said...

Consistency in formal logic means twofold:

1. Consistency of a set of sentences in a formal language.

2. Consistency of a deductive system (formal axiomatic system).

A deductive system is consistent iff it doesn't prove any contradiction, even if some contradiction is a logical consequence of its axioms but is unprovable in the system (which can happen if the system's language is of higher order, because higher order logic is incomplete).

A set of formal sentences is consistent iff no contradiction is a logical consequence of its member sentences.

Remember that p is a logical consequence of the set S of sentences iff all interpretations that make all the members of S true also make p true.

I was referring to consistency of sets of formal sentences. You can formulate a version of Gödel's axioms (including the modal ones) in a formal language (for instance, the language of third order modal predicate logic).
The language will contain just one nonlogical constant, which is the symbol for positive property (say, P). That's the only thing you need to interpret when you look for a model of the axioms.

If you interpret P as the higher order property of being a property of number seven, you get a model for the axioms, which proves that no contradiction is a logical consequence of them. Since God's existence is a logical consequence of the axioms, God's existence is also consistent in this sense.

This has nothing to do with the different ways we have to express consistency of a system in the language of the system (which is, nevertheless, of relevance for Gödel's second theorem). The logical consistency I am talking about is meta-linguistic, it is a semantical concept based on the classical Tarskian approach.

I hope this is helpful.

miller said...

A set of formal sentences is consistent iff no contradiction is a logical consequence of its member sentences.

Okay, so in my definition, I was talking about entailment, but I take it that "a logical consequence of" is different from "entailed by".

Remember that p is a logical consequence of the set S of sentences iff all interpretations that make all the members of S true also make p true.
This is unclear to me. What counts as an "interpretation"?

Nueva Argentina said...

Dear Miller,

let me say first of all that an interpretation of a set of sentences is an interpretation of the language of the sentences (all sentences are supposed to be in the same language).

An interpretation specifies a domain of discourse D for the individual variables and their quantifiers (first order quantifiers). Then it specifies a subset of D for each unary predicate symbol P, Q, ..., a set of ordered pairs of members of D for each binary predicate, and in general a set of members of D^n for any n-ary predicate.

If there are individual constants, they are usually assigned fixed individuals in D.

If the language is second order, their second order n-ary variables and quantifiers are usually taken to range over P(D^n), that is, the powerset of D^n (though this is not so in some nonstandard semantics such as Henkin semantics).

In general, for m-order n-ary variables and quantifiers binding them, the range is P^m(D^n).

Logical constant are interpreted always the same way.

Consider the set

S = {(Ex)Px; (x) (Px -> Qx)}

This set is consistent. Take D=N. Assign the set of primes to P and interpret Q by means of the set N again. Then the sentences say: 'there is a prime number', 'all primes are natural numbers'.

This is a model, i.e. an interpretation that makes all members of S true. The sentence

s = '(Ex) Qx'

is a logical consequence of S (in the usual notation: 'S|=s') because every model of S makes s true as well.

The existence of a model proves S consistent. And this implies that any interpretation of its language (whether it is a model for S or not) that brings in no additional logical structure gives a logically consistent set of propositions.

The requirement that interpretations bring in no additional logical structure is essential and means that logically simple symbols must be interpreted by logically simple objects. For consider for instance that we interpret P by means of the set of all primes, as above, and Q as the negations of P, that is, as the set of all naturals that are not primes. We have used logical composition (by means of negation) to interpret the simple symbol Q.

This interpretation won't yield a consistent set of propositions, since the existence of a number that is and is not prime would be a consequence of it.

Entailment is different from logical consequence. Both are semantical notions but entailment refers to all possible worlds and logical consequence to all models. The former arises from Kripkean semantics and the latter from Tarskian semantics.

That's the essential, I'd say.

miller said...

Interesting...
I guess one way to look at the modal ontological argument is an equivocation between Kripkean and Tarskian semantics.

Thanks for your comments.

Rick Taylor said...

No, we don't have to object to the definition (definitions can be arbitrary) nor need we object to the premise. The mess of logical reasoning falls apart if you write it using modal predicate calculus rather than modal propositional calculus. It's only because most people are too lazy to learn the much more difficult modal predicate calculus that the argument seems solid, but it isn't.

Here's how it translates when done properly:

If God exists, God necessarily exists.

for-all(x) IsGod(x) -> necessarily isGod(x)

It is possible God exists:

possibly there-exists(x) IsGod(x)

From this you can conclude

possibly there-exists(x) necessarily IsGod(x)

And now you're stuck! Notice there's a big fan existential quantifier right in between your possibly and your necessarily. That quantifier does not commute with the necessarily, and so you cannot get the necessarily next to the possibly and the argument fails.

It's only because people rewrite this in Propositional form that it looks like it succeeds. But the predicate calculus was designed to handle existential and universal assertions, and is what you need to make the argument rigorous.

Rick Taylor said...

"I don't know about philosophy, but in mathematics, you can't just define any object you like."

Actually, in mathematics, you can define any object you like! There's nothing preventing me from beginning a proof by saying, define a set to be self-excluding if it consists exactly of all sets which do not contain themselves. Of course I cannot automatically conclude there exists such a set, but the definition is just fine.

I can then even say, suppose that E is a self-excluding set. Of course I can expect to derive a contradiction from that assumption, and I can even then use that to prove the non-existence of a set with that property. But that doesn't make the definition or the assumption invalid.

The proof that 2 is irrational begins, let p/q be a rational number who's square is equal to two.

The only time a definition isn't allowed is if it's gibberish. So for example, the following definition isn't allowed:

Define God to be a really great being that exists necessarily.

The trouble with that definition is that necessarily is an operator that modifies a predicate, and as Kant taught us, existence is not a predicate! If we try to write this out formally, we get

forall(x) IsGod(x) <-> IsGreat(x) and necessarily there-exists(y). .. ummmmmmm

On the other hand, the following definition makes sense:

Define God to be a necessarily really great being!

This yeilds:

forall(x) IsGod(x) <-> necessarily IsGreat(x)

As argued before, if we now assume it's possible God exists:

possibly there-exists (x) IsGod(x)

All we get is

possibly there-exists (x) necessarily isGreat(x)

And the ontological argument fails because of that big fat existential quantifier right between the possibly and the necessarily.

miller said...

Hi Rick,

1. When I said that you couldn't define anything you like, I was thinking about the fact that you can't just list an arbitrary set of properties, and declare a set consisting of precisely the elements with those properties. Perhaps it is more precise to say, as you said, that you can define such a set, but you cannot prove its existence.

2. I think Godel's ontological argument is usually stated in modal predicate calculus form. I usually simplify it to modal propositional calculus because it's easier to explain and I believe that it contains the essential properties and problems of the argument. If I was in error, then you have my attention.

But I'm not convinced yet. My instinct says that "possibly there-exists(x) necessarily IsGod(x)" implies "there-exists(x) necessarily IsGod(x)", even if the proof is not exactly the same. Part of the problem though, is that I can't remember what it meant to say that it is necessary that P(x). x is an object local to this world, so what does it mean to say that in all worlds x has property P? I'll have to reflect on this.

Rick Taylor said...

Heya,

I should have said this before, but I do appreciate your blog. I've only recently been thinking about the ontological argument and learning about modal logic, and it's great to find someone to talk to about these matters. I only just discovered your post on Godel's formulation of the argument; thanks for posting that. Very interesting, but a little sad that assuming God is a good property is such a fishy axiom.

I'm not sure, and I'll write again after I've studied more, but I think saying necessarily P(x) means that P(x) is true in every world where x exists. I'm thinking of a model where a set of objects are imagined to exist independently of the world, and then for each object, there's a set of worlds where it exists. There may be other models, but again I've just barely started reading about this. I just bought First Order Modal Logic by M. Fitting, Richard L. Mendelsohn, which so far looks really really good.

I think that

possibly there-exists (x) necessarily isGod(x)

would mean that there would be an x that (1) existed in at least one world and (2) would be God in every world it existed in, but I don't think it would have to exist in the actual world, so I don't think you can get rid of that possibly.

Rick Taylor said...

"If I was in error, then you have my attention."

Oh no, you posted Godel's proof just fine. As I said, I only just discovered it, but it was wonderful. Second order logic is difficult to parse though. :)

I was saying you gave Plantinga's argument in propositional form, but Plantinga himself gives it that way, so you're representing it faithfully. That's what the problem is, in my opinion (currently); you can't talk sensibly about existence without using predicate modal calculus. It seems to me when you rewrite the argument using it in a way that's natural, the problem goes away. Which might be way Plantinga avoids doing so himself.

miller said...

Cool that you're teaching yourself modal logic. If you learn from a textbook, you might get an idea of the broader context. I don't really have a broad knowledge base, I'm just skilled at debugging proofs.

In retrospect, I think I failed to appreciate the importance of the fact that Godel's argument uses predicate logic. I always felt the proof was relatively convoluted for being essentially the same argument, but on second glance I think the convolution is required for it to be in predicate logic form. Hmm... or maybe it can still be pared down a bit.

Part of the problem is that you cannot say "Necessarily, P(x)". In Godel's argument, the nearest equivalent is "Necessarily, there exists x such that P(x)." Also, Godel needs to do some work to make necessary existence into a predicate.

I conclude that the ontological argument can indeed be stated in predicate form, but it's nontrivial. I also think I was correct that the essential properties and problems are preserved when it is in propositional form.

Rick Taylor said...

There is another problem with the modal ontological argument. We generally assume it defines God to be something that is necessarily existing, but its proponents don't see it that way. Instead, they argue that God is "maximally great" or he is "the greatest possible being," and then argue that something can't be maximally great unless it exists necessarily. It's possible to attack this argument.

To begin with, if God is maximally great, can he create a stone so heavy that he can't lift it? Wouldn't a being who could create such a stone be greater than one who couldn't and therefore wouldn't God have to be able to create such a stone in order to maximally great?

No. The reason is that it would be impossible for God to create a stone heavier than he could lift it; in fact, the idea there would even exist such a stone entails a contradiction. To be maximally great, God doesn't have to do any task we can describe, he only has to do any task that is possible. That's implicit in the second description of God I gave above as the greatest _possible_ being.

So with that in mind, if God is the greatest possible being, he would only have to exist necessarily if that were possible. But suppose for the sake of argument that he only exists contingently. Then it would be impossible for there to be a God who existed necessarily! And so the contingent God would still be the greatest possible being.

From this, it is clear that the problem with the ontological argument is that it equivocates with the word "possible." In assuming modal logic and the S5 axioms, it assumes a meaning for the word "possible" and an underlying metaphysics for its properties (so for example, it assumes it's impossible that it could be impossible for a unicorn to exist). Now I have doubts about the appropriateness of this logic for reasoning about metaphysical possibility. But if you're going to use it, you have to use it! It's not legitimate to use the technical definition of "possible" in one part of your proof and then a different definition of "possible" later, and that's what this argument does. It's possible for the greatest possible being to exist contingently, because if one did, then a being with the same properties that existed necessarily would be impossible.

Nueva Argentina said...

Rick,

you can easily get to

'possibly thereis (x) necessarily isGod(x) & necessarily exists (x)'

Now, assuming a symmetric accessibility relation (i.e. axiom A5), you get:

'thereis (x) isGod(x) & necessarily exists(x)'.

And that's the modal ontological argument.

Nueva Argentina said...

Rick,

necessary existence (existence in all possible worlds) makes perfect sense as a property of an object. Otherwise, one could not state that number 7 has necesary existence, which it surely has.

Nueva Argentina said...

Miller,

from the existence in some possible world of an object x that is necessarily God, you casn't derive God's actual existence. That x is necesaarily God at world w means that x is God at all worlds accessible from w at vwhich x exists.

So x might not exist at the actual world or the actual world might not be accessible from w. To reject the latter, we ussually impose symmetry on the accessibility relation; to reject the former, we must add necessary existence as a property x has at w.

Nueva Argentina said...

Miller,

'necessarily, P(x)' is a well-formed formula in model predicate logic, but it's not a sentence, since it has a free variable.

The point of this discussion is not in predicate vs. propositional logic; the point is whether you accept or not necessary existence as a meaningful property.

Nueva Argentina said...

Rick,

I think in Plantinga's argument maximal greatness entails necessary existence by definition, and, as you claim, definitions cannot de disputed.

The real point is whether S5 (or at least the Brouwerian axiom) is the correct logic for metaphysical necessity.

Rick Taylor said...

I'm limited to ascii here, so some of my arguments might not be clear. When, for example, I write

ThereExists (x) x+x = 4

I intend ThereExists to be the existential quantifier that's normally written as a backwards capital E. The above statement means there exists an element x with the property that x+x=4 (this statement is true because 2+2=4).

When I write

ForEvery (x) If x is a real number then x^2 > = 0

I intend ForAll to be the existential quantifier that's normally written as an upside down A. The above statement means that the square of any real number is at least zero.


Nuevo, you wrote,

'possibly There-Exists(x) necessarily isGod(x) & necessarily exists (x)'

I don't think that makes any sense. What does that last bit, exists(x) mean? Existence is not written as a property, and it doesn't really make sense to. The problem is that if you're writing x to represent some entity, then you're tacitly assuming x exists (even if you intend to derive a contradiction later). So it seems exists(x) ought to be a trivial property that is always true. We can define it this way of course, but then it's obviously useless for proving anything.

Can you show me how to define a property, exists() in terms of the existential quantifier? Something like

Define

exist(x) to mean There-Exists(y). . . ???

I don't see how you do it, and this is what I mean when I say existence, necessary or otherwise, is not a property.

Rick Taylor said...

Nuevo wrote:

"I think in Plantinga's argument maximal greatness entails necessary existence by definition, and, as you claim, definitions cannot be disputed."

I should be more precise when I say a definition cannot be disputed. If H is a new relational variable, we can define H such that

H(x,y,z) <-> Any predicate involving free variables x,y, and z, but not involving the relational variable H

The last part is important because otherwise I could define H such that

H <-> not H

which would of course result in our logic falling apart.

The predicate on the right may also contain free variables other than x,y,z, in which case the definition itself can vary depending on the values of these parameters.

If one can prove

There-Exists(x) P(X)

Then may define and object Obj such that

P(Obj)


If one can prove

For-all(x) There-Exists-Unique (y) P(x,y)

One can use this to define a function F satisfying

For-all (x) (y) F(x)=y <-> P(x,y)


So that's what I mean when I say a definition cannot be disputed. ^^


So we cannot just define

G -> Necessarily G

as this is circular. It must be taken as a premise, or proven from previous assumptions.

I do agree with you that using the S5 modal system involves making metaphysical assumptions that a skeptic need not share (and in fact I think Plantinga used S5 precisely because it allowed for him to talk about a God who exists necessarily in the way he wanted to). Nonetheless, I think there's more than one way to criticize the argument. In fact, I'm beginning to lose count of the number of ways one may legitimately criticize the argument. . .

Rick Taylor said...

Plantinga's Ontological Argument
--------------------------------

Imagine that I define a specific restaurant to be a "kingly restaurant" if it's part of a chain that has restaurants in every city in the country. Suppose I live in Palo Alto CA, and I decide I want to find out if there's a kingly restaurant in Palo Alto.

I visit the local Olive Garden in Palo Alto. Unfortunately it doesn't have a branch in New Haven CT, so the Palo Alto Olive Garden is not a kingly restaurant. I find a Hobee's in Palo Alto. Unfortunately it doesn't have a branch in San Jose CA, so the Palo Alto Hobee's is not a kingly restaurant.

I go on like this for a while, and I wonder if there exists a kingly restaurant in Palo Alto.

Suddenly I have an insight! I don't have to find a kingly restaurant in Palo Alto. I can rigorously prove that if there exists a kingly restaurant in just one city anywhere in the country, then there has to be a kingly restaurant right here in Palo Alto!

While admittedly I haven't proven the existence of a kingly restaurant in Palo Alto, suddenly it seems a lot more likely! Why there are thousands and thousand of cities in the USA, and if even just one of them contains a kingly restaurant, I know there has to be one right here in Palo Alto!

Doesn't that seem like a reasonable supposition? Is there something inherently logically incoherent in the very idea of a kingly restaurant? If not, then it seems like there's got to be one somewhere in the country, given all those cities. On the other hand, the skeptic who doubts the existence of the kingly restaurant suddenly has a huge burden of proof. Why he has to prove there isn't a single kingly restaurant in any city in the country! Has he visited all the cities in the country? What makes him so sure there can't be a kingly restaurant in any city anywhere?

Rick Taylor said...

Nuevo wrote,

you can easily get to

'possibly thereis (x) necessarily isGod(x) & necessarily exists (x)'
------------------

I've been thinking about this more, and I think I see your point. You're not proposing that exists should be a property. You're proposing that necessarily-exists should be a property. The two should be joined into one concept, as it doesn't work if you start with exists with exists separately and then modify it with necessarily. So if x is an object that exists in a particularly world, then necessarily-exists(x) is true in that world, if x exists in all possible worlds. Two things.

(1) I think that solves the problem, as you claim

(2) I think it's cheating.

:)

More specifically, when specifying an alternative logic system, we specify both a syntax and a semantics. The syntax tells the rules for working with the predicates in the new logic. The semantics gives us a way of interpreting the predicates in a model based on our old logic.

Theoretically, all you need to work in a new logic is the syntax. But practically, for building intuition and giving proofs of consistency and completeness, a model is indispensable.

Now, often, looking at the model "from the outside" as it were, you can define concepts which the syntax does not allow you to specify. If you're familiar with non-standard analysis, another example of this is internal vs external sets in nonstandard analysis, where you must pretend the internal sets are the only ones that exist in order for the transfer principle to work.

Now I'm not sure, but I think that while you can define your necessarily-exists property from looking at the model from the outside, the syntax of the S5 does not allow you to include that as a model; the logic isn't suppose be able to look behind the curtain at the nuts and bolts that makes it possible. Of course you can do so if you want, but then all the nice properties of consistence and such you proved for the syntax go out the window, and you're not really working in S5 anymore.

I'm going to have to study more, but I think this is the case, because if necessarily-exists is not given as a property, I see no way to define it syntactically from other elements. It's also weird if necessarily-exists should be a property, when it's obvious counterpart impossible can't be.

So if I have it right, that's yet another fault in the ontological argument. It abuses the modal logic by illegitimately introducing concepts from the semantics into the syntax.

Nueva Argentina said...

Rick,

I think existence is a property on its own; one that, for instance, you and me possess, while the chimaera and the biggest unicorn don't.

You cannot treat the existential quantifier as implying existence, otherwise, we'd have to accept the existence of anything we can name: if c is an individual constant in your language, you'd get:

1. forall x, x=x

2. c=c

3. thereis x, x=c

and you've proven the 'existence' of c, whatever it is. c could well denote the greatest natural number, if you wish; after all, you should be allowed to speak about it, hence give it a name. Incidentally, you would have proven as well that something must exist, so resolving a great metaphysical question. As I see it, it is a huge mistake to assume that existence is somehow denoted by the existential quantifier.

But, in any case, put it this way (as part of a definition of God):

'forall x, isGod(x)-> necessarily (thereis y, y=x and isGod(y))'

So, assuming the actual world is a possible world at all worlds, if God exists in some possible world, there is some being y in the actual world that is God.

Nueva Argentina said...

Rick,

introducing a predicate letter N for necessary existence, as Gödel does in the Dana Scott version of his argument, doesn't touch the properties of S5 in the least.

Gödel uses a higher order modal predicate logic, whose modal segment includes S5 (i.e. includes axioms K and A5, which amounts to R being Euclidian, not symmetric as I wrongly wrote). What predicate letters we add and what sets or properties we choose to interpret them cannot interfere with S5's properties.

Rick Taylor said...

Nueva,

Ok, that does make sense, thank you for explaining it. It still seems to me that you'd need to add more than the predicate N; wouldn't you also need to add some additional axioms reflecting the properties it would have consistent with the semantics? It wouldn't interfere with S5's properties, but it seems like it would add to them, and you'd be working in a more specialized system with the additional letter N and the additional axioms regarding N, so you'd be working a more specialized system; perhaps we could call it S5+ :). Either that or those axioms about the property N would have to be added as additional premises in the ontological argument.

Nueva Argentina said...

Yes, Rick,

of course, an axiom has to be added defining N, something like:

'forall x, N(x) iff Nec thereis y, y=x'

Gödel did it through the notion of 'essence':

'N(x) iff forall phi, if phi ess x, then nec thereis x, phi(x)'

You cannot say this is just S5, of course, but it retains S5's properties, the formal and the informal. Among the latter, the property of being intuitive correct for metaphysical modality.

The idea S5 encapsulates is that metaphysical modality is not contingent. That is:

if nec p, then nec nec p;

if pos p, then nec pos p.

Rick Taylor said...

"You cannot say this is just S5, of course, but it retains S5's properties, the formal and the informal."

That makes sense, appending a new property with axioms would result in a new system that retains S5 properties, just as S5 retains S4's properties. It seems like every step up we go, we're making additional assumptions about the nature of metaphysical possibility. I've only recently bought a book on modal logic, and perhaps after I've had a few weeks to study, I'll be able to speak more intelligently about it.



"The idea S5 encapsulates is that metaphysical modality is not contingent."

I understand that. One thing I've read is that assuming S5 is equivalent to assuming that the access relation between possible worlds is an equivalence relation, dividing them into equivalence classes. Saying a predicate is necessarily true in a given world is saying that its true in every world in the equivalence class of that world.

As I understand it, most treatments of the ontological argument go a little beyond S5, assuming that all worlds are accessible from one another, so that there is only one equivalence class. With this assumption, saying that something is necessarily true in a given world is the same thing as saying its true in all worlds period.

I do wonder how it would work if you were working with a model that satisfied S5, but in which there were more than one equivalence of worlds. It seems then that something could be necessarily true in the worlds in one equivalence class, contingently true in the worlds in another, and impossible in the worlds in a third equivalence class.

I wonder how you'd express that though, except by explicitly referring to these other worlds, as the truth of the proposition in the actual world couldn't be affected by what happened in a world from a different equivalence class. Could we introduce other versions of possibly and necessarily that were allowed to refer to the truth of a proposition in worlds beyond those in the equivalence class of the world being considered?

Nueva Argentina said...

Yes, Rick,

formally, S5 is sound iff the accessibility relation is an equivalence relation.

But most metaphysicians (if not all) understand that a world can only be called 'possible' if it is possible at the actual world (i.e. accessible from the actual world). After analyzing the concept for a while, I can't avoid coming to the conclusion that 'possible' can only mean 'possible at the actual world'.

Assuming this, S5 amounts to all worlds being accessible from each other (i.e. to the existence of just one equivalence class).

On such an assumption, a bit less than S5 is required for the argument: namely, the Brouwerian axiom:

pos nec p -> p

which is equivalent to the accessibility relation being symmetric.

Assuming this axiom, if 'God exists' is necessary at some world, then it is true. And 'God exists' is necessary wherever it is possible, by definition of God.

Rick Taylor said...

"But most metaphysicians (if not all) understand that a world can only be called 'possible' if it is possible at the actual world (i.e. accessible from the actual world)"

I'm a mathematician rather than a metaphysician. If we wanted to, it seems to me we could create two new ontological operators, conceivable and universal. A statement would be universal if it were true in every possible world, not just the worlds in the equivalent class of the actual world, and conceivable if it were true in any world (again not restricted to the worlds in the equivalence class of the current world).

Then even if God were impossible, it might still be conceivable he was possible (and hence necessary). This is all pretty silly, but it would be a way of trying to model a state where we imagine God to be conceivable in some sense (and therefore get around Plantinga's objection), while still asserting he might be impossible.

And of course we could repeat the whole game at a higher level. To me the whole thing illustrates why Plantinga's argument is not convincing.

And of course we could have nested levels of possible worlds, with levels of possibility indexed by the natural numbers, or even by the ordinals. It all seems quite silly.

Rick Taylor said...

"And 'God exists' is necessary wherever it is possible, by definition of God."

I still don't like the idea of being able to define an entity in such a way that it must necessarily exist if it exists, just by definition, which is my main objection to introducing N into the language (even though we can clearly consistently do so). I like to keep it so that definitions can be arbitrary.

Again, I'm currently studying modal logic, so my opinion on this may change as I learn about it.

Rick Taylor said...

nueva argentina: "The real point is whether S5 (or at least the Brouwerian axiom) is the correct logic for metaphysical necessity."

After lots and lots of thought, I've come to the conclusion you were right all along; this is indeed the nub of the matter.

Nueva Argentina said...

Thanks, Rick.

Glad you saw it.

Now I'd say S5 is vastly more likely than not to be right...

Rick Taylor said...

Yes, after even more thought I'd agree. But its correct for alethic modality, not for apodictic modality (which I only just learned about from John Burgess's Philosophical Logic). Alethic modality is the modality of what predicates must necessarily be true by virtue of their form alone, while apodictic modality is that of what predicates we can demonstrate must necessarily be true by virtue of their form alone. In general S5 is assumed to be appropriate for alethic modality, and S4 for apodictic.

Many proponents of the modal ontological argument use the alethic meaning of possibility to establish its validity, then appeal to the apodictic meaning to support the premise "it's possible God exists" by challenging us to prove its impossible for God to exist. But this is specious. The ontological argument is only valid if we use the alethic sense of possibility throughout, and arguing that the concept of God appears not to be logically incoherent is insufficient to establish the metaphysical possibility of God's existence.

In S5, the assertions "P is possible" and "P is either necessarily true or it is possible that P is true contingently" are logically equivalent. So establishing the premise, "It is possible God exists" is equivalent to establishing the premise "Either God necessarily exists or it is possible God exists contingently." Since the proponent of the ontological argument presumably believes it is impossible that God exists contingently, the only way for him to support the premise is to show that God exists necessarily. But this is what the ontological argument purports to prove; therefore it is circular.

Nueva Argentina said...

No, Rick, there is no circularity in the argument.

From the premise 'It is possible for God to exist' you get by means of S5 'God necessarily exists' without any flaw.

If S5 is, as I think it is, correct for metaphysical possibility (alethical and apodictical are perhaps not the adequate adjectives here), then the point is what evidence there is for God's metaphysical possibility.

God's metaphysical possibility is the moot point here.

Rick Taylor said...

No, Rick, there is no circularity in the argument.
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I see the argument is valid in S5. Also, whether an argument is circular is to some degree a judgement call. Still, working in S5, the statements "P exists" and "Either P exists necessarily or it is possible P exists contingently" really are logically equivalent. Being asked to grant a premise that is logically equivalent to "Either God exists necessarily or it is possible God exists contingently," in an argument purporting to prove "God exists necessarily" that assumes as a result of a definition that its impossible for God to exist contingently seems at least arguably circular to me, even if the argument is otherwise perfectly valid.


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From the premise 'It is possible for God to exist' you get by means of S5 'God necessarily exists' without any flaw.
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Absolutely.

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If S5 is, as I think it is, correct for metaphysical possibility . . . then the point is what evidence there is for God's metaphysical possibility.
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I agree. The trouble is that at this point I have no idea how you would give evidence for the metaphysical possibility of God's existence. Epistemic possibility does not prove metaphysical possibility. Yet it seems that if you give evidence for the possibility of something, by definition you're supporting its epistemic possibility, as epistemic possibility is what we can actually justify. Maybe I'm just confused, but I don't see how to proceed.

Of course there is Maydole's argument, but that seemed to me to be completely unpersuasive.

Nueva Argentina said...

Rick,

you say that

'either God exists necessarily or it is possible that God exists contingently'

is equivalent in S5 to

'It is possible that God exists'.

Let me formalize the equivalence this way:

'P(g)<-> N(g) v P(g & P(~g))'

That's right. Adding the premises

'g -> N(g)'
'P(g)'

you get N(g).

We agree that God exists iff God is possible. So, the point is: is God possible? I only know Gödel's argument for that possibility (which you can find at Wiki). Please, inform me about Madole's.

Rick Taylor said...

There's a link to 20 pages of Robert Maydole on the ontological argument here.

http://commonsenseatheism.com/wp-content/uploads/2009/10/Maydole-The-Ontological-Argument.pdf

I admit I haven't read it all, but he makes three assumptions:

1-The negation of a great making property is a great making property
2-Any property entailed by a great making property is itself a great making property
3-Being a maximally great being is itself a great making property

To me it seems there's no reason to grant that any particular property is a great making property before we establish that it's possible for it to hold for something (otherwise, being able to create a rock God could not lift would be a great-making-property). Therefore, before we grant that being a maximally great being is a great making property, we must establish that it is possible for something to be a maximally great great being, which means we need to establish that before we grant this premise, we must establish that it's possible a maximally great being exists. Of course that's what the argument purports to show, and so it is, while perfectly valid, circular.

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We agree that God exists iff God is possible. So, the point is: is God possible?
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This is true, but does this actually get us anywhere? I could equally validly argue,

1. Either God exists or 2+2 = 5
2. 2+2 is not 5
3. Therefore God exists

and then say, ok all we have to do now is to establish that either God exists or 2+2=5. Yes that's true, but given that we know the rules of arithmetic and logic, how would we justify the assertion that either God exists or 2+2=5 without simply arguing God exists directly?

The actual case isn't so different, as the assertion "God possibly exists" is logically equivalent within S5 to "God exists necessarily or it is possible God is contingent." Given that the we've defined God as being impossible to exist contingently, how can we support this in a way that is not just supporting the assertion that God exists necessarily? Perhaps it's possible, but its difficult to see how, especially if we're limiting ourselves to apriori reasoning. Of course if we can support the assertion "God exists necessarily" that will support the premise "God possibly exists," but then the modal ontological argument doesn't actually add anything.

Nueva Argentina said...

Well, Rick, the point for some supporters of the modal ontological argument (like Plantinga) is that they think some people may have independent reasons to believe that God is possible.

Others, like Goedel and Maydole (thanks for the reference) think they can give such reasons. What you report about Maydole (except the first premise, which may contain a typo) reminds of Goedel's.
It seems natural to assume that a conjunction of positive properties is a positive property; so being divine is a positive property; it also seems intuitive that no positive property should entail a nonpositive property (otherwise, how could it be called properly positive?). This (together with the premise that there are some nonpositive premises) implies that being God doesn't entail all properties; hence, it is possible.

I don't worry about equivalence in S5 implying circularity. Arguments are meant to make explicit what previously was only implicit. Some renowned inference rules work on both directions (consider De Morgan's) and this doesn't turn them circular.

But for the moment, I am with you. We also know that Goldbach's conjecture is true if possible (for mathematical untruths are impossibilities); but this doesn't seem to help much to prove it.

Rick Taylor said...

Well, Rick, the point for some supporters of the modal ontological argument (like Plantinga) is that they think some people may have independent reasons to believe that God is possible.
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That's fair enough. I'm skeptical of the possibility of supporting the possibility premise without supporting God's necessary existence, but I can't rule it out entirely, and I'm more than willing to listen to proposals.

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Some renowned inference rules work on both directions (consider De Morgan's) and this doesn't turn them circular.
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It isn't the inference rules that are circular in and of themselves, it's how they're used. Certainly if someone made the argument using De Morgan's Law

Premise: It is not the case that either God doesn't exist or God is not according to his essence a necessarily existing being.

Conclusion: Therefore God exists, and according to his essence is a necessarily existing being.

I think I would call that argument circular; or at the very least, not very helpful.


Similarly:
Premise: it is not the case that God doesn't exist
Conclusion: Therefore God exists

This one's more interesting, because an intuitionist would not hold the premise and the conclusion to be logically equivalent. On the other hand, an intuitionist would not find the argument valid.


Again, this isn't conclusive. the equivalence between possibility and necessary possibility is more elusive than the above laws, and perhaps express a philosophical truth as much as a logical one. And the example of double negation could be thought of as a justification for proof by contradiction; perhaps Plantinga's argument will justify another proof technique. I'm open to being convinced. It would help to have one example from any field where someone argued for the metaphysical possible truth of a proposition that was recognized to be necessarily true if it was true, without just directly arguing for its truth.



On a completely different aspect of the argument, I originally felt that it needed to be recast in predicate logic to be persuasive. I abandoned that for a while, but I'm returning to that now.

The reason is that in it's present form, the modal ontological argument proves that any being conceived as existing necessarily that possibly exists must necessarily exist. As its valid, it's valid for any G that satisfies its premises. But the proponent of the argument does not wish to prove that every such being exists; they don't want to prove that Jehovah exists, and Allah exists, and the God I believed in as a child exists. . They don't want to prove that every God we can conceive of as necessarily existing possibly exists; they want to prove there exists at least one such God that possibly exists. And that means dealing with the existential predicate.

I can't say I'm eager to tangle with the Barcan formulae though. It took me long enough to make sense of S5.

Nueva Argentina said...

Theists surely believe that there can be only one being which is both possible and such that its essence contains necessary existence. For instance, they would deny that an island is possible whose essence contains necessary existence.

But how does this relate to modal predicate logic?

You say:

"It would help to have one example from any field where someone argued for the metaphysical possible truth of a proposition that was recognized to be necessarily true if it was true, without just directly arguing for its truth."

Anyone who supports S5 will be aware that claiming possibility in that context would imply claiming truth. I remember trying to show some friends that modal collapse (i.e. that all truths are necessary) is possible in order to show it is true (roughly, I argued that, whatever the legality that establishes necessary truths (e.g. reason), it cannot limit itself by establishing the existence of contingent truths).

Rick, after reflection, I think you've pointed out something very important: one can hardly regard a property as a great-making one unless one presumes it is metaphysically possible. For instance, in assuming that necessary existence is great-making or that godness is a positive property, we seem to be assuming they are possible. This tastes of circularity.

But a defender of the argument would surely say that the argument goes from the intuition that something is great-making to the conclusion that it is possible.

This is subtle.

Rick Taylor said...

I take back what I said about needing predicate logic. Thinking about it again, I think I see how Plantinga gets around the issue.


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But a defender of the argument would surely say that the argument goes from the intuition that something is great-making to the conclusion that it is possible.
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Yes, but when they consult their intuition, what form of possibility are they implicitly assuming? The notion of metaphysical possibility is extremely counter-intuitive; I've spent about two months figuring out what it was about, it's relationship to S5, and why one cannot legitimately conflate it with logical coherence or conceivability.

I would argue that the ontological argument gains its persuasive force by confusing two notions of possibility. By working in S5, it assumes an understanding of possibility that justifies the validity of the ontological argument.

But when people use their intuition to reflect on the premises, they naturally resort to their intuitive understanding of possibility as conceivability or logical coherence. They assume that because a particular idea makes sense to them, because it doesn't appear to entail a contradiction, because they can hold it in their minds without contradiction, that means it's possible. And it does mean it's possible in a certain sense; just not in the sense necessary for the ontological argument to be valid.

Nueva Argentina said...

You're right that conceivability is not the same as metaphysical possibility. In my ignorance, I can conceive that Goldbach's conjecture is true and I can conceive its being false, though only one is metaphysically possible.

But I also think the defenders of the argument have metaphysical possibility in mind all the time. I'd say, their conviction is ultimately this: if X were impossible, it would not be a perfection (or a positive or a great-making property) because it would entail a lot of bad properties; since it is evident to me that X is a perfection, X must be possible. If in addition necessary existence and godness are perfections to you, then S5 gives you God.

The argument may be demanding something like a logic of perfection. Besides S5, that's the underlying issue.

Certainly, that logic is not yet available.

Rick Taylor said...

You're right that conceivability is not the same as metaphysical possibility. In my ignorance, I can conceive that Goldbach's conjecture is true and I can conceive its being false, though only one is metaphysically possible.
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I actually don't mean conceivability in that sense. I'm not using it in the sense of "for all we know." I'm using it in the sense of what is logically demonstrable, or what we can coherently imagine.

So if we assume for the moment Goldbach's conjecture is false, then I would deny that it would be conceivably true in the sense I'm referring to. Because if Goldbach's conjecture is false, it is demonstrably false. There really is an even number greater than four that's not the sum of two odd prime numbers, and we can't coherently imagine such a number as the sum of two primes when it isn't.

On the other hand, if Goldbach's conjecture is true, it might still be conceivable it's false. It might be the case that Goldbach's conjecture is true, but that it's impossible to prove it true, and that we can even add as an axiom that there exists an even number greater than 4 that is not the sum of two primes to number theory without ever being able to derive a contradiction.

So I'm talking about conceivability in the sense of what can be demonstrated or coherently imagined using reason alone (John Burgess called it apodictic modality), not in the sense of for all we know. Both of these are different than metaphysical possibility.

Rick Taylor said...

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I'd say, their conviction is ultimately this: if X were impossible, it would not be a perfection
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Well I would agree with them. I agree a property must be possible in order to be a perfection. I would say that if being a maximally great being is impossible, then it's not a perfection. But then I question what the justification is for presuming being a maximally great being is a perfection, before we've established it's metaphysically possible.

And if the justification for accepting that "being a maximal great being" is a perfection is an intuition, then what is the nature of this intuition? In particular, does this intuition contain the knowledge that it is metaphysically possible that a maximally great being exists? Because if it does, then there's no need for Maydole's argument. We can justify the premise in Plantinga's argument by saying we have an intuition that a maximally great being is possible and go from there. Then Maydole's justification is a big red herring; our real justification is our intuition.

On the other hand, if the answer is no, then the intuition must have been on the order of, I really don't know whether being a metaphysically great being is metaphysically possible, but I can see it must be a great making property whether it is metaphysically possible or not! That makes no sense to me. The intuition that even if it were metaphysically impossible, being a maximally great being would be a great making property seems like nonsense to me. In fact it seems clearly false.

But if that's not the intuition they had, if their intuition was that being a maximally great making being would be a great making property only if it were metaphysically possible, then they need a separate intuition that it's possible to be a maximally great being. And if they have that separate intuition, they don't need Maydole's argument at all, they can use that separate intuition to directly justify the premise of Plantinga's argument.

My point here isn't to deny the possibility of genuine intuition. If someone tells me they have an intuition that a maximally great being exists in all possible worlds, how can I object?

But then I question what Plantinga's ontological argument and Maydole's proof add to that intuition? It seems people want to deny they have an intuition that it's necessarily the case a maximally excellent being exists. The want to say, No, I don't know that such a being necessarily exists, I just have intuition that it's possible it necessarily exists. Or no, I don't have an intuition such a being is even metaphysically possible, I just have intuition that it's a great-making property to be such a being.

In each case it seems to me the intuition doesn't make any sense without an intuition that a maximally excellent being exists necessarily. Which is fine. But if one has such an intuition, then what does one need any of these so called "proofs" for?

Nueva Argentina said...

Rick, let me comment on this:

"On the other hand, if Goldbach's conjecture is true, it might still be conceivable it's false. It might be the case that Goldbach's conjecture is true, but that it's impossible to prove it true, and that we can even add as an axiom that there exists an even number greater than 4 that is not the sum of two primes to number theory without ever being able to derive a contradiction."

You can only add this to first order Peano arithmetic and keep consistent if Goldbach's conjecture is not provable in it (though this is surely entailed but your 'impossible to prove').

But if the conjecture is true, it is a logical consequence of the axioms of second order arithmetic because those axioms induce a categorical theory: all of its models are isomorphic. So you cannot add to those axioms any sentence contradicting the conjecture and still have a consistent theory (provided the conjecture is true).

(Though you could have a consistent second order deductive system because the conjecture could be unprovable in second order arithmetic: although it would be a logical consequence of its axioms, second order logic, being incomplete, could perhaps not derive it from them)

Anyway, even in your sense of conceivability, it's clear that it need not be equivalent to possibility, since you admit the conjecture could be true and its negation be conceivable (although impossible).

Nueva Argentina said...

Rick, let me quote you:

"...if the justification for accepting that "being a maximal great being" is a perfection is an intuition, then what is the nature of this intuition? In particular, does this intuition contain the knowledge that it is metaphysically possible that a maximally great being exists? Because if it does, then there's no need for Maydole's argument. We can justify the premise in Plantinga's argument by saying we have an intuition that a maximally great being is possible and go from there. Then Maydole's justification is a big red herring; our real justification is our intuition.

On the other hand, if the answer is no, then the intuition must have been on the order of, I really don't know whether being a metaphysically great being is metaphysically possible, but I can see it must be a great making property whether it is metaphysically possible or not! That makes no sense to me."

For all I can see, you're quite right. The attempt at proving God's possibility made by Goedel and Maydole is not clear at all. They start with the assumption that being God is a perfection and it makes little sense assuming that unless you're also assuming that being God is possible, for nobody (not certainly Goedel or Maydole) would count as a perfection what entails all imperfections.

This is clarifying.

Thanks.

Rick Taylor said...

Thank you. I've also find it very clarifying to have someone to talk with about these matters who has some real background in the subject matter.