Tuesday, April 21, 2009

A modal ontological argument

Update 2015: I wrote a new series on ontological arguments.

If you wish to understand this post, you should probably read my exposition on modal logic first. You may also wish to read my analysis on a much simpler ontological argument.

To briefly review the simpler ontological argument: The argument says that we can define God as a necessarily existing being. Therefore, by definition, God exists. However, this takes the power of definition too far. The most we can say is that if there exists an object which we can properly call God, then that object, by definition, exists. If God exists, then God exists.

The problem with the argument is that it's rather useless to define God as a being which exists. But hold on! I never defined God as a being which exists. I defined God as a being which necessarily exists. This definition is not quite as useless. We can say that if there exists an object which we can properly call God, then that object, by definition, necessarily exists. If God exists, then God necessarily exists.
By definition of God: g g
The letter "g" represents the specific statement "God exists". Note, however, we did not completely define God. In fact, we could replace God with all kinds of absurd objects, and the proof would still hold. This is actually quite a problem for the ontological argument (as well as many other proofs of God). The proof is valid for any statement p as long as pp is necessarily true. For instance, we could replace the object God with "the invisible pink unicorn which necessarily exists" or "a left shoe which necessarily exists".

Nonetheless, let's continue onward. I'm going to go through the proof, step by step.
• Theorem 1: ¬g ¬g (The contrapositive of the definition of g)
• Theorem 2: ¬g ¬g (Using the definition of to substitute into Theorem 1)
• Theorem 3: (¬g ¬g) (Application of Axiom N to Theorem 2)
• Theorem 4: ¬g ¬g (Application of Axiom K to Theorem 3)
• Theorem 5: ¬g¬g (Axiom 5 applied to ¬g)
• Theorem 6: ¬g ¬g (Combining Theorems 4 and 5)
• Theorem 7: ¬¬g ¬¬g (The contrapositive of theorem 6)
• Theorem 8: gg (Using the definition of to substitute into Theorem 7 twice)
• Theorem 9: g g (Combining Theorem 8 with Axiom T)
You may have noticed that I did not prove what I set out to prove. I only proved g g, which says that if g is possible, then it is true. That's not quite the same as saying that g is true, but it's still something. Now, all we have to do is prove that g is at least possibly true.

And here is where the problems begin. All the previous work was purely logical manipulation, and is necessarily true if you accept the axioms of modal logic. I thought the axioms were pretty reasonable, and rejecting them would be too high a price to pay. However, it seems we need another premise, g. To support this premise, lots of arguments have been offered by various people, but I don't think they're nearly as fun or rigorous as the modal logic section.

One common argument for g is that g is self-consistent. I can conceive of a God without having any contradictions. Based on my current knowledge, it is entirely possible that God exists. It's possible that God exists, therefore God exists.

The problem is that this same argument seems to apply to the statement ¬g. ¬g is a self-consistent statement. I can conceive of a world in which God does not exist without having any contradictions. Indeed, I can conceive of a world where nothing at all exists, where there could not possibly be any contradictions since there is nothing around to contradict. Based on my current knowledge, it is entirely possible that God does not exist. Therefore we can take the premise ¬g. By Theorem 2, we conclude ¬g: God does not exist.

Obviously, the premises g and ¬g cannot both be true. Either God exists in all possible worlds, or God exists in none of them. Since we used the same argument for both both g and ¬g, that argument must be fallacious. But where exactly did it go wrong?

The error, I think, is in the concept of . g does not quite mean "g is possibly true." In fact, it means "Among all possible worlds, there exists at least one in which g is true." The concept of "All possible worlds" was never exactly defined. In fact, the definition is arbitrary. I could have declared "all possible worlds" to be our world and our world alone, and you never would have been able to prove me wrong from the axioms.

I also could have declared "all possible worlds" to be the set of worlds which are metaphysically possible (a rather complex philosophical concept). Under this definition, we would not be able to prove g or ¬g.

However, if I had declared "all possible worlds" to mean the set of all worlds which are self-consistent, then we would run into problems. Because under this definition, both g and ¬g appear to be true (unless the concept of God is inconsistent). And they contradict each other.

Similarly, if I had declared "all possible worlds" to mean the set of all worlds which are epistemologically possible (meaning, it may be true, for all we know), then we would have the same contradiction.

And what happens if we construct a pathological definition of "all possible worlds" such that g is true and ¬g is not true? Then I might question Axiom T, since it is no longer obvious that our world is included among this so-called set of "all possible worlds."

And it goes on and on. Well, wasn't it clever, at least at first? I think so. It seemed for a moment that we arrived at a paradox, a sort of 1=2 moment. Ontological arguments tend to be that way. Most people immediately recognize that it is a little too clever, that it proves a statement which is a little too strong to be true. Similarly, most philosophers think that ontological arguments fail, though they may disagree on exactly why they fail.

In my opinion, ontological arguments are merely interesting philosophical curiosities. It's rather silly when an apologist actually tries to use one as a serious argument.

Scott said...

Sorry to keep hectoring you, but are you familiar with Goedel's ontological proof? I think it captures something more substantial than many ontological proofs, although of course it still has problems. The exposition here is excellent, if you can get past the cheesy pictures:

http://www.stats.uwaterloo.ca/~cgsmall/ontology.html

miller said...

Why yes, yes I am. I've been meaning to write about it.

Chuck said...

"In fact, we could replace God with all kinds of absurd objects, and the proof would still hold."

That's actually not true. The statement, "If God exists, then God exists necessarily". This is a true statement whereas "If a pink unicorn exists, then it exists necessarily" is a false statement. Your objection would hold only if the statement was "If God exists necessarily, then God exists necessarily." Conditional statements are not always tautologous, otherwise, there would be no reason to say that denying the antecedent and affirming the consequent are fallacious.

Furthermore, one need not rely on the premise "<>g" in order for the argument to hold water. All you need is the premise, "~g-->~<>g" and the premise that whatever cannot possibly exist in logically contradictory. Thus, you form a dichotomy between necessity and impossibility. Since God is possible, you can infer by modus tollens that "~g" must be false. Here's a good formalization of Charles Hartshorne's modal argument which you did not write an argument about:

(1) (~Rg-->~<>Rg)
(2) (x)(~<>Rx == Cx)
(3) (~Cg)
(4) (~<>Rg-->Cg)
(5) ~(~<>Rg)
(6) ~(~Rg)
::. Rg

Inferences used: MT, UI, Equiv, Simp, DN.

Rx = x refers to a real thing outside the mind

g = the idea of God

Cx = x is logically contradictory

The Barefoot Bum said...

Since God is possible...

You are confusing epistemic and modal possibility. In modal logic, this statement is: We definitely know that there is some possible world where God actually exists. This is not a sound assumption: we do not know there is any possible world in which God actually exists.

miller said...

First, I'm pretty sure I did not say it was true for a pink unicorn. I said it was true for the object, "the pink unicorn which necessarily exists." Not sure which pink unicorn you're talking about, but it's not the one I was talking about.

Second, I know that g => []g is not tautologous. However, I defined g to be an object with the property that g => []g, so it's true by definition of g.

Third, I know that you don't need to take "<>g" as a premise. But the particular argument I'm considering does indeed take "<>g" as a premise. I couldn't possibly cover every single variation of the ontological argument, I would get bored first.

Charles Hartshorne's argument is basically equivalent. You've just replaced "<>g" with "~Cg". I don't necessarily accept that premise. Would you like to discuss it more?

Chuck said...

Again, the premise is stating, "If God exists, then God exists necessarily", not "If God exists necessarily, then God exists necessarily", which would have to be a premise in order for your objection to hold any kind of ground. Remember, you said that "we could replace God with all kinds of absurd objects and the proof would still hold", but I've just demonstrated that if you replaced "God" in the premise "g-->[]g" with "unicorn", then the premise would have a completely different truth value. So I ask you: What modal ontological argument has the premise, "[]g-->[]g"? That's definitely not part of Plantinga's, Godel's, or Hartshorne's.

God is *not* defined as a necessary being. God is *defined* as the greatest conceivable being and ontological necessity is a logical implication of that. In the same way, "p-->q" is not *defined* as "~p v q", but the latter is a necessary implication of the former.

Obviously, you are attacking versions of the argument which are weak, though to be fair, I haven't read your critique of Godel's argument.

Hartshorne's argument is not equivalent. Saying that X is a contradictory idea is *not* the same thing as saying that there are no possible worlds where X. Again, you continually conflate tautologies with implications and they are *not* the same.

I'd love to discuss it further. How is "God" a contradictory idea?

The Barefoot Bum said...

God is *not* defined as a necessary being. God is *defined* as the greatest conceivable being and ontological necessity is a logical implication of that.

You don't show a proof of this statement.

In the same way, "p-->q" is not *defined* as "~p v q", but the latter is a necessary implication of the former.

Excuse me? In the same way? The latter is a logical transformation rule. Call me naive, but I don't know of any actual rule of propositional calculus or any-order logic that transforms "greatest conceivable being" to "ontological necessity".

Furthermore, as Kant and Godel already noted, we have to add the property of actual existence as a "greatest" to make the argument work. You can't prove a premise, and there's no logical contradiction to denying it.

Saying that X is a contradictory idea is *not* the same thing as saying that there are no possible worlds where X.

Why not? To deny this interpretation says either that there is at least one possible worlds where at least one contradictory idea is true, or that there are some non-contradictory ideas that are not true in every possible world. This seems to contradict ordinary possible world semantics.

And what, then, does premise (2) [(x)(~<>Rx == Cx)] mean in your earlier comment? Premise (2) seems to explicitly state that non-existence in any possible world is equivalent to logically contradictory.

The problem with Hartshorn's argument is that g is in some way a modal statement; it is not a non-modal statement. The domain of modality must include only non-modal statements; if we want to have a domain of modal statements, we have to create a meta-modal domain to talk about their possibility. A self-referential domain of modality would enable Russel-like paradoxes.

I've never met a philosopher or philosophy student who grasped even the basics of self-referential paradoxes. Computer programmers who understand recursion seem to grasp the idea quickly.

The Barefoot Bum said...

... if we want to have a domain of modal statements, we have to create a meta-modal domain to talk about their possibility.

... if we want to have a domain of modal statements, we have to create a meta-modal system to talk about their meta-possibility.

As noted, this is theory of types 101. If you want to create a different way of handling self-reference than a theory of types, you're free to do so, but simply ignoring well-understood self-referential paradoxes is insufficient.

The Barefoot Bum said...

If it is formally possible to draw any modal conclusions from "greatest conceivable being", then "greatest conceivable being" is part of a modal system, and we must create a meta-modal system to discuss it.

For example,

(1) "A blog called 'Skeptic's Play' exists"

is non-modal statement. There's no modal operator. I'm not saying (1) is possible, or necessary; it's just a bare statement.

The statement

(2) "It's possible that (1) is true"

is a modal statement. These are different statements. It's equivalent to

(2') "There exists a possible world in which (1) is true"

It's also intuitively plausible that there are possible worlds where (1) is false:

(3) "It is possible that (1) is false"

which is the same as

(3') "There exists a possible world in which (1) is false"

It doesn't matter how we conclude (2) and (3) (or 2' and 3'); they are modal statements, and they are necessarily true (or necessarily false) in all possible worlds within the domain of modality.

We happen to know that (2) is true: (1) is indeed true in this possible world. We do not, however, know that (3) is true: It is logically possible for it to be true or false, since neither its truth nor falsity would entail a logical contradiction.

Therefore, if we allow a self-referential domain of modality, we must be able to say:

(4a) There exists a possible world where (3) is true
(4b) There exists a possible world where (3) is false

However, this construction leads to a contradiction: There exists a possible world where (1) is true AND There exists no possible world where (1) is true.

I know you're not going to understand this, Chuck, but I hope miller will get it.

Chuck said...

Proof of what? That ontological necessity is a necessary implication of greatest conceivable being? It's simple, a necessary implication of "greatest conceivable" is eternal, as anything which is not eternal may either begin to exist or stop existing and that would require it to be sustained by contingent states of affairs, which cannot apply to a being with attributes fully actualized. Go ahead and read up contemporary Thomism.

Why would there be an axiom of logic which states the attributes of God? That's being disingenuous, as you know that this is not what anybody is proposing.

Kant *never* proposed that you had to add the property of existence to make anything work. Kant proposed that Anselm treated existence as a predicate, which is the *reason* that the argument does not work. Plantinga, Hartshorne, and others avoided this.

Contradictory ideas and impossibilities are reducible to one another, but they are not contained in each other's definitions. An impossibility is "(Ex)~<>x", while a contradiction is "(Ex)(Px & ~Px)".
A biconditional stipulates that both the antecedent and the consequent are implications of one another. For example, "1 + 1 iff 2" is a true statement but "2" is not *defined* as "1 + 1".

"g" is an individual constant. The fact that it is included in the premise on its own attests to the fact that existence is not being treated as a predicate. Otherwise, the statement would utilize "Ex" or "x has existence". It is not a modal statement, it is a statement about the present state of affairs as opposed to what could or could not be the case, which are modal statements.

Self-referential paradoxes are stupid things like "This statement is false" and it has been resolved by philosophers like A.N. Prior.

What exactly is the correlation between formal logic and computer programming?

Chuck said...

I really do not see the point in being condescending. If you do not think I will understand, why not make an effort to help me understand? Or are you not out to inform anyone but simply to show off your proverbial penis size?

Anyway, here is how I've interpreted your argument:

s = blog called "Skeptic's Play"

(1) S
(2) <>S
(3) <>~S
(4) <> (<>~S)
(5) <> ~(<>~S)

(2) and (3) are both consistent with (1). No problem.

I'm not sure what you mean by "necessarily true (or necessarily false) in all possible worlds within the domain of modality". Is this an application of axiom S5?

We *do* know that (3) is true and we also know that it is *not* logically possible for it to be false because that would violate axiom S5. Otherwise, you are saying that there are possible worlds where possible things are not possible. That's a logical contradiction.

(4) and (5) are false statements. They both violate axiom S5.

The Barefoot Bum said...

I really do not see the point in being condescending.

You wouldn't, would you.

Like I said, I'm not trying to convince you, I'm trying to convince miller, who I believe to be an honest seeker after the truth.

The Barefoot Bum said...

Skill in philosophy consists of obscuring the fallacies and unsupported assumptions in one's argument.

miller said...

Hello,
I haven't had the time to read these comments yet, but I will. I also plan to revisit the ontological argument in another post, because a friend of mine had a similar question. Patience.

For now, I would like Chuck to support this assertion:

"We *do* know that (3) is true and we also know that it is *not* logically possible for it to be false because that would violate axiom S5."

Okay. Prove to me <>~S.

Chuck said...

There are possible worlds where this blog does not exist. Are you saying that your blog exists by logical necessity? If so, why can I conceive without contradiction a world where your blog does not exist?

miller said...

Are you saying that your blog exists by logical necessity?

Nope. I'm just challenging you to prove otherwise. You can't do it, not without taking new premises.

miller said...

Okay, here are some responses to Chuck. I will intentionally leave out things that I want to include in my upcoming post.

...if you replaced "God" in the premise "g-->[]g" with "unicorn"...

Didn't I already say? I was talking about "the pink unicorn which necessarily exists," not just any old unicorn. You're making me repeat myself.

What modal ontological argument has the premise, "[]g-->[]g"?

None. I suspect this particular statement is a response to an objection that I did not make (ie Kant's second response shown here). I agree with you here; Kant's objection does not apply to modal forms of the ontological argument.

Here's Godel's Ontological Argument. Godel's argument also removes the "<>g" premise, replacing it with something even more clever than "~Cg". "~Cg" is an intermediate step of the proof. But the central problem is that "<>g" does not really mean what it intuitively seems to mean, nor does "~Cg". When you properly understand what the premises mean, there is no basis to assume that they're true.

But I think my upcoming explanation will be better written than that one.

Hartshorne's argument is not equivalent. Saying that X is a contradictory idea is *not* the same thing as saying that there are no possible worlds where X.

I didn't say they were the same. I said they were equivalent, as in logically equivalent. The main difference is that the premise "<>g" has been replaced with "~Cg". These premises are logically equivalent.

Another minor difference in Hartshorne's argument is that he replaced one of the premises with its contrapositive, and replaced modus ponens with modus tollens. This amuses me, because it only succeeds in making the proof more cluttered.

Chuck said...

Didn't I already say? I was talking about "the pink unicorn which necessarily exists," not just any old unicorn. You're making me repeat myself.

And I've addressed it. The premise, "If a necessarily-existent unicorn exists, then it is necessary" is a useless tautology that has nothing to do with any modal ontological argument that I've seen. In fact, I'll try it with Charles Hartshorne's form:

(1) If a necessarily existent unicorn does not exist, then it is impossible.
(2) If it is impossible, then it is logically contradictory.
(3) A necessarily existent unicorn is not logically contradictory.
(4) It is not impossible.
(5) A necessarily existent unicorn exists.

Here's the problem: The first premise itself is self-refuting. It is stipulating what would be true if an existing being did not exist. To really make the objection get off the ground, you could stipulate in the antecedent that the unicorn is *eternal* but then comes to second objection:

We have to examine the metaphysical issues involving the nature of material things such as unicorns and see if they are truly capable of being self-existent (hint: they are not, read up on form/matter, substance/accident, nature/essence, etc.)

"None. I suspect this particular statement is a response to an objection that I did not make."

No. You said "we could replace God with all kinds of absurd objects and the proof would still hold" and I've shown you that this is just not true if you represent the argument accurately. You are attempting the same reducio ad absurdum that others have tried against Anselm and it did not work for them. What makes you think that it will work for you?

"I didn't say they were the same. I said they were equivalent, as in logically equivalent. The main difference is that the premise "<>g" has been replaced with "~Cg". These premises are logically equivalent."

So you agree that if something is not logically contradictory, then it is logically possible?

"Another minor difference in Hartshorne's argument is that he replaced one of the premises with its contrapositive, and replaced modus ponens with modus tollens. This amuses me, because it only succeeds in making the proof more cluttered."

I don't quite see how that is the case. Hartshorne's version comes across to me as the most parsimonious of the modal proofs.

miller said...

So... You say that (1) is a useless tautology... and it's self-refuting? Either you are making big errors, or you communicate poorly. This is not a productive discussion, and I have little reason to think it will become productive any time soon.

If we are to continue, I think we should go back to statement S, which is "A blog called 'Skeptic's Play' exists". Consider "<>~S". I will continue when you prove it, or admit that it is impossible to prove.

The Barefoot Bum said...

My point of using (S) was to illustrate domains. (S) as I wrote it is a non-modal statement: it says nothing about possible worlds.

"A necessarily existent being exists" or "A necessarily existent being does not exist" subtly equivocates "exists".

It's helpful to think of modal logic in terms of set theory.

A possible world is a set of "entities", and there's a set of all the possible worlds (a set of sets). <>p is the same as saying at the set of all possible worlds contains at least one possible world that contains p.

A necessarily existent being is not an entity: it is a set of entities, the set of entities that exist in different possible worlds, with a one-to-one mapping between the elements of the necessarily existing "being" set and the set of all possible worlds.

If we define g as a "necessarily existing being" we cannot say there exists a possible world that contains g; possible worlds contain only entities; they do not contain sets of entities, and they do not contain any entities that exist in other possible worlds.

All of Plantinga's bullshit about "transworld identity" is an attempt to flim-flam around the theory of types and make the flim-flam so complicated no one can figure out how to construct a Russel paradox.

miller said...

Thanks, Barefoot Bum. Clearly spoken.

I think that it at least makes logical sense to say that it's possible that g exists. However modal statements on g have counterintuitive meanings. Modal possibility on g cannot match up with the idea of epistemological possibility, because g is itself a modal statement.

Chuck said...

"A necessarily existent being exists" or "A necessarily existent being does not exist" subtly equivocates 'exists'"

Fortunately, the modal ontological argument makes no such mistake. It only presumes that if God exists, then God cannot possibly not exist. It makes no provisions under which "existence" is contained in the God concept. It merely posits that an eternal being can exist only under certain preconditions, namely, that this being can neither have begun nor can this being end.

Possibilism itself is a self-refuting worldivew insofar that it ostensibly creates a real ontology for possibilities beyond the idea that our understanding of what *could* be is based on our knowledge of how the actual world operates and thus on our ability to predict its behavior. Accordingly, any observation we make would not only tell us something about the actual world, but would tell us things about a completely *different* world that we do not occupy, even though we did not make an observation withint he context of this world. This is neither tenable nor refutable, but pragmatically, it is useless. I see *possibility* as extending to one reality that exists and that any statement we make directly or indirectly says something positive about the one reality that we live in, even if we do not have the capacity to understand it exhaustively. But the main point that I wanted to make is that possibilism itself is a view of an actual state of affairs, namely, that possible worlds are ontologically real instead of being imaginary. Thus, it is self-refuting.

The Barefoot Bum said...

Congratulations, Chuck. You've used two hundred forty one words to say nothing at all.

miller said...

For the interested: I have revisited the modal ontological argument.