Ontological arguments

[erislover]

Ultrafilter, it is [(G->G)&(<>G)] => G.

@hauss

It’s only an intellectual phenomenon. Do you not see this?

I’m an idealist. Everything is “only” an intellectual phenomenon. Sorry to disappoint!

[erislover]

@ultrafilter

One thing that’s been bugging me about this whole thing…

What we are offered is a proof in S5, not of G, but of (G -> G) -> G. Now, unless S5 is very different from traditional propositional calculus, there’s a problem here. If G is true, clearly (G -> G) -> G is true, as it’s an implication with a true conclusion. But if G is false, G -> G is still true, but (G -> G) -> G is false: it has a true hypothesis and a false conclusion.

(~G->~G)->~G?

[hauss]

@erislover

Ultrafilter, it is [(G->G)&(<>G)] => G.

I’m an idealist. Everything is “only” an intellectual phenomenon. Sorry to disappoint!

If that is what an idealist is, than it is not saying much about you, here is why: There are different intensities of intellectual phenomenons and this is all that matters in life.

If it is proven that a being exists in someone’s mind, such as this statement does, what is that to me? But if it is proven that there is a being who will call for an early summer, that is everything to me.

Both of these above cases are indeed intellectual phenomenons, but for different reasons. The first is intellectual because the being originated in someone’s mind. The second is intellectual because an outside force altered the mind, through physical sensory cues. It originated outside the mind, and infiltrated the mind.

Which do you think has more substantial existence? Which has more relevance to your life and your percepted environment? The mind-originated, faceless, non-proactve being, or the mind-altering, proactive being? Why do you think I am respecting this statement only as much as I respect any other clever mind puzzle?

It is worth noting the difference, that is all I am saying.

This statement has it’s place. It’s place is not realistically relevant.

[ultrafilter]

@erislover

Ultrafilter, it is [(G->G)&(<>G)] => G.

Yeah, I missed that.

If we take as an axiom that A -> A for any sentence A, we can show that <>A -> A. That leads to a much cleaner proof of G.

Proof of G in S5:
<>G (assumption)
<>G -> G (from below)
G (modus ponens, 1, 2)

Proof of <>A -> A in S5:
<>A & ~A (assumption)
~~A & ~A (modal quantifier expansion)
~A (&-elimination, 1)
~A (modus ponens, 3, axiom of S5)
~~A (&-elimination, 1)
~A & ~~A (&-introduction, 4, 5)

Every characterization of S5 that I’ve seen uses the axiom in question, so I think this proof is OK.

However, this still rests on the assumption that <>G is true. I haven’t seen any logical arguments on this matter, just metaphysical handwaving. At bottom, we’re back at the old debate as to whether God exists, but we’ve clarified that the only way to show that God doesn’t exist is to show that it’s not possible that God exists. I’ve known that since long before I encountered this argument.

Now, if someone could state G in the predicate calculus (first-order should be OK), we’d have something to debate. As is, there’s nothing new here.

[erislover]

What it means to be a metaphysical idealist is a pretty long and complicated question. But roughly the tenet states that all existence is [at least] mind-correlative. This can seem trivial, as in, “To know something, we must have grasped it intellectually,”; or, it can be stronger, as in, “Reality is wholly constituted of [the] mind.” I don’t intend to address the topic further in this thread, though if you’d like to start another to discuss it I’d be happy to participate.

[erislover]

@ultrafilter

However, this still rests on the assumption that <>G is true. I haven’t seen any logical arguments on this matter, just metaphysical handwaving.

Well, the matter of the second assumption is pretty clean. We’re restricting the set of beings under consideration to only those that are not logically forbidden. This is not, to me, a suspicious step.

In (K), the base modal logic of Kripke upon which S5 is built, there is a rule that demands
A -> A
called “the necessitation rule.” It is used for every theorem. Roughly, all theorems are necessarily the case. There is possibly the matter of equivocation here by using the necessitation rule as a “property” of god. Seen in this light, we might be suggesting that God’s existence is already a theorem, else why assign the necessitation rule? An interesting question I’d like to hear Lib comment on.

At bottom, we’re back at the old debate as to whether God exists, but we’ve clarified that the only way to show that God doesn’t exist is to show that it’s not possible that God exists. I’ve known that since long before I encountered this argument.

It does seem to drive a wedge between strong atheism and the more wishy-washy positions. If we accept the premises, the conclusion definitely follows, and it seems only strong atheism allows us to reject the premises.

[hauss]

@erislover

What it means to be a metaphysical idealist is a pretty long and complicated question. But roughly the tenet states that all existence is [at least] mind-correlative. This can seem trivial, as in, “To know something, we must have grasped it intellectually,”; or, it can be stronger, as in, “Reality is wholly constituted of [the] mind.” I don’t intend to address the topic further in this thread, though if you’d like to start another to discuss it I’d be happy to participate.

I was just using your position to make a point about the irrelevance of Libertarian’s ontological statement. Actually I don’t know if he realizes that the being he has proven exists is only an idea… he might think he has proven an actual being, which is absurd.

[erislover]

hauss, I fear you are missing the subtlety of modal logic. It distinguishes three modes of existence: possibility, actuality, and necessity. Possibility is the case where something is not logically forbidden. It would be analogous to a purely mental idea, like unicorns. There seems to be nothing which forbids their realization, except that they are not realized… which brings us to actuality. These are possibly-existent beings that we find to have, I suppose one could say, manifested. This is the horse, which is neither logically impossible nor missing from our experience: it exists here. Then we would have the third mode of existence, which is necessary existence. That which necessarily exists is not only actual and possible (and some feel actuality demands possibility, not without justication IMO), but actual in every possible world.

Here’s where we have to sort of take a moment to consider what we’re talking about in modal logic. What we are doing are constructing, in principle, all the various “worlds”, one might say permutations, where the rules of logic under consideration hold. Each world exists (possible existence) and has things in it which exist (more possible existence). Our world is one of many possible worlds: the actual world. Given all these permutations, that which exists necessarily exists in every single permutation.

Roughly, it encapsulates the English language’s differentiation between, “can/could [exist],” “does [exist],” and “must [exist]”.

[Liberal]

@Eris

The rejection of <>G indicates that the definition is improper in some way.

That is the most clear and cogent wording I’ve yet seen about the problem of rejecting <>G — and that includes Plantinga, Hartshorne, and the whole lot.

[Liberal]

@hauss

By “shed light”, I meant it exposed the statement for what it really was, taking out the distractors. I understood it before, for the most part, but after the N*m post, it’s clear that it is somewhat “circular”, or “useful of itself”. I’m sure you know this?

I am not sure why you seem to think that you can create information (conclusion that God exists) from no information, but rather a definition and an assumption? No matter how much you deny this, it is truly what you are doing here… but by all means carry on. With a brain as big as yours, you are bound to stumble upon something substantial sooner or later. Just don’t make someone else’s gimmick your foundation.

Yeah, but you ignored everything I said. Peano arithmetic is equally circular and, in that sense, every deductive argument is circular — including the one you’re making to defend your position.

[Liberal]

@Eris

In (K), the base modal logic of Kripke upon which S5 is built, there is a rule that demands
A -> A
called “the necessitation rule.” It is used for every theorem. Roughly, all theorems are necessarily the case. There is possibly the matter of equivocation here by using the necessitation rule as a “property” of god. Seen in this light, we might be suggesting that God’s existence is already a theorem, else why assign the necessitation rule? An interesting question I’d like to hear Lib comment on.

The necessitation rule (RN) essentially states that necessity obtains from necessity — that is, any truth derived from a necessary truth is itself a necessary truth. Thus, if A -> B, then B. The RN is itself a metatheorem. But A -> A is not a premise in any system of K. The reason it succeeds in the MOP is that it is reasonable to posit that if God exists in actually, then He exists necessarily because, owing to His ontological nature as defined, He is the B of A -> B.

[erislover]

Gotcha.

[Liberal]

@Eris

That which necessarily exists is not only actual and possible (and some feel actuality demands possibility, not without justication IMO), but actual in every possible world.

Regarding the underlined portion, yes! In fact, the Brower Axiom of modal logic is A -> <>A. It’s a symmetric accessibility relation.

[Liberal]

@ianzin

Existence is not a property that a thing has or does not have.

Nah. Kant’s been dead for a couple hundred years now.

A prime stumbling block to this conclusion is the objection raised by Hume, Kant, and many others that Socrates’ existence could be a real property only if it added something to him, which it clearly does not. The obvious assumption here is that, if his existence were a real property, it would have to be like his other real properties such as his wisdom; and since they do add something to him, his existence should either do the same or forfeit all claim to being a real property. Underlying this suggestion is the further assumption that the relation between Socrates and his existence must duplicate that between him and his wisdom not merely in some respects but in all respects.

Now, a condition of wisdom and existence being properties of Socrates is that they each be individuated by him. That is to say, just as the wisdom-of Socrates must differ at least numerically from the wisdom-of-Plato, so too must the existence-of-Socrates differ at least numerically from the existence-of-Plato. (Although bundle theorists of all persuasions would deny this inference, their denials can be discounted, since it can be argued that each version of the theory entails the self-defeating conclusion that in some cases even individuals could themselves be instantiated in other individuals.) It is one thing, however, to claim (correctly) that existence and wisdom have both to be individuated by Socrates, but quite another to say that both individuations must have the same ground. In the case of his wisdom and many other properties, he individuates them in being their recipient. Yet, although he clearly could not be the recipient of his existence, that need not preclude there being some other way in which he does individuate it. If there were, his existence would be no less entitled to be called a property than is his wisdom.

Is there such a way? It has been argued that there is, for Socrates would individuate his existence if he were not its recipient but its bound. (Cf. B.Miller, The Fullness of Being, chapter 4) The notion of a bound is more than a merely a spatial one, for there are also bounds of thought, bounds of desire, artistic bounds, and so on. Socrates would be the bound of his existence in respect of all human areas. In being bounded by him, his existence would be individuated and distinguished from Plato’s no less than, in being received by Socrates, his wisdom is individuated and distinguished from Plato’s. Moreover, since no bound can be real unless what it bounds is also real, the fact that Socrates is real would entail that what he bounds (his existence) must be real as well. The salient point, however, is that it makes no sense to speak of a thing that is bounded being added to by its bound. Thus, even though it makes no sense to say that Socrates’ existence adds anything to him, that would not detract from its being not only a property, but a real one to boot.

http://plato.stanford.edu/entries/existence/

[hauss]

Libertarian: Ok, since this idea is hundreds of years old, I’ll lend a little more respect. Here’s a simple question for starters: What if the being that you say exists [to be just as real as your arm] killed him/herself, or at least made themselves cease to exist in the form they once did in order to be called the necessary supreme being?

To state the obvious, your statement would have been correct one moment and incorrect the next, and you don’t know if the being has killed itself between the origin of this statement hundreds of years ago and now.

Could there possibly be a successor to THE NECESSARY SUPREME BEING???

I’m trying to keep my head out of the clouds and on the ground here…

By the way, how is saying that [information cannot be created from no information] circular deductive reasoning? Formation of information can only happen with other information!

[Liberal]

@Ultra

Proof of <>A -> A in S5:
<>A & ~A (assumption)
~~A & ~A (modal quantifier expansion)
~A (&-elimination, 1)
~A (modus ponens, 3, axiom of S5)
~~A (&-elimination, 1)
~A & ~~A (&-introduction, 4, 5)

Ultra, I’m not following this. For one thing, how does modus ponens formulate ~A? In other words, what has implied ~A such that, it being true, ~A is true? And how does ~A & <>A imply <>A -> A (which was to be proved)?

[Liberal]

@hauss

What if the being that you say exists [to be just as real as your arm] killed him/herself, or at least made themselves cease to exist in the form they once did in order to be called the necessary supreme being?

Necessary existence cannot not exist. That means that if the being commits suicide, all of existence goes with it.

Could there possibly be a successor to THE NECESSARY SUPREME BEING???

No, because if there were, then it would not be the necessary supreme being.

By the way, how is saying that [information cannot be created from no information] circular deductive reasoning? Formation of information can only happen with other information!

Well there you are. You just said it yourself!

[hauss]

About the Reductio Ad Absurdum… this is the 3rd contention:

“If something exists only in the mind (i.e. does not actually exist) but is possible (in the sense that was defined in 2), then that something might have been greater than it is. For example, a majestic mountain that exists only in the mind (i.e. a nonexistent, imaginary mountain) might have been greater: the mountain existing in reality.”

Question: Why is it that if I imagine in my mind a perfect mountain, Anselm says that there could be a more perfect mountain in reality? Why can’t I imagine the ultimate? Is my mind limited compared to reality or something? What would compel someone to assume that the moutain in reality “might have been greater” than my imaginative one?

[hauss]

My other question: Could you please explain why, if God does exist, that it is necessary?

[ultrafilter]

Every once in a while, you really hate the inner-quote stripping feature.

@Libertarian

Ultra, I’m not following this. For one thing, how does modus ponens formulate ~A? In other words, what has implied ~A such that, it being true, ~A is true? And how does ~A & <>A imply <>A -> A (which was to be proved)?

Here’s the proof again, mostly for my ease in writing this:

<>A & ~A (assumption)
~~A & ~A (modal quantifier expansion)
~A (&-elimination, 1)
~A (modus ponens, 3, axiom of S5)
~~A (&-elimination, 1)
~A & ~~A (&-introduction, 4, 5)

The axiom is your necessitation rule. Given ~A, we can use B -> B to conclude ~A. I trust this is not controversial.

I think this is not clear because I didn’t explicitly state that it’s a proof by contradiction. The premise is the negation of the statement <>A -> A. Does that make it a bit easier to follow?

I’ve been looking over S5 (using the description here) and I think that I have an alternate presentation that I prefer. I’ll work on it and see what I can get.