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

Great Debates
161

:frowning:

Alright, I need to back up a bit and focus.
[list=1][li]First off, Lib, you’ve been quite a trooper here, and flattery is not my strong suit. I either gush or say nothing, so this very paragraph is as good as it is going to get. :)[/li][li]Secondly, you can’t have it both ways. Either the definition of God is contingent upon how it would be defined in all possible worlds or it is not.[/li]***If it is not, then we literally have no idea what Tishammer’s proof demonstrates because G, for all our mumblings about it, is underfined in any particular world, including our own.
**If it is, then in any possible world where Tishammer’s proof can exist, God can be proven to exist. Here is where I toss my “logic-only” world and show that Tishammer’s G maps explicitly with the perfect logic system, and Tishammer’s proof shows that the perfect logic system must exist. Neither the proof, nor our conception of it, forbid this operation. (afaict)
[li]Thirdly, the idea of God’s perfection has subtly shifted somewhere. Is it or is it not the “most” whatever of any property which can have a most?[sup][/sup] [/li]***If it is not then there is nothing in the proof that compels us to use the word “being” to describe god. In fact, we may replace “being” with “idea” or “concept” and the proof still holds. It is possible that we haven’t described anything whatsoever
***If it is we return to point 2b.[/list=1]

Being an erislover and an atheist all in one I am neither troubled by accepting nor by rejecting this proof. I have no internal motivation to believe nor disbelieve it. That is, the fact that we are attempting to utilize “God” doesn’t bother me. Just wanted to toss this out. :wink:

*[sub]and, btw, your rejection of “nothing less than no contradictions” not being in the definition is false; “most” and “least” are a matter of our convention; there is no reason to say that we couldn’t consider lack of contradiction to be a thing, in which a perfect logic system has the most “lack of contradiction”… hey, you wanted an a priori proof, there’s no a priori reason to consider “most” along any synthetic a posteriori scale (unless you’re a fan of Kant, which I doubt), especially since we are discussing all possible worlds, including those in which “lack of contradiction” is the essence in question instead of “contradiction”.[/sub]

162

I believe because it is absurd!

163

Eris

I appreciate your declaration of disinterest. I agree with Newton that few if any people who do not believe in God are going to examine this proof, slap their foreheads, and go, “Well, then! I think that I shall now begin an epistemological inquiry into this being.” Most atheists, I think, will shrug it off from nothing more than prejudice, just as most theists will likely buy into it without even knowing what it is.

In the middle, there are the rare individuals, like you and Newton, who put aside their personal prejudices long enough to listen, examine, and discuss rationally what the argument means and what it implies. I harbor no illusions that you will see things my way. I am merely thankful that you allow me the courtesy that I am no fool.

The definition of God is necessarily not contingent. If it were, it would not be defining God, because God is not analytic. But I think you’ve slightly misstated your (2b). Tisthammer’s God is provable for (but not necessarily in) all possible worlds. There might, as you’ve pointed out before, exist a world of contradictions where God, though existing, might be impossible to conceptualize.

For me, the most important ontological implication of Tisthammer’s proof is that of an “objective existence”. It certainly could be the filters that I’m looking through because that is how I’ve seen God all along, as the Objective Reality (and the reason why I proudly adopt the label of Objectivist, despite Ayn Rand’s pathetic mutilation of her own philosophy).

At any rate, I think this is pretty much wrapping up unless some new ideas are tossed out. I stand where I stood at the beginning, in awe of this young man and his grasp of sound principles across multiple philosophical disciplines. I wish he would become a scientist (according to the definition of science that you offered in science thread). We need more scientists who understand the nature of knowledge and existence, scientists who understand that their own discipline is rooted in philosophy.

164

Has anyone emailed the author of the proof you linked to in your OP a link to this thread, by the way?

165

Let me say right off the bat that I have very little experience in formal logic so please be gentle. :slight_smile:

Due to historic (and probably technical) reasons, logic formalisms normally define possible to be the case that something is possible in at least one world. Thus, possible is inclusive of any subset of worlds greater than zero as well as all worlds. This makes sense because usually things may exist in some but not all worlds or they could exist in all worlds. However, there is no reason that we couldn’t use a form of logic that treats possibility in SOME worlds as distinct from possibility in ALL worlds. However, for the being that is defined as God it makes no sense for the being to exist in SOME but not all worlds. Can the proof under discussion be modified to treat possibility of existence in some worlds as distinct from possibility in all worlds? Would the proof still be valid?

If nothing else, I think that the proof might make more sense to me in this modified form.

At the risk of making a fool of myself, I will give an example of what I am thinking. Feel free to point out any glaring errors of logic but also please address the general gist of the alternate proof.

I’ll take Lib’s English explanation of Tishammer’s proof and highlight my attempt to change it to use separate statements about SOME possible worlds and ALL possible worlds.

Definition: God is a being of maximal perfection in every possible world.

Given:

  1. If He does exist, then He exists necessarily.

  2. God’s existence is possible.
    ===============================
    2A. God’s existence in ALL worlds is possible
    2B. God’s existence in SOME worlds is impossible
    2C. God’s existence in NO worlds is possible
    ===============================

Derived:

  1. If God exists necessarily, then He exists.

  2. Suppose for the sake of argument that if God exists necessarily, then it is possible that He doesn’t exist. (This is the Law of the Excluded Middle.)

  3. If it is possible that God does not exist, then it is necessarily true that He cannot exist in every possible world, since He is a being of maximal perfection. (This is the application of Becker’s Postulate.)
    ===============================
    **If it is possible that God does not exist in ANY possible worlds, then it is necessarily true that He cannot exist in SOME possible worlds, since He is a being of maximal perfection. **
    ===============================

  4. Therefore, if God exists necessarily, then it is necessarily true that He cannot exist in every possible world. (This is the Transitive Law.)
    ===============================
    Therefore, if God exists necessarily, then it is necessarily true that He cannot exist in SOME possible worlds.
    ===============================

  5. If God cannot exist in every possible world, then God necessarily does not exist. (This is an inference from modus tollens.)
    ===============================
    If God cannot exist in SOME possible worlds, then God necessarily exists in NO worlds or He necessarily exists in ALL worlds. QED
    ===============================

This at least is consistent with most people’s concept of God and the possibility of His existence.

166

Hmm, no one responded about my re-wording of the first assumption. I’ll proceed as if it is correct.

We now have

  1. If God exists in any possible world, then God exists in all possible worlds.

  2. God exists in some possible world (i.e. God is possible).

Then it follows as a syllogism that

  1. God exists necessarily.

This makes the logical argument much easier to follow. It also makes the assumptions clearer - specifically, assumption 1 is tantamount to assuming the conclusion.

Or so it seems to me.

167

rsa, I’m not sure you’re using “possible” correctly. It can’t be defined as you do:

because you use the word in its definition. Rather, something is possible (in this world) if it is true in some world.

I have to admit I’m not clear on what these “worlds” must satisfy. Apparently they don’t include all imaginable worlds. In “normal” modal logic, they must be at least logically consistent. (Or so I gather from the discussion above.)

168

You’re quite right of course FriendRob. Poorly worded on my part.

169

Something for almost everybody (sorry TruthSeeker and erislover, not enough time or philosophical expertise to either agree or disagree with your positions yet).

ethnicallynot:

I think your criticism is well placed. Let me point out that it is possible (though not fruitful since it raises bigger difficulties) to dodge two of your points:

This is only valid in a logic with a Euclidean accessibility relation (Euclidean is both symmetric and transitive).

And this is only valid in a logic with a reflexive accessibility relation. However, if one denies that accessibility is reflexive, symmetric, or transitive, then they are not able to prove much of anything.

Libertarian:

I suggested rejecting one of the three required assumptions (the two axioms and the symmetry of accessibility). You ask (for all three): on what basis? Well, on the basis of refusing to accept the conclusion, of course :). A theist might cynically see it as the desire to defend the actual world from God. An atheist would probably see it as taking care to not assume that God exists.

I have been intimating that the two axioms say something about the accessibility relation, but this is not really true. To be precise, the intended accessibility relation affects how palatable we will find the two axioms.

Imagine that the first axiom declares a kind of “kudzu” God. His existence in any world spreads out to include existence in every world he can reach from there. As I said before, I’m inclined to accept this axiom. And I would further argue for acceptance of either of the other two assumptions, but not both.

If I accepted the kudzu God and allowed accessibility between worlds to be symmetric (to work "both ways), then the only way to still insist that there might be some non-God worlds somewhere is to keep (at least some of) the non-God worlds completely unreachable from the God worlds (and vice versa). Then, it would be reckless to assume <>G is true of this world since that would place us squarely in the God camp.

If I accepted the kudzu God and also allowed God to be possible here in this world, then the only way to keep this world even possibly free of God is to insist that accessibility is not required to be symmetric (there are at least some “one way” accessibility relations) to keep the kudzu out.

I’m not really actively advocating either of these, just trying to show exactly where and how the conclusion “gets into” the system. Hopefully I have illustrated what a non-theist might be committed to rejecting should they reject the conclusion, or which fundamental beliefs they might be required question should they accept the argument.

FriendRob:

Material implication is wierd. It’s the two argument binary valued function defined by (A->B) == ~A OR B. As far as I can tell, that function is used only because of the 16 two argument binary valued functions, it’s the closest to what we mean by if…then… statements. It has it’s problems when used to model every single conditional in our discourse. It’s not entirely intuitive that something that is false should imply everything, or something that is true should be implied by anything. Even after resolving that, there are some stronger paradoxes. The following are all valid material conditionals.

|- (p -> q) V (q -> r)
If I win the lottery then spinach tastes wierd, or if spinach tastes wierd then Al Gore is the next president.

(p & q) -> r |- (p -> r) V (q -> r)
If the Red Wings win Saturday and Monday then they will go the the finals. Therefore if the Red Wings win Saturday then they will go to the finals, or if the Red Wings win Monday then they will go to the finals.

~(p -> q) |- q -> p
It is not the case that if she is Miss America then she looks horrible in a bathing suit. Therefore, if she looks horrible in a bathing suit then she is Miss America.

p -> q |- (r -> q) V (p -> s)
If I lives in Los Angeles then I am a Californian. Therefore, if I earn my Ph.D. in philosophy I am a Californian, or if I live in Los Angeles then I ride my bicycle to work.

(p -> q) V (r -> s) |- (p -> s) V (r -> q)
If you’re in London then you’re in England, or if you’re in Paris then you’re in France. Therefore, if you’re in London you’re in France or if you’re in Paris you’re in England.

C. I. Lewis (not C. S. Lewis) was mostly responsible for reviving ideas of modal logic and possible worlds in the early 20th century by trying to find alternatives to material implication. He arrived at the idea of strict implication (which contains its own paradoxes): an implication is true not just if it happens that the consequent is true or the antecedent is false, but if it is required that the consequent is true or the antecedent is false. Seen in a modal framework, that means that the implication is not contingent, but necessary.

means that in every world like this one (that is, accessible to this one), Gods existence (in that world) implies his necessary existence (his existence in every world like that one).

Very, very close. “If god exists in any possible world like this one, then god exists necessarily in that world (i.e. in every possible world like that one)”

Later, you said:

Yes, you’ve got it, nearly exactly. The first assumption by itself is not enough, we also must also dot the i’s (assume that God exists in at least one world like this one), and cross the t’s (assume that if a world is like this one then this one is like that one).

Pretty much correct, again. We often use different notions of necessity. If I’m talking about the analytic necessity of “2+3=5”, then I do mean every imaginable world. If I’m talking about the impossibility of not traveling faster than 186,000 miles/second, then I don’t mean every conceivable world, but every one that obeys the same physical laws as this one. If I were talking about the moral necessity of not killing another human, I would mean every world where I do the moral thing. If I were talking about temporal necessity, then I would mean every possible future state of affairs that could hold given the current actual world.

rsa:

You speak of the distinction between possibility in some world and possibility in all worlds. If I understand you correctly, you are drawing a distinction between something that is possible: <>P, and something that is necessarily possible <>P. The two are equivalent at a world only if the accessibility on worlds is Euclidean (symmetric and transitive). A Euclidean relation is one where you can complete the triangle: given 0R1 and 0R2, you can conclude 1R2 (and 2R1).

Your proof sketch seems OK. If you think of your argument in terms of possible worlds, you are partitioning the set of possible worlds into God worlds and non-God worlds, and requiring that they cannot possibly pollute each other (with their God or absence thereof). Since you cannot tell which set the actual world lies in, you would need an extra assumption (such as Tisthammer’s <>G) to put yourself firmly in one camp or the other.

kg m²/s²

170

I’m not sure what comment to make, Rsa. (Not that you solicited one.) But your axiom-set 2A-2C seems contradictory, unless they are disjunctive (“or” rather than “and”). How can it be possible that God can both exist (2A) and not exist (2C)? Yet, if they are disjunctive, they are tautological.

171

Newton

I don’t think Rsa can construct his argument as is. It must be decided whether God’s existence is possible or not. It can’t be both.

With respect to your once again incredibly honest and candid description of how you make your assumptions, I wouldn’t even presume to talk you out of thinking the way you think. But I would like to avail myself of the same opportunity to explain honestly why I think the way I think.

When I was an atheist, my biggest problem with God was that He wasn’t necessary. And necessity, of course, is a key component of William of Ockham’s famous Razor, a component seldom even intimated, I might add, in most pop-culture paraphrases of it. The one that irks me most is “All things being equal, the simplest solution is the best”. Ackkk! What the hell? Ockham wrote, “Entia non sunt multiplicanda praeter necessitatum.” Beyond necessity. He rightly in no way advocates that we discard necessary entities just because doing so might simplify matters.

Thus, I suppose, it is normal that when I experienced God’s presence for myself (as it so happens while on a quest to prove the incompetence of biblical scholars) He became suddenly necessary in a very personal way. He is more real to me than the hand in front of my face.

Therefore, I can see that I approach the argument from a foundation of prejudice, just as you do. And you’re right that controversy over the proof is more appropriately likely at its beginnings, rather than in its bowels. The argument helps show what our prejudices are.

Frankly, I see Tisthammer’s argument more as uncontestable proof that a rational man can believe in God than as uncontestable proof that God exists, proof that can only be acquired subjectively because of our nature and the kind of beings we are. We know nothing that our own brains haven’t processed through the filters of our own narrow experiences.

172

Newton meter, thanks for your comments. If I understand the terms correctly, I was not trying to draw a distinction between <>p and <>p. Rather I was trying to split apart <>p itself.

For example:

><p (possible in some, but not all worlds)
( )p (possible in all worlds, but not just in some worlds)

Worlds should be taken to mean “worlds like this one”.

So for most things, we would have for example:

><toaster
( )toaster

if we assume that it is possible that a toaster may exist in some, but not all worlds and that it is also possible that a toaster may exist in all worlds.

In the case of God it seems that only ( )God is valid and ><p is not valid based on the definition of God (maximal perfection).

I was wondering if there would be something to be gained by analyzing a God proof with my two types of possibility rather than the conventional <>p which combines my ><p and ( )p.

Lib, you’re right in that I should not have included 2C. It doesn’t belong there. Does it make more sense now? And I welcome any comments you may have. My point is not to construct a particular argument but rather to analyze the proof that God must exist since even many theists would not claim that the proof actually “proves” what it purports to.

Personally, I believe that a rational man can believe in God and that the proof is unnecessary.

173

Why, thank you, Rsa! Your respect is much appreciated! :slight_smile:

Sorry to butt in on your question to Newton, but I’d like to say one thing about it. You can’t split <>P because it’s an atomic unit that says simply “it’s possible that P”. It’s not a compound proposition.

I’m afraid that if you drop 2C, then you have a problem with your 7th inference. You haven’t shown that it is possible (and you’ve discarded stating it axiomatically) that God exists in no possible worlds. You have to drop that statement from the disjunction, leaving you with “If God cannot exist in SOME possible worlds, He necessarily exists in ALL worlds.”

But… :wink:

That negates your axiom 2B, unless you qualify it to say that God’s existence in some worlds implies His existence in all of them. And if that is the case, then your conclusion is a circulus in demonstrandum (circular argument, drawing the same conclusion as the axiom it posits).

174

Now this gets even stranger the more I read into ethic’s post. I’ve never considered meaning to be a part of logic per se (that is, necessarily), but rather a game with specific rules for manipulating symbols.

What propositions we may map to these symbols are not themselves contained in the logic. There is no symbol set which describes the meaning of itself completely. There are symbols and rules for manipulating them, and there are our interpretations of the symbols, but we are not obligated to say that the symbols actually represent the meaning we put in them.

That is, we are not compelled to do so due to the symbol-game itself. That we do it is not a function of logic. P->P doesn’t mean anything. Neither does G for all I know. When I approach strict symbol manipulation, I think of it as strict symbol manipulation.

Then I ask: is it worth our while to assign meaning to these symbols? Well, yes, I think it is.

Gah, I’m sorry, but this is too big of a hijack for me to ramble on about here. Let me just say that I think that assigning meaning to this proof is not obviously required by modal logic; in fact, since it is analytic by nature, it should be possible to assign many meanings to the proof, some of which would be synthetically true and some of which would be synthetically false. Nothing in modal logic forbids us from saying: “Cheese is the average rate of butt water” and calling that G. Assigning that meaning to G just makes the proof become nonsense, even though the same symbol-manipulation rules were followed the whole time. Assigning a meaning to G which doesn’t allow the proof to make sense doesn’t invalidate the proof, and neither does assigning a meaning to G which gives the proof a sense mean that the proof proves the meaning we’ve assigned to G.

Of course, the above is not able to be represented in logic very well, because I’ve just said that



I.     **P**->**P[sub]1[/sub]** (the proposition P *means*P[sub]1[/sub])
II.    **Q**->**Q[sub]1[/sub]** (the proposition Q *means* Q[sub]1[/sub])
III.   **P**->**Q**
IV.    **P**
IV.    ~(**P[sub]1[/sub]**&**Q[sub]1[/sub]**)

Which should be impossible, but such is logic. Meaning is not contained in the system, and we must be very cautious when trying to assign meaning into it. Not that it can’t be done, but that since the system itself has no way to contain meaning, when we do so we cannot expect synthetic statements to be analytically true even if they seem to be.

Probably why philosophers so often seperate analytic statements as meaningless but certain (necessarily true) and synthetic statements as meaningful but uncertain (possibly true).

175

Wow, Eris, good stuff about the mapping!

The rate of consensus on the axiom that “Cheese is the average rate of butt water” when divided into any arbitrary number might likely be undefined. :wink: I don’t follow your coded list, though (and I don’t mean the misnumbering). Is it a list of axioms? Because I don’t understand how any of the inferences follow from the others.

176

I’ve got to run, but I humbly submit that I did split <>p. It’s right there in black and white. :slight_smile: It may not accomplish anything and what I’m getting at could probably be stated in conventional notation but I don’t know how. Perhaps you could explain how to do that.

As far as not showing that it’s possible that “God exists in no possible worlds”, isn’t that implicit in <>p? Saying that something is possible doesn’t make it necessary. As long as I don’t say p, then the possibility that God exists does not eliminate the possibility that God exists in no possible worlds.

Doesn’t <>p simply mean that it is possible that p may exist in 0, 1, or more worlds?

177

–God is defined in Tisthammer’s proof as the “greatest possible being”.—

Ah, so now we aren’t using the concept of “perfect” any longer? That, at least, is more sensible. But that hardly resolves the issue: greatest in WHICH respects? Again “greatest” is not a meaningful appelation by itself unless it is a subjective ranking.

You still haven’t tackled the problem that, even accepting the idea that you can tease a proof of existence out of a definition, your greatest possible being seems to lack any positive characteristics other than existence, which is paradoxical. Existence cannot be the only known characteristic of a being. Existence of WHAT being? What are you saying exists? A bunch of lacks?

As long as you stick to your “risk nothing” negative definitions, this proof cannot establish anything about the existence or non-existence of any particular being.

—Even an armchair logician can see the absurd contradiction in any conclusion that states that which is possible is not possible. —

And only someone evading the issue can massage that out of what has been said.

178

The point is, that existence itself does not follow from the definition. What follows from the definition is that IF god exists, he exists necessarily because of symmetry. Because necessary existence is the greatest form of existence. Therefore you could argue ANY kind of existence for god, and they would all be equally valid as long as they were necessary. For example:

Definition: God is a being of maximal perfection in every possible world.

Given:

  1. If he is nonexistent, then he is nonexistent necessarily.

  2. God’s nonexistence is possible.

Now you can use the proof you have provided to prove god’s nonexistence.

Derived:
3. If god is nonexistent necessarily, then he is nonexistent.
4. Suppose that if god is nonexistent necessarily, then it is possible that he exists.
5. If it is possible that god exists, then it is necessarily true that he cannot be nonexistent in every possible world, since he is a being of maximal perfection.
6. Therefore, if god is nonexistent necessarily, then it is necessarily true that he cannot be nonexistent in every possible world.
7. If god cannot be nonexistent in every possible world, then god necessarily exists.
8. Therefore, if god is nonexistent necessarily, then god necessarily exists.
9. If it must be true that god is nonexistent necessarily, then it cannot be true that god necessarily exists; therefore god necessarily is nonexistent.

Therefore, since god is nonexistent necessarily, god is nonexistent.

179

The point is, that existence itself does not follow from the definition. What follows from the definition is that IF god exists, he exists necessarily because of symmetry. Because necessary existence is the greatest form of existence. Therefore you could argue ANY kind of existence for god, and they would all be equally valid as long as they were necessary. For example:

Definition: God is a being of maximal perfection in every possible world.

Given:

  1. If he is nonexistent, then he is nonexistent necessarily.

  2. God’s nonexistence is possible.

Now you can use the proof you have provided to prove god’s nonexistence.

Derived:
3. If god is nonexistent necessarily, then he is nonexistent.
4. Suppose that if god is nonexistent necessarily, then it is possible that he exists.
5. If it is possible that god exists, then it is necessarily true that he cannot be nonexistent in every possible world, since he is a being of maximal perfection.
6. Therefore, if god is nonexistent necessarily, then it is necessarily true that he cannot be nonexistent in every possible world.
7. If god cannot be nonexistent in every possible world, then god necessarily exists.
8. Therefore, if god is nonexistent necessarily, then god necessarily exists.
9. If it must be true that god is nonexistent necessarily, then it cannot be true that god necessarily exists; therefore god necessarily is nonexistent.

Therefore, since god is nonexistent necessarily, god is nonexistent.

180

Rsa

No, sorry. To be possible, a statement must be true in at least 1 world.

Apos

Yes, we are. The greatest possible being is that which is perfect in every possible world.

Well, we aren’t using it by itself. It’s used with other words. (Look for the squiggly marks.)

Normally, unless we’re all drunk, existence is the topic when speak of ontological matters.

A being Whose existence is the maximum perfection in all possible worlds.

It proves that a being exists Who is greater than any other.

You’ve caught me. Looking back, I see that I have shunned the entire debate. I was hoping by page 4 to morph the thread surreptitiously into meaningless prattle over piddly shit. Thanks for helping out.