Yes, quite a few Dopers are physicists, many with advanced degrees. I might recommend that you post your topic as a new thread since physics-types aren’t all that likely to visit this one. At least not more than once.
Hrumph!! I shall not update my book in the “light” of such misguided tomfoolery.
A logical, rational understanding of God is as a bird with one wing: no matter how precisely your terms are defined and how logically you form your argument, it still cannot fly for it needs the other wing of faith!
Done…thanks.
David.
—Then I apologize. What are you?—
I have no idea what you might call me: I am concerned about existence. It is possible that god exists: to say that god is spiritual conveys, to my understanding, no extra information about what god is. Likewise, to say that god is material. Indeed, to classify any being as material/spiritual seems empty, not least of all because it is not clear what the basis of the distinction is, other than one is the negation of the other.
—I think Jesus spoke of the Spirit quite positively: the whole being born again thing in His conversation with Nicodemus. He linked it to truth: “God is spirit: we must worship Him in spirit and in truth.”—
This doesn’t tell me what spirit is, distinct from other things. To use it in a gramatically correct sentance does not always give insight into a word’s meaning. The very problem I am describing is that “is spirit” conveys nothing intelligible about God, unless it is one of the metaphorical senses (i.e. God is an emotion/conceptual).
—There is no sense in which materialism is more valid than spiritualism, given that the very discernment of material is made by — other material.—
I don’t think either are valid characteristics by which things can be discerned. Clearly, it’s quite possible that we may have limited capabilities by which we can discern certain things that exist and not others: but this is no cause to label those we can currently discern as “material” and through negation all things that we cannot “immaterial.” (or, if that is all that is meant, surely those are not the right words for it)
—And belief? Isn’t that a non-concept, too?—
No.
I think “strong” free will is a non-concept, not because it is abstract or “immaterial” but for the reason that it negates any nature of a given will… leaving what? How do particular choices get made instead of others? The concept seems self-anihilating: any attempt to explain or describe how any particular choice is made positively, even in theory, negates the freeness component of the will. How, even in theory, can free will be conceptualized as a noun, verb, adjective, or anything else?
Lib: can you point me to some articles or textbooks on Bouer’s Theorem? I can’t find any web resources on it so far, and I’d like to be able to cite it.
Apos
It’s Brouer (two Rs). 
Mabye that will help.
It’s nothing vague or mysterious. The plain dictionary definition (incorporeal consciousness) is fine.
It matters whether a frame of reference is absolute or dependent. A will that is contingent on some other will is not free; but a will that trumps all others, even God’s own will, is definitively free. Unless you can show that you are aware of my consciousness in the same way that you are aware of your own, then yours is absolute since will is the exercise of what a consciousness seeks.
—It’s Brouer (two Rs). Mabye that will help.—
Hey: I just copied and pasted the name from your original mention of it. 
That pulls up several hits, but the only relevant ones are all just restatements of the modal ontological proof itself, and don’t go into specifics about the theorem. Are you aware of any of the books/articles that discuss it in greater depth? It certainly seems common sense to me, but it would be nice to have a more formal explication/discussion handy.
—The plain dictionary definition (incorporeal consciousness) is fine.—
Sadly, that definition is just more of the same: “incorporeal” is simply a permutation of the same problem. Sure sure: it’s not “corporeal”… but what is it? And HOW does it differ from something that is “corporeal?”
—A will that is contingent on some other will is not free; but a will that trumps all others, even God’s own will, is definitively free.—
The problem is that the characteristics ascribed to the strong sense of “free will” are not simply that it not be determined by another will: but that it not be determined. And that includes it’s own nature.
The question it faces is: why would one beings make one choice, and another a different choice, in the same situation? It seems as if any sort of answer negates the idea of free will, because it would necessarily involve some account of how the wills are different to begin with, and that this difference determines the difference in choices.
—Unless you can show that you are aware of my consciousness in the same way that you are aware of your own, then yours is absolute since will is the exercise of what a consciousness seeks.—
I fail to see what relevance that has to demonstrating the conceptual workability of “free will.”
Dammit. Looking back, I see that I’ve misspelled his name three different ways. It’s all those vowels together, I suppose. This time, I’m copying and pasting: Brouwer. You should be able to find the Brouwerian systems in practically any text dealing with the fundamental axioms of S5 modal logic, since his Theorem 33, <>P -> P, when added to S4 forms S5. Search for variants, like Brouwer Axiom or even Brouwerische (in case the German was transliterated). I got a bazillion hits.
L. E. J. Bouwer founded the Intuitionist school of logic (or math), based primarily on the notion that P -> ~~P is valid, but ~~P -> P is not. He thought that the use of the Excluded Middle was suspect at worst and unnecessary at best. (Substitute necessity for negation, and you can see clearly why.) There is some controversy over whether the theorem might have only a tenuous connection to Brouer, but historically it has been designated as such.
It differs from corporeal by not having a physical body. I’m not sure what your problem is, so I don’t know how to help you. You seem to be saying on the one hand that material and spiritual are both meaningless, and yet, on the other, you seem to accept only material things as a priori analytic. Maybe the problem is that you define consciousness as dependent on a brain. Consciousness can also be defined in terms of identity.
I think it’s self-evident that no two situations are ever the same. There is exclusivity of identity as well as exclusivity of temporal space. Either the time or the place or the person or the mis-en-scene will always be different. I guess I should say, for whatever it might be worth, that freewill in the context that I use the term here is applicable only to moral decisions.
Well, it has to do with the idea that neither of us can know the other’s moral circumstance in the way that we know our own. God has given you a moral reference frame like His own: absolute, known only to you, and inviolable. As Ian Percy said, “We judge others by their behavior. We judge ourselves by our intentions.”
Apos, if you continue to try to hunt down a reference for Brouwer’s theorem, could you post the proofs you find of it here? Or just the reference, and I’ll post the proof, if I find something to say about it. My guess is it will involve a “stacking” of modal operators which yields nonsensical statments given the possible worlds interpretation we are using for them. At least the only proof I have found does, as I discussed in previous posts.
Possibly of interest to anyone who appreciated my arguments in this or the other thread.
—It differs from corporeal by not having a physical body.—
What DOES it have? And HOW is it different from “physicality?”
—I’m not sure what your problem is, so I don’t know how to help you.—
My problem is this: when you define something purely by negation, it becomes consistent with just about anything.
—You seem to be saying on the one hand that material and spiritual are both meaningless, and yet, on the other, you seem to accept only material things as a priori analytic.—
Tsk: who is calling them material? You are. They are things. I don’t doubt that the things we commonly think of as existing may be a very biased sample of reality, biased by our limitations of looking.
—Maybe the problem is that you define consciousness as dependent on a brain.—
I don’t think I necessarily define it that way. But the brain is at least a known thing that OFFERS a means of explanation. One can’t really say the same about a “not-brain” or a “not-material” thing. So what? What is it? How does it make consciousness go?
—Consciousness can also be defined in terms of identity.—
But where/how is the association of identity taking place? Back to the begginging we go…
—I think it’s self-evident that no two situations are ever the same. There is exclusivity of identity as well as exclusivity of temporal space. Either the time or the place or the person or the mis-en-scene will always be different.—
That the person is different is the whole point! If you are going to explain the difference in terms of the person, then the idea of free will is anihilated. If you are going to explain it in terms of the situaiton(temporally or spatially), then likewise. It you re going to explain it at all, then you seem have a problem. “Free Will” only seems to work when one ignores the question of what is being willed and why.
But you don’t have to grant the reality of two different choices given the same situation: only the in theory question.
And if you deny the question even in theory, then what reason do you have to claim that anything special is going on at all? All we have are a bunch of different choices in different situations.
—I guess I should say, for whatever it might be worth, that freewill in the context that I use the term here is applicable only to moral decisions.—
Why?
—Well, it has to do with the idea that neither of us can know the other’s moral circumstance in the way that we know our own. God has given you a moral reference frame like His own: absolute, known only to you, and inviolable.—
This still doesn’t address the conceptual problem with “free will.” The problem isn’t related to not knowing each other’s situation: it is related to the question of why/how one makes one choice as opposed to another.
Still searching for something by Brouwer stating that (p->p)->(<>p->p)
I have found A-><>A: is that a start?
I have also encountered people who claim that the arguement begs the question because one can beg the question even if the question beggining is split up among the premises.
Their case is that in modal logic, <>A->A, and that G=G and <>G is exactly that with just one susbstitution. My response is that <>G can’t be split into <> and G to do a substitution of that sort, and anyway even if they were right, doesn’t accepting the premises, however unconvincing it might make that particular proof, still mean that one already accepts G?
But I’m not quite sure how to address it in the most convincing way. Exactly why is this criticism wrong? I’m not sure if <>A->A is valid, though that seems to be a contention here, and it certainly seems to follow (prior to my criticism of it).
Apos writes:
"Still searching for something by Brouwer stating that
(p->p)->(<>p->p)
I have found A-><>A: is that a start?"
Well, if that is the start, we’ve already got a nonsensical statement. What does <>A say in our possible worlds interpretation? It says “IN every possible world, it is true that ‘there is a possible world in which A is true’” But in our world, and other possibilities relavent here, we will never come across a possible world. There are no possible worlds living IN our world. Thus this “stacking” of modal operators leads to nonsense.
Apos writes:
"I have also encountered people who claim that the arguement begs the question because one can beg the question even if the question beggining is split up among the premises.
Their case is that in modal logic, <>A->A, and that G=G and <>G is exactly that with just one susbstitution."
(Ok, it looks like you are changing your notation, putting the operator on the right. I am assuming <>A is what you meant by <>A before. Please correct me if I am wrong.) I think discussing an argument which defines G=G is not worthwhile; I don’t think G=G is a valid definition. See Kitarak’s comments on this, which I think echo my objections. But for the sake of clarity: look, G is a sentence and G is a sentence. You are saying the sentence “God exists” is the SAME SENTENCE as “It is necessary God exists”. But they are not the SAME SENTENCE, even if G=>G and G=>G. Then G<=>G would be valid, but we might as well assume the weaker definition “G is a being with the property G=>G.” Of course you can substitute G for G and vice versa if G=G, that is if they were the same sentence, but they are not the same sentence. What is really of interest is G=>G, or G<=>G, and these don’t substitute that way.
—I am assuming <>A is what you meant by <>A before.—
A-><>A was taken straight off the page of Brouwers axiom, found here:
http://plato.stanford.edu/entries/logic-modal/#2
From http://plato.stanford.edu/entries/logic-modal/#2
“One could engage in endless argument over the correctness or incorrectness of these and other iteration principles for and <>. The controversy can be partly resolved by recognizing that the words ‘necessarily’ and ‘possibly’, have many different uses. So the acceptability of axioms for modal logic depends on which of these uses we have in mind. For this reason, there is no one modal logic, but rather a whole family of systems built around M.”
The iteration principles are about how “stacking” of modal operators works in the various systems refered to. So in other words, I don’t think S4 and S5 are valid systems with our uses of “necessarily” and “possibly”, since they allow you to “stack” modal operators.
From the same webpage
“A summary of these features of S4 and S5 follows.
S4: …= and <><>…<>=<>
S5: 00…= and 00…<>=<>, where each 0 is either or <>”
I don’t think these stackings make any sense with our use of “necessarily” and “possibly”.
Can one prove Brouwer’s Theorem in a system built around M other than S4 and S5 (which does not allow stacking of modal operators)?
Trent Dougherty, in the presentation of his argument here
http://www.abarnett.demon.co.uk/atheism/ontol.html
writes:
“There is an interesting theorem in S5 called Brouer’s Theorem… This theorem is derivable in weaker systems as well.”
He (annoyingly) does not give details on these other systems.
My guess is that these other systems would allow stacking of modal operators, and so would not be applicable with our uses of “necessarily” and “possibly”.
The only proof I have found of Brou(w)er’s theorem is given here:
http://www.angelfire.com/mn2/tisthammerw/rlgn&phil/ontological.html
and it uses stacking of modal operators (in particular ~G=>~G). But ~G (English translation: “IN every possible world, it is true that ‘G is false in one of the possible worlds’”) makes no sense. Possible worlds do not live IN individual possible worlds. It would be like saying “In our world it is true that ‘one of the unicorns flies’” . But there are no unicorns IN our world. We can imagine them, as we do possible worlds, but statements about them are neither true or false IN our world. They can be true or false in our imaginary worlds where unicorns exist. In the same way, statements about possible worlds are not true or false IN our world (or a given possible world). In our imaginary metaworld containing the possible worlds the statements can be true or false, but not IN our world (or a given possible world). Any time you stack modal operators you get such nonsense statements.
—My guess is that these other systems would allow stacking of modal operators, and so would not be applicable with our uses of “necessarily” and “possibly”.—
Um, what does it matter that these senses are not the same as the plain enlgish uses? What matters is simply that our usage be well defined and consistent.
The rest of your criticisms are exactly what I have suggested about the proof: that existence is a characteristic tied to each specific world, and no being in any particular world is thus a necessary being, because that is a definition outside the scope of the world. You can’t find a necessary being in a possible world: you can only do so by generalizing about what’s in all possible worlds.
Apos responds to:
—My guess is that these other systems would allow stacking of modal operators, and so would not be applicable with our uses of “necessarily” and “possibly”.—
with:
“Um, what does it matter that these senses are not the same as the plain enlgish uses? What matters is simply that our usage be well defined and consistent.”
I wouldn’t describe our use as “plain english”. Plain english can be used to assign these operators all sorts of differing meanings. For instance could mean “at all points in time it is true that” and <> could be assigned “at some point in time it is true that”. The specific meaning we have attributed to them, using plain english, is important. Some assignations (is that a word?) might be consistent with S4 and S5. My point is, when we assign to mean “in all possible worlds it is true that” and <> “in at least one possible world it is true that”, stacking them makes no sense. I think to APPLY modal logic to anything you have to first think if the operations you will perform with the operators will always make sense in terms of how you assign meaning to them (which happens to be in plain english).
Apos writes:
“The rest of your criticisms are exactly what I have suggested about the proof: that existence is a characteristic tied to each specific world, and no being in any particular world is thus a necessary being, because that is a definition outside the scope of the world. You can’t find a necessary being in a possible world: you can only do so by generalizing about what’s in all possible worlds.”
Yes, I think fundamentally we agree. That is what I said in my first post addressed to you. You are saying the definition is out of the scope of the world. I think you mean you object to saying IN a possible world W it is true that “some being exists in all possible worlds.” Which is a special case of my objection, since when you stack modal operators you get statements which say IN a possible world W it is true that “<some sentence involving possible worlds>”. Thus you say the definition is out of the scope of the world (and I agree), and I am saying stacking modal operators yields statements which assert some sentence is true in a world, where that sentence is out of the scope of the world.
Fascinating thread. I have read every word, but I cannot promise that they have all remained with me. Since I am late to the party, I will do my best to not simply rehash what others have said. My apologies in advance if this feat is beyond me.
Lib
I find the proof (in several variations) offered to be unsound. I disagree with the axiom <>G. I also find the application of S5 to this particular formulation unsettling. Please understand, I have no objection to S5 per se–but I do think that the manner in which S5 treats necessity and possibility is not entirely congruent with the specific model which you have presented. I do think I have a perspective on each of these issues which has not yet been raised by others, but I’ll address those later. They are not new points so much as different means of approaching teh same idea.
I am more interested in looking at the Proof of Necessary Existence which you have supplied. I have a couple of questions about it. Now, I am by no means an expert on modal logic, so I am hoping that these are questions you will be able to answer within your self-imposed “quick posts only” restriction. Having just finished a self-imposed posting restriction of my own, I will of course understand if that is not the case.
[ol][li]My first question concerns the formulation itself. You have written:[/li]"[symbol]"[/symbol]x[][symbol]$[/symbol]y(y = x) (Necessary existence exists)"
Now, I understand that this is the name does not originate with you, but the way I parse this statement is: “For all x, it is necessary that for some y (y=x) is true.” My question is:
Assuming you agree with the formulation above, how might “x” be understood if we did not pair it with “for some y=x”. It seems to me that without such a pairing the formulation [symbol]"[/symbol]x leads directly to the predication of existence. Now, I’m not saying that is necessarily (sorry, couldn’t resist) a flaw, but since you have stated that treating existence as a predicate was not a requirement of this proof of G it would seem that teh proof of existence was therefore integral to the proof. Do you agree, or do you find my formulation in error?
[li]My second question conerns the use of teh quantifier axiom in the proof. Step (2) is:[/li][symbol]"[/symbol]y(y != x) → (x != x) (the Quantifier Axiom)
Now, I admit that I have never been entirely comfortable with quantifier axioms, but setting aside my personal misgivings I note (from one of the excellent sites which you cited): *However, in a language that treats non rigid expressions as genuine terms, it turns out that neither the classical nor the free logic rules for the quantifiers are acceptable. * Now, the definition of G which you argue is clearly a non-rigid expression, but the Proof of Necessary existence requires a classical axiom of quantification (complete with free variables, even). This would seem to have very real consequences for the suitability of the proof. (i.e. the suitabilility of this semantic model for addressing this question.)
[/ol]
Now–one small quible on another matter:
This is not true. Necessary existence does not imply actual existence in K, but you are using S5 which explicitely contains M ([]A -> A). If you wish to assert this result using only K, then I believe you will have to abandon both the Proof of Necessary Existence and Brouwer’s Theorem.
That ends the orginal thoughts portion of my post. I’ll now return to looking sideways at things people have already said.
**~<>G is not a suitable axiomatic position given the definition G->G. **
The implication of this statement is clear: [sub]def[/sub]G does not allow for the possibility that no world exists which contains G. In other words–while not strictly assuming it’s conclusion the argument assumes that it’s negation cannot be true. But is that statement formally true? Looking at it, it seems to be a clear restatement of Brouer’s Theorem: A → []<>A (substitute “A necessarily existent object” for A). But, of course, this can be attacked on two grounds:
(1) A cannot be asserted at this point in the proof. We have no grounds for asserting that our definition specifies an actual object.
(2) “A necessarily existenct object” isn’t really the form of teh definition (traditionally–Lib I must join those who feel that G=G is not a meaningful sentence. G->G is the definition: if G exists then it exists necessarily. But this makes it clear that ~<>G is not a contradiciton of the definition. ~A does not contradict A->B.
In short: I see no valid reason to prefer <>G to ~<>G as an axiom based solely upon the definition and the rules of modal logic (even S5).
Also, I find the idea that one cannot contribute to a discussion of a proof unless one is willing to assert unconditional support for a particular axiomatic set to be yet another statement I will not accept axiomatically. It is often very interesting to note the differeing results yielded by a substitution of axioms. Shall I be banished from a discussion of topology because I will not take a stand and “accept” a particular parallel postulate?
S5 is the proper modal logic for examining the existence of G given the definitions Lib used on page 1 of this thread.
(1) S5 incorporates the axioms: 000…[]=[] and **000…<>=<>[/], where 0=v<>. (Or alternative axioms which yield those results.) Lib’s definition of G was G. Now, the axiom <>G thus becomes <>G becomes G. Thus, we are axiomatically asserting that G is necessary. Since S5 also contains M, this is an axiomatic assertion of G. Thus, Lib’s definitions and S5 amount ot an axiomatic assertion of G.
(2) the traditional definition for G (G->G) avoids (1), but the applicability of the axioms mentoined above to the definitions:
**Possible means truth in at least one world.
Necessary means truth in every world.
A world is a set of statements. **
yields statements without meaning.
For intstance: []<>[]G would parse: it is true in every set of statements that it is true in one set of statements that G is true in every set of statements But this can be true only if every set of statements contains statements about “every set of statements”. Is this truly something we wish to assert is true? (sorry–it was necessary for me to type that). Understand, this is not an argument that S5 is based upon meaningless axioms. It is an argument that the concepts of necessity, possibility, and world in S5 are not, fundamentally, identical to the definitions given by Lib. Lib, that’s not a slam on you, I know that the definitions you offered are common, perhaps even standard, in discussions of modal logic. I maintain, though, that the axioms of S5 do not support that simple English formulation. (I think I hear Russell and Wittgenstein whispering things into my ears. But it might just be preguntas) Speaking of which:
Erl, I’ll see you in the Wittgenstein thread tomorrow (work permitting). After my hiatus, I figured I should tackle something easy first. 
And I think pretty much most of the dissenters tend to focus their argument on some variation of this point. It is good to see we agree that the proper focus for attack is more or less on the declaration of possibility, specifically where ~<>G is not allowed (there was another thread on a very similar proof where I bantered at great length on it… dunno if a link to that is contained in this thread, though)
Always with semantics. And then who can say that logic isn’t a normative science?
Well, I think both critiques have merit. Finding fault with the axiom is the “cleaner” position. The logic is valid in S5; I don’t think that position is in dispute. However, the correspondence of the proof in S5 to the “real world” formulation is also a reasonable area of examination. If is a fact that for any G deemed to possess the quality in S5 G will “exist” if it is deemed “possible”. This is nothing more than a simple restatement of Brouer’s theorem: <>[]A -> A.
It holds for any A. The question is whether the specific qualities being attributed to G, and in particular the natural language explanations for G being offered along with the proof, map accurately into the logical space determined by the S5 axioms. To my mind, they do not. I readily admit that for any A, if a it both necessary and possible in S5 then it “exists” in every w[sub]S5[/sub]. Again, that is axiomatic. If it were not true, we would not be using S5. That does not mean that God, expressed in natural language as the “maximal possible being” exists in our world.
I do agree that attacking the axiom is the more direct approach, as well as the more traditional logical critique. I find the argument that having formed a definition one must then axiomatically assume that theobject of the definition has possible existence to be unconvincing. In this particular case, I find translating the axiom into natural language to be illumuniating:
"Let us assume axiomatically that in some possible world exists a God who exists in all possible worlds". No, I don’t think that I will.