What if I posited the existence of a being containing the set of properties {A, ~A}. To deny that entity’s existence would be most logical. I’m positing that it’s not even possible such a being exists, and it is indeed not possible. So where’s the distinction?
Surely some things can be posited and rejected without having to say that they “possibly” exist (putting aside for the moment that I and other determinists reject the notion of “possibility” as mere fiction.)
Nightime wrote:
One of those who has stated that several times is I. The most recent being just above where you wrote: “Before anyone jumps the gun, I will pre-concede for the hundredth time that this proof does not prove the existence of Christ.”
Well, that’s three questions:
My question is, exactly what do you believe that the statements that make up “possible worlds” represent?
Whatever they are said to represent. There is an abstract world of algebra. If I offer that 5x = 10, then I may prove that x = 2. But until I say whether I am talking about 5 apples or 5 oranges, the proof remains abstract. Modal logic is no different in this regard. By rules of manipulation, any arbitrary subject can be predicated with respect to its modality.
In particular, do they represent a world that exists in the same way as our own?
Not necessarily. If statements are merely “possible”, then they might not be “actual”. For example, if I represent a circle as A, and define it as having a circumference that is a ratio of pi to its diameter, then only if the universe is flat space could A exist in actuality.
Does each set really represent an actual world that exists in the same way as our own?
Not necessarily, if each set is representing merely possible things. But if they are representing necessary things, then they must exist in our world the same way as our own, but not necessarily in some other world the same way as our own, depending on the accesibility relation, R, on K.
In other words, if by the actual world we mean the physical world, we can speak deontically of a relative necessity to accomodate the physical world and other worlds. Let [symbol]w[/symbol] be the physical world, and let [symbol]w[/symbol]’ be some other possible world. Let R be a frame that is merely transitive and symmetrical (but not reflexive). Let be deontic necessity, and let A be a true statement in R. Then:
[sub]R[/sub]: A[[symbol]w[/symbol]] iff [symbol]"w[/symbol]’, if [symbol]w[/symbol]’ is accessible from [symbol]w[/symbol], then A[[symbol]w[/symbol]’].
The distinction is that you do not deny a prior the existence of {A, ~A}; rather, you first posit {A, ~A}, and then you show that, by the law of noncontradiction, {A, ~A} cannot exist. Without first positing the possibility, the contradiction can never be shown.
The modal ontological proof behaves no differently. It posits <>G (it is possible that God exists). Then, logical inferences are drawn. If at any step, something is deduced that contradicts <>G, then — and only then— may G be considered impossible.
Again, just to make sure we’re talking about the same thing, are you making a distinction between denying the possibility of existence and offering the possibility of non-existence? Because I don’t think they’re the same thing. If non-existence is possible, then existence is not necessary, right? Because if existence is necessary, then non-existence would not be possible.
I’m making a distinction between denying the possibility of existence a priori and contradicting the possibility of existence by inference. The former is a logical fallacy; the latter is logically valid.
Positing “SOMETHING’s existence is not possible” implies that SOMETHING’s existence is contingent. But if SOMETHING’s existence is contingent, then SOMETHING’s existence is possible.
I know. What gives you the indication I think A -> A holds anywhere outside of this proof?
Absolutely correct.
Again, what do we mean by “existence”, and what do we mean by “possible”?
I see people making all kinds of statements involving these terms, yet no one seems to have any clear understanding of what the terms mean. At least, no one’s offered an explanation of them.
If you understand what you mean by these terms, I challenge you to explicitly define them in your next post.
Libertarian, let me put it one other way for my own edification. Do you disagree with either of these sentences as stated, without any further qualification:
If existence is necessary, then non-existence would not be possible.
If non-existence is possible, then existence is not necessary.
The Vorlon, I am sure that my statements are correct by my definitions. My purpose in asking Lib is that if he disagrees with either statement then I know we are using different definitions for existence and possible, and therefore my input into this discussion would be irrelevant.
Their meaning is in how they are used.
I have an issue with every single one of the premises. But this is my biggest sticking point too. One of the premises of the “proof” is that G->G. I see no reason to take this as a given.
So then how could we determine if any of you are using the terms inconsistently?
That’s a fairly trivial method for ensuring that you can’t be accused of error: no matter what you say, you’re correct!
Why are any of you arguing about whether God exists when you can’t even explain what you mean by existence?
zwaldd, that’s why I want to know what your definitions are. Simply asserting that your statements are consistent with them isn’t enough.
But it is the definition. Is it a good one? Well, that’s a good question. I, for one, don’t take with it. The set-up of the proof requires the ontological argument IMO. But if we accept the ontological argument there’s no need to resort to symbology.
Not really the same thing. Not all words can be defined with the degree of accuracy you seem to be seeking. Of course this doesn’t stop us from using them.
Is “stop” referencing a sign, is it a command, is it a question? Well, how am I using it?
No, it is a word. That word can be used as a command, as a reference to a sign, as a question. In that quote it is being used as a word. That’s what it “means”.
So before I continue, am I to believe you do not use the word “exists” or “possible”?
Logic is trivial.
I have been trying to discuss the definitions.
I disagree with that statement, particularly in this case.
** But you can supply (I presume) an explanation/definition for each of those usages. All I need to do is figure out which sense is being used from context.
[quote]
**No, it is a word. That word can be used as a command, as a reference to a sign, as a question. In that quote it is being used as a word. That’s what it “means”.
I use both of those words. I have a clear understanding of what they mean. Would you like me to state their meanings explicitly?
That’s not my impression of what you think. My impression is that you think that the proof is claiming that a general A -> A holds here, and is defining God into existence. It doesn’t and isn’t. The G -> G axiom follows from the proof of necessary existence, the definition of God, and the 4-Axiom in the ways I’ve described.
Somebody wrote:
“Their (i.e., certain words in this philosophical discussion) meaning is in their use.”
Somebody responded:
Presumably, the first quote refers to the later work of Wittgenstein. If so, the response – that this erases criteria of correctness – is not valid.
Wittgenstein’s appeal to how we use words is an appeal to what every competent speaker of the language already knows; W called it “grammar,” and went on to say that “Grammar reveals essence,” and “Grammar tells us what kind of object anything is.” PI, 373.
In Wittgenstein’s view, traditional philosophy is marked by a tendency to abandon the shared criteria that allow humans to communicate in the first instance, such that philosophy speaks in a void.
When W asks, “what should we say (what would we call) …?”, he is not predicting what will be said in certain circumstances; he is asking something which can be answered by remembering what is said and meant, or by trying out his own response to an imagined situation. The problems of philosophy are not solved by “hunting out new facts; it is, rather, of the essence of our investigation that we do not seek to learn anything NEW by it. We want to understand something that is already in plain view. For this is what we seem in some sense not to understand.” PI, 89.
Wittgenstein’s later work is of course formulated in part as a critique of his own Tractatus, which attempted to find the logical structure of language and thereby the world, and in so doing overlooked the way in which humans actually speak, i.e., relate to the world. Or, as W stated in PI 129,
“The aspects of things that are most important for us are hidden because of their simplicity and familiarity. (One is unable to notice something – because it is always before one’s eyes.) The real foundations of his enquiry do not strike a man at all. Unless THAT fact has at some time struck him. --And this means: we fail to be struck by what, once seen, is most striking and most powerful.”
This thread reminds me once again of why Wittgenstein is the most important thinker of our era.
If existence is necessary, then non-existence would not be possible.
Agree.
If non-existence is possible, then existence is not necessary.
Agree.
Do you see a reason to accept A -> C here?:
**If A = B, then
If B -> C, then by substitution of A for B, A -> C.**
If not, then we have nothing to discuss. If so, then why do you not take G -> G here — i.e., how is it different from the above)?:
**If G = G, then
If G -> G, then by substitution of G for G, G -> G.**
p -> p is a fundamental axiom of 4-level and higher modal logic.
One of my favorite quotes is from Wittgenstein: “The riddle does not exist. If a question can be put at all, then it can also be answered.” — Tractatus Logico Philosophicus