[QUOTE=begbert2]
Okay. I spent much of last evening (whilst doing other things) working myself into a tizzy, trying to understand not just how modal logic works, but the purpose thereof. (If one looks at it crosswise it begins to appear that there is not a distinction between necessariness and truth…which is bad. Also not correct, but if it were correct it would unfold the entire thing into a scam.)
…
So, just because a statement is true, it’s not necessary. What then does it mean to be necessary? Based on my newfound (and possibly grossly incorrect) new understanding, to be necessary means it can’t even be imagined to be false. We can imagine a scenario, a ‘possible world’ wherin Germany didn’t lose the war; ergo the assertion that it did isn’t necessary. The same goes for any bald fact.
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I’m glad you’ve begun to acquaint yourself with modal logic, begbert2, and have realized the distinction between contingent truths (those which are true, but not necessary) and necessary truths. However, I have some small points to make about what appears to be your understanding of modal logic.
First of all, it is a common misconception to think that modal logic is just about necessity and possibility. Modal logic is brimming with all sorts of operators beyond those two. Things like “Person P believes that”, or “The theory T proves that”, or “At some point in the past, it was true that”, or “It is very probable that” are also modal operators as well. In general, a modal operator M is one that is non-truth-functional; i.e., such that the truth status of M(A) is not determined by the truth status of A alone. Knowing whether something is true or false doesn’t tell you if it’s necessary, if someone believes it, if it’s provable in some system, etc.
The particular part of modal logic dealing with necessity and possibility is called alethic logic. However, there are various sorts of alethic logics as well. In general, the alethic systems posit a multiverse of “possible worlds” (“possible worlds” is a technical term, don’t read anything into it), and also that some of these worlds can “see” other ones. A statement is considered necessary in some world W if it holds in every world W can see, and considered possible in world W if it holds at some world W can see. You seem to be thinking that alethic logic forces us to take a statement to be possibly true if there’s some logically consistent, conceivable state of affairs where it holds, and to be necessarily true if it holds in every logically consistent, conceivable state of affairs. This is a “plenitude principle” (of “possible worlds”), corresponding to taking the set of “possible worlds” to be every logically consistent, conceivable state of affairs, and saying that each one of these worlds can “see” all the other ones. (“plenitude” basically means we don’t leave anything out; we add in everything we can grab). That’s certainly one natural and intuitive system to formulate in alethic logic (incidentally, since every world can see every other world in that system, it’s called an S5 system), but it’s not the only way. Alethic logic is capable of handling subtler notions of necessity and possibility as well. Depending on the particular notion of necessity and possibility you want to capture, you can set the alethic logic up differently (i.e., with a different conception of the multiverse of “possible worlds” and the “sight” relation between them). Hopefully, my post #439, where I discuss the semantics of alethic logic in more detail, may be of some use to you.
That having been said, the particular system of alethic logic you are concerned with (the “possible worlds” being all conceivable states of affairs, and with all of them being able to “see” all the rest) is important, because it is the only real way to justify some of the premises in the modal ontological argument, and your analysis of the problems with this argument’s premises is pretty spot-on. I’ll just comment on them a bit more:
[QUOTE=begbert2]
So. Presuming <> to approximately mean “it can be imagined to be the case that”, and to approximately mean “it cannot be imagined not to be the case that” , what does this mean for the premises of The Ontological Argument?
Version 1: (Defining G to mean “God exists”.)
P1: <>G: It can be imagined to be the case that God exists.
P2: G -> G: If God exists, then it cannot be imagined to be the case that God does not exist.
Uner this argument, with nothing else at all implied by the term ‘God’ but ‘thing’, I can accept P1 readily enough. “A thing called God might exist.” Sure, sure.
However I reject P2. There’s nothing that exists that I can’t imagine not to exist, absent other presumptions that might contradict such an assumption. In fact, the only way I could possibly accept P2 would be if God was presumed or defined to necessarily exist.
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Yes, I basically agree with you here, although, I should note, I think P2 should be (G -> G), or else the ontological argument doesn’t work. If P2 is just G -> G, then we can satisfy both premises by taking God to be something which doesn’t exist in reality (thus satisfying P2 by making the antecedent false), but which could possibly exist (thus satisfying P1). So I’m going to carry on discussion assuming P2 was meant to be (G -> G).
There’s no good reason for us to accept <>G except via the plenitude of “possible worlds” principle: whatsoever is conceivable is possible. If we interpret G as something conceivable, we see that G is possible. But this PoPW principle also commits us to accepting the existence of a “possible world” in which nothing at all exists, and thus forbids us from simultaneously accepting the second premise (that it be logically demanded that if G is true at all, then G can’t be conceived to be false). So, as you say, there’s no good reason to accept both premises here.
[QUOTE=begbert2]
Which brings us to the other version of the argument:
Version 2: (Defining G to mean “God, which exists necessarily”, aka “God, which cannot be imagined to not exist, exists”.)
P1: <>G: It can be imagined to be the case that God, which cannot be imagined not to exist, exists.
P2: G -> G: If God, which cannot be imagined not to exist, exists, then it cannot be imagined to be the case that God, which cannot be imagined not to exist, does not exist.
Well, under this argument P2 is darned near a tautology. However, P1 has now become unacceptible. I can’t even imagine something that cannot be imagined not to exist. Anything can be imagined not to exist, absent other presumptions.
So, I regect P1 if you explicitly include necessary existence into the definition of God, and I reject P2 if you don’t explicitly include necessary existence into the definition of God. Therefore, under all possible definions of God, I cannot accept all of the premises of this argument; ergo, I find it to be unsound, proving nothing. Q.E.D.
(Unless there’s something I’ve seriously misunderstood about the premises or how modal logic is used, anyway.)
[/QUOTE]
Again, you’re pretty much on here. P2 is a tautology in this formulation, and thus we must admit it (it needn’t even be a premise, it can be proved from the definition of God in this formulation). However, P1 is unsupported now; we might want to get P1 through the PoPW principle, saying if we can conceive G, then we should derive <>G. But can we conceive G? Well, G says that God exists necessarily. Can we conceive of God existing necessarily? Well, you may be tempted to say yes, but don’t forget that we’ve adopted the PoPW principle to get this far, and thus need to remember what we mean by necessarily: logically/conceptually demanded. Can we conceive of God’s existence as conceptually demanded? No, certainly not (since, again, it’s easy to conceive of a world in which nothing exists). And thus we have no reason to accept premise P1.
Now, for my wholly own thoughts on the argument.
Basically, what the apparent soundness of the argument comes down to, I think, is a sleight of hand, where we find P1 reasonable under one interpretation of necessity/possibility, but not the right one to make the argument go through, but are tricked into conflating the two interpretations. P1 (it is possible that God exists necessarily) seems reasonable on some sort of plenitude-based account: certainly, we could imagine a world W which was part of a multiverse M such that God happened to fall into every “possible world” of M. But if that multiverse M isn’t the full multiverse F of things visible from our actual world (which our commitment to a plenitude-based account of possibility tells us should contain all consistent states of affairs), then God’s necessity in W is entirely dependent upon W being unable to see any worlds outside of M (in particular, W can’t see the conceivable worlds in F where God doesn’t exist). Thus, we are committed to saying that some worlds may be hidden from other worlds if we want to obtain P1 on a plenitude account. Specifically, the manner in which we allow worlds to hide from each other corresponds to something no stronger than an S4 system (worlds can see themselves, and sight is transitive, but sight won’t always be symmetric). The modal argument will go through fine and get us <>G (we can see some world which can only see worlds where God exists), but we won’t be able to move from <>G to G (and thus to G) without the full system S5 (which adds the symmetry condition on sight). So that is the problem with the argument: in order for us to convince ourselves of the truth of the premise <>G, we use a plenitude principle whose validity is based on an understanding of alethic logic as allowing asymmetry in the sight relation between possible worlds; however, to then, later in the argument, move from <>G to G and then G, we must use a logic which demands symmetry on the sight relation.
What I’ve said above may be all rather dense and unclear, so I’d be happy to explain it further, but it is late and I really must sleep.
(One point to note: I’ve used the evocative term “sight” many times above, and in previous posts on modal logic, but the usual term for this concept is the more dull sounding “accessibility”).
