Friday, March 25, 2005

Possibly Necessary

DTWW discusses using modal logic to prove that God exists. In the modal system S5, if something is possibly necessary then it logically follows that it is actual. Think about it: X is necessary if it is true in all possible worlds. X is possible if it is true in some possible world. So if Nec(X) is possible, that means there is some possible world for which X is true in all possible worlds. That could only happen if X really was true in all possible worlds (including the actual one).

Dan then justifies the premises:
Surely it's analytic that God is a necessary being (if God exists at all). And it's equally obvious that God is possible; after all, God is conceivable. Therefore, God exists.

Now, this is a terrible argument for God's existence. We could just as well apply it to any conceivably necessary entity. For example: it seems possible that there could be a necessary blob, therefore the blob is actual.

This is patently ridiculous. I think the problem is due to an equivocation on the modal terms. People don't realise what they're saying when they call something 'possible' or 'necessary' (let alone 'possibly necessary'!) within the S5 system.

Let me use the term "quasi-necessary" to refer to any entity that is necessary if it exists at all; but without prejudice to the question of whether it actually does so exist.

It seems we can conceive of such quasi-necessary entities, be they Gods or blobs. But it would be a mistake to call them 'possible' on these grounds. To make a claim about possible necessity is not just to make a claim about some possible world, but rather the entire collection of all possible worlds. To say that Nec(X) is true at some possible world (i.e. "possibly true") just is to say that Nec(X) is true, simpliciter. (See here.)

Now, we can of course turn the argument around and point out that if it is possible that a quasi-necessary entity not exist, it actually must not exist at all. After all, to say it's possible that not-X, is just to say that X is not necessary. Since it is patently obvious that it's possible for God to not exist -- atheism is not inconsistent -- it follows (via the same logic as before) that God actually does not exist.

Theists will claim this argument is question-begging -- and I agree -- but it is no different from the earlier argument for God's existence. My point is that it is question begging to make any claims about the possibility of quasi-necessary entities. Claiming it's possible is just the same as claiming it is actual; and claiming it's possibly false is the same as saying that it's actually false. No progress can be made here.

Dinner Table Donts highlights what I think is the core confusion here:
I find it hard to think that God is not at least possible in our given context. Are the intuitions of millions of people, over millions of years suddenly just wrong about a Higher Power? I know the evolutionists will eat me alive for this, but doesn't the intricate design of our world and our universe at least allow for the possibility of a creator?

This confuses epistemic with metaphysical possibility. Given our evidence, it is possible that God exists, and also possible that he doesn't. Either is consistent with the evidence. (Of course, that isn't to say that both positions are equally justified.) But, as already noted, any quasi-necessary claim cannot be both (metaphysically) possibly true and possibly false, for that would make it both actually true and actually false, which is a contradiction.

Going back to my quasi-necessary blob, it seems possible to us that it might exist. At least, it is epistemically possible. (I certainly can't rule out the possibility of a necessary blob.) But, assuming that it does not actually exist, it follows that, due to its property of quasi-necessity, it isn't even (metaphysically) possible. The same goes for God, if understood as a quasi-necessary being. His actuality as a quasi-necessary being would rule out even possible non-existence, and similarly his actual non-existence would rule out his possible existence.

This can be overcome if the blob is not quasi-necessary. Your average blob is metaphysically possible, even if it isn't actual. The same would go for God if we stopped conceiving of him as quasi-necessary. A quasi-contingent God, though actually non-existent, could possibly exist.

But a quasi-necessary God (or blob)? Sorry, not a chance.


  1. You certainly say a mouthful, Richard. I would like to join in. First, the use of modal logic to prove the existence of God was famously voiced by Plantinga at the end of his book, Nature and Necessity. I assume that this is what Dan is referring to. (Either this or a similar argument by Charles Hartshorne.) This argument harkens back to the old arguments of Anselm and even Descartes. Your attempt to talk about a necessary blob is an analogue to Guanillo's attempt to refute the ontological argument by talking about a necessary island.

    I think you're right to point out equivocations between the terms in this argument. If you want to read my post on conceivability arguments for dualism, you will see that a fairly similar thing is going on here. That is, Plantinga (and Dan) are equivocating on the word 'possible', as you point out.

    I have another method of attack, as well. I figure that a modal argument will go something like this:

    (1) If there is a logically perfect being (God) then that logically perfect being is necessary.
    (2) It's possible that there is such a logically perfect being.
    (3) Therefore, there is a necessary logically perfect being.
    (4) Therefore, there is actually a logically perfect being.

    I would first attack premise (1). I would say, "I don't know about this 'logically perfect being' talk." Presumably, the theist would say, "Ahh, that is just a stipulative definition. You cannot attack that premise!" At that point, I would simply deny premise (2). That might leave us at an impasse, but that would be fine with me.

    Here is an interesting sidenote. A notably famous philosophical atheist has a forthcoming article saying that time was caused by a necessary and essentially timeless point. See here if you're interested.

  2. My justification of the ontological argument is excessively short; you'd be better off reading Plantiga if you want the real thing. I don't buy the modal argument either, and so I've glossed over much of the ink which has been spilled.

    Recall that MLp>p is only a theorem in any system that includes KB. The move from "possibly necessary" to "actual" tends to surprise people; they think it's a trick. That might suggest that KB and its extensions don't adequately mirror the process of modal reasoning.

  3. Chris, did you mean for your link to point here? (I recented linked that very article on my sideblog, actually!) I recall enjoying your post on conceivability, which I found resonated nicely with my own thoughts. And I agree with your "method of attack", in that we should first deny quasi-necessity, or in face of stipulation simply deny that it is really possible.

    Dan, good point about how this modal system might not match up to 'folk' modal reasoning. I'd like to learn more about the latter, I think...

  4. I assume that there are possible worlds with no God in them. Here are two:

    World 34729492340: Contents: one football.

    World 34729492341: Contents: two footballs.

    If these worlds are possible, God doesn't exist. Is this right?

  5. Jim -

    It depends. If those worlds are possible, then one of three things follows:
    1. God is not a necessary being.
    2. God is a necessary being, and God doesn't exist.
    3. While you thought you were imagining a world with one football, you actually imagined a world with one football and God. God doesn't do much in football-world, though, so He's easy to miss.

    That last one probably sounds weird, but similar arguments have been made regarding the identity of indiscernibles. The canonical counterexample to the identity of indiscernibles is a world which contains two featureless iron balls, each precisely one meter across and two meters apart. Now destroy one: does the other vanish? Well, why should it? So what we have are two indiscernible objects that are not identical. One response goes like this: actually, what you were imagining in your thought experiment was one sphere, in a very small universe. The geometry of this universe is curved in such a way as to make it seem like there are two spheres.

  6. I'd thought of your 3 before, Dan, but never an argument for it. Thanks. Still, I'd need to be given evidence that my two worlds are cases of the kind that your iron ball case exemplifies.

  7. I think Jim made an excellent point (I should think so, I was dying to make it but had my hopes dashed when I saw that he did).

    There are two issues here. First, what are we to make of S5? Second, what are we to make of appeals to S5 to argue for philosophically contentious claims? I don't have any problem granting that there is p.f. justification for the thought that there is a possible world with a God in it. I'm quite willing to grant S5. Still, it seems that whatever justification one might have had for believing God is actual is lost however once one recognizes that it is just as easy to conceive of a Godless world which if metaphysically possible rules out the existence of God in all worlds. Since the atheological and theological arguments appear to be on par, it seems the honest thing to do is admit that S5 hasn't improved the prospects of the ontological argument in the slightest.

  8. premise 1: It is possible that logical perfection exists in the idea alone

    Premise 2: God is definitionally a non contingent logically perfect entity

    There exists a possible world in which God can not exist

    therefore as "God" is a noncontingent being "God" can not exist

  9. Hello,

    With respect to necessity and possibility, what do you make of the "all possible worlds" used by modal logic? It cannot be a set, because there is no set of all sets (forbidden by Rusell's paradox). What is it then?. Is it the Cantorian Absolute Infinity?. That's risky because that "collection of all sets" is ridden with paradoxes. Could that be the source (partially) of the strange conclusions of the Ontological Argument?

    Hari Seldon.

  10. I wonder about the claim that "God is conceivable." What precisely is the cognitive content of "God?" If you ask ten different people you'll get ten different answers, so who should you believe?

    I'm sure I have no idea what is meant by "God" (where God is an entity, and not just a term).

  11. It's true that with regards to necessary entities, if they even possibly exist, it follows that they do in fact exist-that's the modal ontological argument in a nutshell. Now as you point out, it's also conversely true that if it's even possible that a necessary entity (God in this case) does not exist, it would follow that a God does not exist necessarily (and therefore would not be a maximally great being).
    So if God's existence is either necessary or impossible, which is it? On one level, it's tempting to think that it's possible that God exists, and it's possible that he doesn't, so the modal ontological argument fails to prove anything one way or another-this is what Richard thinks. I disagree.
    This is to confuse epistemic possibility (God's existence and nonexistence both seem conceivable) with metaphysical possibility (what can actually be the case). Only one of the two can actually be the case (God either does, in fact, exist, or he doesn't). So the question comes down to this: is it more plausibly true that (1) It is possible that God exists or (2) It is possible that God does not exist? I think based on both our intuition that God's existence is possible and the other arguments we have for God's existence that (1) is more plausibly true than (2), so the modal ontological argument is sound.


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