Yeah, I too am skeptical of logic's metaphysical implications.
That "ontological proof" seems to work just as well for any conceivably necessary entity. It seems possible that there could be a necessary blob. Therefore the blob is actual.
What a stupid argument!
I agree with your approach, Dan; the formal structures of the various modal systems are useful for modeling types of reasoning; there is nothing in the system itself that says under what circumstances (if any) the system is a model of good inference (beyond the obvious point that it has to be a circumstance in which the axioms could be applied).
The conceivability issues, though, are not so convincing; in general people who actually have put forward ontological arguments have rather precise notions of conceivability in mind, and so we can't merely assume our (generally rather mushy and vague) concept of conceivability is the one they are actually using.
Brandon -
My notes about the relation between conceivability and possibility wasn't aimed at any particular version of the ontological argument. I'm not really familiar with the details of any such arguments (esp. not their modal varieties). The subject happens to be something I'm interested in, and this seemed like a good place to jot down a few preliminary notes.
Fair enough; you're right that it is an issue (sometimes the issue) for the argument, and would have to be considered for any given case.
Help me out here. I'm sticking on
MLx > x
which you attribute to B and 'duality'. B is
x > LMx
and 'duality' isn't given. Can you throw me a line about it?
Oops. That's what happens when you copy from yor notes. The weak dual of any conditional-free formula is taken by changing all conjunctions to disjunctions, all L's to M's, and vice versa. Let A* be the weak dual of any conditional-free formula A; if A>B, then B*>A*. I can provide a proof if desired.
You can also derive MLp>p from B relatively easily; the proof goes like this:
1. ~x>LM~x—B, substitution
2. ~LM~x>~~x—1, contraposition
3. ML~~x>~~x—2, modal negation
4. MLx>x—3, double negation
It should be clear that you can do similar proofs for other conditional-free formulas. Duality justs lets us skip a bunch of steps.
I'm curious about the 'g->Lg' bit - and in particular whether it makes sense to read that necessity as de dicto.
I could understand someone who argued that if God existed then God would necessarily exist (read de re), but I'm not sure it would follow that if God existed 'God exists' would have to be a necessary truth. And g->Lg read de dicto sounds far, far less plausible than if it's read de re.
Or am I just missing something?
I'm not really conversant with the details of the modal ontological argument. I take it that the L(g>Lg) part is supposed to represent the claim that the concept of God is a concept of a necessary being. The argument might go like this: If God exists, God necessarily exists. But this just follows from the concept of God, and so it is analytic that if God exists, God necessarily exists.
Another approach might go like this: God is a necessary being, so if God exists, God necessarily exists. But it seems to be nonsense to say that a necessary being might not have been a necessary being. We might consider the converse: could a contingent being have been a necessary being? I'm not sure how to answer that question, but someone might push in this direction.
In any case, I can't quite shake the suspicion that that clause is a way of disguising a question-begging argument.
Well - I understand (though I think it ridiculous) that premise when put the way you've put it, but it seems to me precisely to be a de re reading. We are attributing to God necessary existence, if he exists.
But doesn't the argument require something more like "if the sentence 'god exists' is true, then the sentence 'god exists' is a necessary truth"? And I don't know that that follows quite as easily.
Of course, it has been a rather long time since I've done any serious logic, and it's hardly my strength so I could be easily wrong here. I'm just not sure.
Actually where I part company is your claim that god is conceivable. Sure some omnipotent being or big man in the sky is conceivable but if one accepts g>Lg one has rolled into your concept of god that he is logically necessery if he exists.
So the right question is it concievable that a logically necessery god exists? I don't really have any handle on whether I can conceive of this or if I am incorrectly conceiving of something else.
I suspect with any reasonably appropriate notion of conceivability, i.e. one which doesn't make it conceivable that 2+2=5, this will come out to be conceivable only if god really exists just as statements about mathematics (the only place we frequently encounter things which if true are necessery) turn out to be conceivable only if true.
Dan wrote: " Let A* be the weak dual of any conditional-free formula A; if A>B, then B*>A*. I can provide a proof if desired."
I think you'll have to. If I read that aright, and I think i do, you've left something out of your description of dual. Maybe "...and negate each simple proposition".
Is the dual of Lx & My
Mx | Ly
or
M~x | L~y
?
Dan wrote: "...But this just follows from the concept of God..."
Let me stop you right there. 'the' concept of God entails that there is one, singluar, universally accepted concept of God. Which isn't the case.
I'm strongly reminded of Keith Burgess-Jackson (http://analphilosopher.com) who recently threw out a common, powerful (in the sense that his definition implies lots of stuff) and not universally held description of God.
Looking at your syllogism, I'm well convinced of its logical soundness. At the same time, I'm far from ready to accept either of "g>Lg" or "Mg". Which is just what you'd expect, given that I don't accept the conclusion.
Craig -
I'll provide a proof of this property of weak duals in a future post. Also, you're correct to say that there isn't just one concept of God. I should have said that it follows form the traditional conception of God in the Western Christian tradition.
Sorry liberal arts major here; my head is spinning from the mathematical suggestionsthat I am unable to comprehend. Can we get warned before numbers and/or logic is invoked? Mahalo!
Malia -
I'll put a disclaimer on future logic posts. I typically abbreviate those posts, since I suspect that most of my readers aren't particularly interested.
It's not about being interested, but about being capable of comprehending what you are saying - which, really, is a measure of the inadequacy of your readership, not *you* LOL
I too am unable to follow the logic trail here, not my forte. But I liked the title of the post "Logic is for Tricking People".
You referenced God's existence and the necessity of the existence of God as two related issues. You also referenced the fact that there is no single universally accepted concept of God, that 'God' always comes packaged with cultural and historical imagery attached. Could it be that 'God' is an image we need to use in order for our intellect to conceive of ourselves and our place in this life? Do other animals need to have a 'God' image, or do they live at a level of consciousness that does not require such an entity? Where does having a need for a 'God' image come from? Is it necessary for our mental and spiritual health to worship this image? To contemplate this image? To study and understand the development of this image in a larger social and historical context? To question the validity of the need for this image?
I've been interested in this "Question of God" for some time and still don't have it figured to my satisfaction, though I've come to some foundational concepts that seem to not be quite so offensive to my sensibilities as the original ones I had crammed into me in my youth. The logical approach to the question would likely be interesting for me to pursue at this point, but requires more study than I have time to do right now. Damned responsibilities!
Later.
That is, it's an unexpected result of a series of moves in a formal system, and it's got some curious properties—but it doesn't mean anything. In general, I take this position w/r/t the whole of logic and mathematics. In fact, I see the lack of meaning to be the great strength of logic and math. The reason mathematic representations of the world and formal logical representations of thought have been so effective is that we are free to reinterpret the semantic content of the relevant symbols as appropriate for a given context.
It's probably right to say that pure mathematics is meaningless, and perhaps this applies to logic as well. However, meaninglessness alone doesn't give us effectiveness. It has to be applicable to the situation at hand, and then we actually have to apply it. That is, we have to interpret it with a meaning in order to use it, and the question is just whether this meaning is appropriate to the axioms used in the relevant bit of mathematics (or logic) that is being applied.
Thus, in order to apply this argument and show that God exists, one would have to show first that a modal logic at least as strong as KB is applicable to the relevant concepts. There may be one sense of necessity in which one can grant the premise that if God exists, then it is a necessary proposition that God exists. And there may be a sense of possibility in which it is possible that God exists. And I am probably willing to grant that each of these senses of necessity and possibility is usable as part of some system at least as strong as KB.
However, it is not at all clear that the sense of necessity and possibility needed for these two premises are dual to one another as required by the interchange laws that are tacitly used in step 3 of the argument. Thus, I can accept the validity of the formal argument while rejecting its applicability, without saying that the logic is meaningless. If the senses of necessity and possibility involved clearly did have the interchange property, then I would accept the conclusion of the argument. The formalism just gives us an easy way to make the argument - if applicable in the situation under consideration, we could have made the entire argument directly in English, without any use of formalized logic.
Hello,
This post is more than two years old, I hope you're still reading it. I've always found the Ontological Argument interesting but deep down, smoke and mirrors.
However, I was thinking about it, and I'm curious about what is meant by "all possible worlds" in modal logic. Surely it is an infinite "collection", but is it something that can be conceived of or used as a foundation of modal logic?. I mean, it is known that the set of all sets cannot be considered a set, and in fact, it's hard to say anything meaningful about this Cantorian absolute infinite. Therefore, is "all possible worlds" really a good foundation for modal logic?. Wouldn't it be better to say "necessary or possible with respect to universe U(D)" (where U(D) encompasses all relevant possible worlds for a particular domain D, but it falls short of being "all possible worlds").
Hari Seldon
Hi, Dan! I've just started a blog at www.blog.holycyclops.com, and I've linked to this page in one of its first few posts. If that's not OK, please e-mail me to let me know. The reason for the link was that I used your version of the ontological argument in one of my nascent blog's first posts. You might be interested in it.
I hope all is well!
Keith Brian Johnson
OK, apparently I can't put two URL's in the "URL" box and still have the "homepage" button work. Sorry.
Keith Brian Johnson
So you have a colleague who takes the possibility of the modal ontological proof as a reductio for the systems in which it is possible...
This I would call: ATHEISM AS A FUNDAMENTAL PRINCIPLE OF LOGIC.
So there must be an atheistic logic, a theistic logic, a male logic, a female logic, a white logic, a black logic, an aryan logic, a jewish logic, a proletarian logic ...
Poor logic!
Commenting by HaloScan
