Friday, October 24, 2008

Possible Worlds II

We were talking about possible worlds being relative to other possible worlds, not relevant, and this is a technical term.

First, for most philosophers, all possible worlds are relative to all possible worlds. This has a consequence that it is governed by a formal logical system called S-5. If not all worlds are possible to all possible worlds, then weaker systems of logic govern our inferences. This is precisely what Kripke showed in his published work in the 1950's, when he was 18(!).

It follows in S-5 that anything which is possibly necessary is actually necessary. this has been used to prove God's existence in a version of the Ontological argument. God, if he exists, has the property of being a necessary being. It is possible that there is a God. Therefore, it is possible that a necessary being exists. By S-5, God is actually necessary, and and since anything which is necessary is true, God exists in this world, too.

Other philosophers deny that all possible worlds are possible relative to all possible worlds. Aristotle might not have existed. So there is a world at which there is no Aristotle. At that world, is it possible that Aristotle's son existed? What does it mean to say that some non-existent thing might have existed? In response to these kinds of questions, they deny the universal connection between possible worlds. Worlds in which Aristotle exists are not possible from worlds in which Aristotle does not exist, though the reverse does not hold. If that is right, the logic of modality is not S-5, but something weaker (S-4, for those who are counting) and it no longer follows that just because something is possible that it is necessarily possible, nor does it follow that something that is possibly necessary is actually necessary.

I am looking at my car. It is possible for it to start. But is it necessarily possible for it to start? On one way of looking at things, no, because it is possible for the engine block to be totally ruined. Spelling that out in possible worlds means some possible worlds are not possible from all possible worlds. It is possible that Aristotle exists. But is it necessarily possible? What if human beings never develop? What if the world blew up before humans ever appear on the world stage? Then there would still be a possible world in which Aristotle exists ("Aristotle might have existed" is true), but that world would not be relatively possible from worlds where the world blows up before humans arrive on the scene. Hence we can say Aristotle could not exist if the world blew up before humans arrive on the scene, even though he might exist in other circumstances.

Hope that helps.

I was going to post more on the metaphysics of modality, but ce side tracked me. Blame him. Will try to do that Monday.

15 comments:

ce said...

I readily accept that blame. S-4 seems far more acceptable IMO. It is both possible that there is a God, and that there is not a God, after all, and to force a God upon a Godless world just seems...bizarre to say the least.

Anonymous said...

Here is the rub as I see it: Using S5 we might argue:

If It is possible that it is necessary that God exists then God exists.
It is possible that it is necessary that God exists.
Therefore, God exists.

but then again, as ce points out, using S5, we might argue:

If it is possible that it is necessary that it is NOT the case that God exists, then it is NOT the case that God exists.
It is possible that it is necessary that it is NOT the case that God exists.
Therefore, it is NOT the case that God exists.

Does this consequence --the seeming contradiction-- show that S5 is suspect? I don't think so. Why? Because the second premise in each argument is supposed to be a metaphysical claim (not an epistemological claim) and only one of them is true --so one of the arguments is unsound. BUT, which one is it? Plantinga says that only God could know for sure. But the problem might be deeper than that. What sort of observation would indicate that physical object X is metaphysically necessary or not? Is talk of metaphysical necessity of physical objects nonsensical?

Help us Mr. Wizard!

tmf

Hanno said...

What if we suppose that a possible world cannot be created without a God? Not a bad assumption, and then every possible world will have a God that makes it. If we get that far, its easy to make it the perfect God. Then we have a being at every possible world, and hence a necessary being.

ce said...

Then we have a being at every possible world, and hence a necessary being.

Which with a mere play of words becomes necessarily not the case. The fact that the two statements are metaphysical and not epistemological in nature only further aggravates the problem. It can't be that we're simply mistaken. The two cases are simply incompatible. It's not a matter of knowing because according to the usage of S-5, we already know (the mere possibility makes it such).

It's possible that a given world is of such a nature that it could not be created by God (any God). As such, the creator God could not have created that possible world, because the possibility of said world makes it so. So how does this help S-5?

so one of the arguments is unsound.

But via S-5, aren't they actually considered sound? By the S-5 conjecture they are both possible, and relative. Because they are possible, it is such. So either reality (or some realities?) need not adhere to logic or this is simply the wrong logic to use.

tr said...

The S5 axiom is if p, then necessarily possibly p. All that entails is that if God exists in one possible world then he exists in all of them, in which case Todd's first argument is sound. But God could not exist in any possible world, in which case Todd's second argument is sound.

S5 is sound and complete, so there's nothing technically wrong with the system.

ce said...

tr

A system that allows contradictions to be sensibly formed is not at least suspect?

What sort of observation would indicate that physical object X is metaphysically necessary or not?

None, which we have.

Is talk of metaphysical necessity of physical objects nonsensical?

What makes you think that God is a physically extant being? Angels are supposed to be spirits, which can make themselves known, and have a physical impact on the world, but are not themselves physical beings. Why would God not be capable of existing in a form somewhat like that?

Maybe "God" simply doesn't entail what people would like it to entail.

Here's a question: if God is everywhere, is there a possible world where God could not be? If it is possible for a world N to exist without space (therefore there is no "where" as such in that world) then perhaps so.

Let's jettison that silly science stuff for a moment, since it's all wrong any damn way (obviously scientists just haven't read The Book). Is that an allowable limitation to put on God? But what if this spirit thing is the real deal? I can imagine a ghost. Such a being seems to totally lack traditional spacial limitations.

But doesn't that wind up making God quite unlike "us"? But I'm not sure that's a problem. Is it sufficient that God is merely a rational, ethical being? Maybe "in His image" means something else entirely. Unless we think He really does hold a strong resemblance to Alanis Morrisette, and that's somehow quite important.

Hanno said...

Well, lets be careful here. It is true that the S-5 (and S-4) are sound and complete, and it does follow that there is a model of both, and hence that they system are consistent.

It is inconsistent, however, in S-5, (ie you can prove a contradiction) to hold that it is possible that there is a necessary being, and it is possible that there is not a necessary being, precisely because in S-5, it does follow that if something is possibly necessary, then it is necessary.

That is not necessarily (heh) a problem with S-5, but shows that the initial premise, that it is both possible to have a necessary being and possible not to have a necessary being, is inconsistent and hence simply false. No big deal.

The apparent problem comes in when we seem to agree that both things are possible. The agnostic says "There might be a God, but there might not be as well." But if we understand that claim not about metaphysical necessity, but about epistemology (For all I know there might be a God, but maybe not) then there is no problem, since certainty clearly does not follow S-5. No one thinks that it follows from possible certainty that we are actually certain.

So the problem only applies to metaphysical necessity: The person has to hold that it really is possible that a necessary being exists, which holding that it really is possible that no necessary being exists. Todd then asks, What does that mean?

Anonymous said...

Thanks for getting us back on track Mr. Wizard....

So the problem only applies to metaphysical necessity: The person has to hold that it really is possible that a necessary being exists, [while] holding that it really is possible that no necessary being exists. Todd then asks, What does that mean?

And I am not sure that it makes sense to talk about metaphysical necessity of a physical being since I don't know what observable difference there would be between the two... (And this concern has nothing to do with God and whether or not He is a physical being.)

tmf

ce said...

I do feel compelled to state that switching from a metaphysical inquiry to an epistemological one is rather dodgy, even if pertinent.

No one thinks that it follows from possible certainty that we are actually certain.

Methinks Hanno does not have a gamefaqs account.

This sparked another question, if anyone feels compelled to field it. Todd's question of what this entails is certainly not new. We ask the same thing when something is logically possible, but rails against science. What kind of universe would this have to be if little g is not little g elsewhere in the cosmos? If something in someway can accelerate beyond C (which is a most excellent letter, I might add)?

And we knee-jerk against it even more strongly in cases of out-and-out inconsistencies. What in all the nine Hells is a five sided square? What does it mean to have an aardvark, which is not in fact an aardvark? A married bachelor?

The point of this little tangent in relation to possible worlds is that I'm not convinced statements of what is or is not possible is so much a statement about metaphysical potentialities so much as it is about our capability to reason, understand, and rationalize. Saying simply that "I don't get it", or rather "I don't know what that means", seems to flatly state far more about "us" than it does about "it". Or, to be brief: nonsense is only nonsense in relation to certain sensibilities.

Are we we sure the limitations we put on possible worlds are limitations on possible worlds, or limitations we have and thus place upon them?

ce said...

I'm not seeing this:

It follows in S-5 that anything which is possibly necessary is actually necessary.

Is there a proof constructed for this? An argument given? Something?

I can easily couch this in different language. If it "might be the case", then it's "possible". If it "is necessary", then it "must be" the case. But if it "might must be the case", does that actually make it so that it "must be the case"?

How did we cleverly eliminate the "might"? Why does it just drop away? I don't see how that follows.

It might be the case that McCain eats babies (apparently, Obama eats puppies). In some other world, it might must be the case that McCain eats babies. So, it must be the case that McCain eats babies? Huh?

Where did the possible (might) go, and how is eliminating it justified?

I don't see that it's unreasonable to simply reject that premise. Hanno is right in that the possibility of certainty does not automatically grant certainty. Why in all the possible worlds does it follow that the possibility of necessity automatically entails necessity? That sounds like balderdash and chicanery.

Hanno said...

I'm not seeing this:

It follows in S-5 that anything which is possibly necessary is actually necessary.

Is there a proof constructed for this?


Yes.

Hanno said...

We can do this semantically or formally. Suppose every world is possible relative to every possible world. Suppose that it is true that P is possibly necessary. So in some possible world, P is necessary. But for P to be necessary, it must be true in all possible worlds. And since all worlds are relative to all possible worlds, we cannot narrow that. P must be true in all posbible worlds. So P is necessary. Now if its necessary, it must be true in this world, too. So it is actually true. And it since P is true in every possible world, it is necessarily true in this one, the actual world, too. So if P is possibly necessary, then it is actually necessary. QED

ce said...

H

TY.

Suppose every world is possible relative to every possible world.

So, perhaps that is the premise that's really chaffing me. If Illyria really visited a universe composed entirely of shrimp (and I have no reason to believe she was lying), then I have to wonder in what way we would consider that world somehow relative to our own in any interesting or useful sense of the term.

Isn't "they're all relative" just a baseline assumption? We needn't imagine a shrimp world for this. What about a universe in which the earth never formed? Heck, just watch an episode or two of Sliders. So doesn't this just wind up begging the question? If we're willing to concede, then it can be discussed, obviously, but how would one actually go about arguing for this?

ce said...

Addendum: Now where is that post on the metaphysics of modality?

Hanno said...

Well, that is the key assumption that underlies the formal system called (unpoetically) s-5. When you deny it, you get a less strong system.

There are good reasons for denying it, but the underlying foundation of your own intuition seems to be that for something to be possible in this world, you must be able to connect it to this world, give some story about how from some situation in our world, this other world could have come to be. I didn't go to medical school. But I could have because I could have taken the tests, and had I taken them I would have done well and my reason for believing those claims will be entirely factual. i have taken similar tests and done well, etc. This ties possibility to this world, and makes modal claims not tied to this world nonsensical or false. Uh, for more on this, read my dissertation.