Philip Osborne

Here's the proof that if something is possibly necessary, then it is necessary.

1. If x is not necessary, then it is necessarily true that x is not necessary. [~Lx => L~Lx] (By Modal axiom S5, which states that the modal status of all propositions is necessary)

2. If it is not necessarily true that x is not necessary, then x is necessary. [~L~Lx => Lx] (Contrapositive of 1)

But "not necessarily not" is logically equivalent with "possible," so by a few simple inferences, we get that if x is possibly necessary, then x is necessary. The proof above only works in the S5 system of modal logic, which contains the crucial axiom used in (1). Other systems do not use this axiom. It used to be popular to oppose the ontological argument by suggesting it is unpersuasive in as much as it relies on a controversial axiom of modal logic. In recent years, however, much of that controversy has died down.

Synaesthesia

Learning modal logic should be one of my projects for the summer.

I wonder though, how one would use modal logic to deal with the idea of it being *possible* that it is possible that X is neccessary.

By *possible*, I of course am refering to the "possibility" of everyday parlance, where X is possibly necessarily false, possible false, possibly necessarily true, possibly true.

Koyaanisqatsi

To hell with modal logic, Syn, it just ultimately ends up in this kind of technical masturbation.

Apply <a href="http://home.netcom.com/~trifonov/" target="_blank">Topos theory</a> instead! Much better...

mibby529

There are no sound theist arguments. The only gods that cannot be proven nonexistent are those that aren't even defined. But those same gods cannot be proven existent, either, since they're undefined.

rainbow walking

And remains a mammoth exorcise in question begging.

Necessary to what?

Using modal logic my little Case pocket knife is God.

1. By the rule of S5 my pocket knife is not necessary...unless, I find that my continued existence is contingent upon that little pocket knife. For instance if I find myself completely enshrouded in about 2oo mils of shrink wrap with half a minute's oxygen left in my lungs, suddenly that pocket knife becomes absolutely necessary.

2. Yet, it is, indeed, the case that my pocket knife, in most any other circumstance is not necessarily necessary.

3. So my pocket knife is possibly necessary and therefore demands your immediate conversion to the religion of Bladeianity and requires a weekly donation of 3-in-1 oil and a good whetstone.

Theli

Well put, Koy!
Since there are not just one proposition that is possibly necessary, we get huge complications as several propositions suddenly becomes neccessary (including contradictions).
It is possibly necessary that god exists, and it is possibly necessary that god does not exist.

According to this form of modal logic, it is necesary that god exists, and don't exist at the same time.
How can then this modal argument be applicable in reality?
Too bad Kenny refused to tackle my complaint at the argument.

Tercel

Theli,
Allow me.

Modal logic deals with "possible worlds". Imagine a multiverse containing every single world that is logically consistent.
If X is true in all of those worlds, then its status (according to modal logic) is "<strong>Necessary</strong>".
If X is true in some of those worlds, then it is called "<strong>Possible</strong>".
If X is true in none of those worlds, then it is called "<strong>Impossible</strong>".
Nothing difficult so far, that's just definitions.

Hence what does it mean to say that something is "possibly necessary"? We are talking in modal logic speak here, so by "possibly" in "possibly necessary" we do <strong>not</strong> mean that it might be "necessary" and we just don't know.

Remember "possibly X" means that X is true in some possible world. So if we say "X is possibly necessary" then it's the same as saying "There exists some possible world in which X in necessary".
Since X being "necessary" means that X is true in all possible worlds, if there exists a possible world in which X is necessary, then we know that X must be true in all possible worlds.

In short: If there's somewhere where its true to say that X is true everywhere, then it follows that X is true everywhere.

Hence "possibly necessary" = "necessary" in modal logic speak.

Theli

Tercel...

Yes, so far I understood it.

We don't?
You never said what it does mean. What does it mean?

no. no. no. no. no!
You did excacly the same thing as Kenny did. You flushed the last "possibly".
X is only possibly true in all worlds, as it was only possibly necessary in one world.
If you claim that "necessary in one world" equals "necessary in all worlds", then why did you call it "necessary in all worlds" to begin with?
The "neccessary in all worlds" is ofcourse unfounded. It's a premise, aswell as the conclution of the argument. That makes this argument a tautology, doesn't it?

"Somewhere" in this case is everywhere.
If you (for some reason unknown) can say that X is necessary everywhere in a certain possible world, then you can say the same in all possible worlds.

As you can just aswell assume the opposite, this argument becomes useless. It is not applicable.

[ July 25, 2002: Message edited by: Theli ]</p>

Kenny

Simple, you make a distinction between something’s being epistemologically possible and something’s being logically possible -- the former being with respect to what we know and the latter with respect to logical coherence. It is the latter respect that modal logic refers to when it says the word “possible.”

That’s all I’ve got time for right now…

God Bless,
Kenny

[ July 25, 2002: Message edited by: Kenny ]</p>

Clutch

Automaton, Kenny,

I know the Plantinga paper, and I can assure you that the confession of non-proof, and the comparison to a circularity, is applied by AP to his own best reconstruction of the OA.