[FOM] A minor issue in modal logic
Michael Lee Finney
michael.finney at metachaos.net
Sun Jul 4 11:40:51 EDT 2010
It would seem to me that Np means that p is necessarily true without
distinction of world, or that p is true in all possible worlds. In
either case it would be true in the actual world because in one case
the world was not distinguished and in the other case surely the
actual world is possible. So you have Np => Np[a] no matter how you
look at it.
I do not see how the converse would hold, in general, no matter how
you look at it. If you have Np[a], p may be necessarily true in the
actual world, but I don't see how that ensures that p must be true in
any possible world. There could be worlds which cannot be an actual
world - perhaps they violate actual laws of physics and are made of
half real matter and half anti-matter.
For a believer we have g in the actual world, for an atheist we have
~g in the actual world. For an agnostic we have g v ~g in the actual
world and for the completely indifferent we don't even have g v ~g.
That is entirely a matter of assumption and a biased assumption in the
case of St. Anselm and others trying to prove the existence of god.
If you want to prove the existence of god you cannot start with
g, Ng[a] or Ng. I would even argue that P[g v ~g] cannot be assumed
without biasing the argument.
We could have a world ruled by the Norse gods and in that world ~g
would hold. Given the possiblity of a world in which ~g holds, Ng
cannot hold because Ng => ~P~g, and therefore by Modus Tollens you
have P~g => ~Ng.
I think the only real question is if it is possible to show a valid
rule of inference in the arguments of those trying to prove the
existence og god that doesn't depend on their assumptions of god and
which would be different from current modal principles.
Michael Lee Finney
KBJ> All and sundry: I have encountered an issue in modal logic
KBJ> that I haven't seen and don't know where to look for a resolution
KBJ> of, and my local philosophy department's members haven't been
KBJ> much help. Perhaps FOMers can help.
KBJ> The fundamental question, I think--I'll give its origin in
KBJ> a moment--is this: For any proposition p, where "Np" means "It
KBJ> is necessarily true that p" and "Np[a]" means "p is necessarily
KBJ> true in the actual world," are Np and Np[a] equivalent--or, if
KBJ> not, does Np[a] at least entail Np?
KBJ> Keith Brian Johnson
KBJ> FOM mailing list
KBJ> FOM at cs.nyu.edu
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