[FOM] A minor issue in modal logic

[laureano luna]

Mon, 05 Jul 2010 Richard Heck wrote:

"This is not quite accurate. In S5, you have that, if P is necessary at 
w, then it is also necessary at any world w' that is connected to w, 
i.e., hereditarily accessible from it. But you can have models in which 
there are disconnected clusters of worlds, and in those models you can 
have P necessary at some worlds and not necessary at others. Of course, 
the disconnected worlds are irrelevant to what is true in a given world, 
so they are customarily ignored."

Quite right. But it is not only a matter of customary carelessness: for me and many others (I think) possible worlds that are not possible in the actual world make little sense.

Richard Heck also wrote:

"What is more important is to distinguish (in this notation) "Np[a]" from 
"N(p[a])". I.e.
     It is true at a that it is necessary that p
from
     It is necessary that p is true at a.
The former trivially entails Np, if a is actual; I took the question to 
concern the latter, which does not entail Np."

I think your reading is less straightforward. The original wording is: 

""Np[a]" means "p is necessarily true in the actual world"".

I take it that 'necessarily true' and 'necessary' are synonymous.

Best regards,

Laureano Luna