[FOM] A minor issue in modal logic

[Richard Heck]

On 07/04/2010 07:06 PM, laureano luna wrote:
> On 3 jul 2010 Keith Brian Johnson wrote:
>
> "The fundamental question, I think--I'll give its origin in a moment--is this:? For any proposition p, where "Np" means "It is necessarily true that p" and "Np[a]" means "p is necessarily true in the actual world," are Np and Np[a] equivalent--or, if not, does Np[a] at least entail Np??"
>
> Definitely 'Np' is not usually read as Np[a] i.e. as 'p is necessarily true at the actual world' but as 'necessarily p' or 'p is true at all possible worlds'. I wouldn't advise mixing in one sentence the 'necessary-possible' terminology with the 'possible worlds' terminology, at least in this context.
>
> If you wish to pass from the 'necessary-possible' language to talk about possible worlds, you can read N and P as quantifiers over possible worlds, respectively as the universal (Aw) and the existential (Ew).
>
> Then you can interpret 'Np' as 'Aw p[w]' and 'Pp' as 'Ew p[w]'. Therefore, 'N(g->  Ng)' can be read as 'Aw(g[w] ->  Aw g[w])'. This entails
> 'Aw (~Aw g[w]) ->  Aw ~g[w]' or '~Aw (~Aw g[w]) v Aw ~g[w]', which through S5 entails the desired 'Aw g[w] v Aw ~g[w]'.
>
> Anyway, 'Np[a]' would entail 'Np' in a suitable system containing S5, since S5 implies that the necessary propositions are exactly the same at all possible worlds.
>
>    
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.

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.

Richard


-- 
-----------------------
Richard G Heck Jr
Romeo Elton Professor of Natural Theology
Brown University