[FOM] A minor issue in modal logic

[laureano luna]

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.


Laureano Luna