[FOM] A minor issue in modal logic

[Richard Heck]

On 07/03/2010 01:53 PM, Keith Brian Johnson wrote:
> All and sundry:  I have encountered an issue in modal logic that I haven't seen and don't know where to look for a resolution of, and my local philosophy department's members haven't been much help.  Perhaps FOMers can help. 
>
> 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?
>
>    
The semantics of such operators is well understood. The answer, in both 
cases, is "No".

Let's use "Ap" to mean: p is true in the actual world. The operator "A" 
is known as the "actuality operator". Let's define it carefully. A model 
is a triple <W,V,@>, where W is a set of worlds, V is a map from worlds 
to sentence-letters (true in that world), and @ is the actual world of 
the model. Then "A\phi" is true in a world w \in W iff \phi is true in 
@. Note that the parameter w does not appear on the right hand side. So 
it follows immediately that "A\phi" is true in all worlds or false in 
all worlds, as \phi is true or false in @. That is, "A\phi" is always 
necessarily true or necessarily false at _every_ world, which of course 
implies that "A\phi" is equivalent to "NA\phi". Moreover, since \phi 
implies "A\phi" (in the sense that, if \phi is true in the model, i.e., 
at @, then "A\phi" is also true in the model), we have that \phi implies 
"NA\phi". So \phi is equivalent to "NA\phi", in the sense that they are 
always true or false in the model (i.e., at @) together. So "NA\phi" 
certainly does not imply "N\phi", since otherwise \phi implies "N\phi".

So the simplest sort of counter-example is: Take a two world model; let 
every world be accessible to every other. Let p be true at @ but false 
at the other world. Then Ap is true at both worlds (since p is true at 
@), so NAp is true at both worlds, but of course Np is false at both worlds.

Richard

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