On 6 March 2014 22:06, Bruno Marchal <[email protected]> wrote:
>
> On 05 Mar 2014, at 23:31, LizR wrote:
>
> Let's take 3 worlds A B C making a minimal transitive multiverse. ARB and
> BRC implies ARC. So if we assume ARB and BRC we also get ARC
>
>
> Right.
>
>
>
> (if we don't assume this we don't have a multiverse or at least not one we
> can say anything about.
>
>
> This, or something like this ...
>
>
>
> []p in this case means the value of p in A is the same as its value in B
> and C (t or f).
>
>
> What if p is false in A, and true in all worlds accessible from A?
>
> Well that means ~[]p, doesn't it?
>
> This also means that in A B and C, []p is true, hence we can also say that
> in all worlds [][]p.
>
>
> Correct.
>
>
> (And indeed [][][]p and so on?)
>
>
> Sure. at least in a multiverse where []A -> [][]A is a law. In that case
> it is true for any A, and so it is true if A is substituted with []A, and
> so [][]A -> [][][]A, and so []A -> [][][]A, and so on.
>
>
>
>
> So it's true for the minimal case that []p -> [][]p
>
> But then adding more worlds will just give the same result in each set of
> 3... so does that prove it?
>
>
> Not sure.
>
Me neither, as will now be demonstrated.
>
>
>
> No, hang on. Take { A B C } with p having values { t t f }. []p is true in
> C, because C is not connected to anywhere else, which makes it trivially
> true if I remember correctly. But []p is false in A and B. So [][]p is
> false, even though []p is true in C. So []p being true in C doesn't imply
> [][]p.
>
>
> I might need to see your drawing. If C is not connected to anywhere else,
> C is a cul-de-sac world, and so we have certainly that [][]p is true in C
> (as []#anything# is true in all cul-de-sac worlds).
>
>
A ---> B ---> C
and
A ---> C
where ---> means 'can access' - so C is a cul-de-sac and { A B C } is
transitive.
OK, []X is true in C where X is anything.
So if []p isn't true in A, then [][]p isn't true for { A,B,C } (though it's
true in C treated as a multiverse)
But for []p to be true in A, that means p is true (or false) in all worlds
accessible from A, including C. That is, p has the same value in A B and C.
So does that imply []p is true in all worlds accessible from A? Yes, I
think so. And that implies [][]p for all worlds accessible from A,
including C (trivially).
Isn't that what I was trying to prove? Or have I just wandered off into a
cul-de-sac myself?
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