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 (if we don't
assume this we don't have a multiverse or at least not one we can say
anything about. []p in this case means the value of p in A is the same as
its value in B and C (t or f). This also means that in A B and C, []p is
true, hence we can also say that in all worlds [][]p. (And indeed [][][]p
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?
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.
So that seems to disprove it, because C is in its own little multiverse.
There's nothing in the definition that says ARB and BRC entails CRA or CRB,
is there?
Unless I have the "trivially true" thing wrong...
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