On 08 Mar 2014, at 06:20, LizR wrote:
On 6 March 2014 22:06, Bruno Marchal <marc...@ulb.ac.be> 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?
That would contradict Kripke semantics. it says that []p is true in
alpha IFF p is true in all worlds accessible from alpha (and in this
case alpha does not access to itself, and so the falsity of p in alpha
does not entails ~[]p is true in alpha. Indeed, as p is true in all
beta such that alpha R beta, we do have []p true in alpha.
I think you were just thinking with Leibniz semantics.
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.
OK :)
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.
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)
You lost me, here. You suppose R transitive, and I guess you are
trying to prove that []p -> [][]p has to be true in all the worlds A,
B and C, and this for any valuations V.
It is simpler to assume that you have a counter-example (a world in
which []p -> [][]p is false), and get a contradiction from that (by
absurdum).
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.
In your little structure, but is it clear if that is preserved in all
transitive multiverses?
And that implies [][]p for all worlds accessible from A, including C
(trivially).
Isn't that what I was trying to prove?
Not sure. A bit fuzzy. The question is more "are you convinced
yourself by your reasoning?".
Or have I just wandered off into a cul-de-sac myself?
No worry. It is very good that you seem aware you have not yet make a
proof.
More on this in the sequel.
Bruno
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