On 08 Mar 2014, at 06:32, LizR wrote:
On 6 March 2014 22:06, Bruno Marchal <[email protected]> wrote:
Liz, meanwhile you might try this one, which is a bit more easy than
the transitivity case:
Show that (W,R) respects []A -> <>A if and only if R is ideal.
(I remind you that R is ideal means that there is no cul-de-sac
world at all in (W,R)).
OK, I consult my diary and...
Ideal is as you say, yes! :-)
Excellent :)
So []A -> <>A means that A is some proposition universally true in
an illuminated, accessible multiverse, and this implies that A is
possible in that multiverse.
Not at all, and you know that, as you show below.
Hang on I must be missing something.
OK, I hang on.
That seems trivially obvious! Maybe you could point out what I've
misunderstood here...
It is not "misunderstanding", it is precipitation.
Let me try again.
OK.
[]p means that for any world alpha, p is true in all worlds
accessible from alpha.
That is much better.
(Doesn't it? Well if p is a proposition, which might be 'x is false'
then that seems reasonable).
You lost me here, but it looks like non relevant, even for you.
And <>p means that, ah, ~[]~p iirc. Which is to say it isn't true
that there is a world accessible from alpha in which ~p.
~[]~p means that it is not true that ~p is true in all worlds
accessible from alpha. That is, it means that there is a world beta,
with alpha R beta, such that beta verifies p. <>p true in alpha = I
can access, from alpha, to a world where p is true.
But isn't that implied by []p?
You fall again back in Leibniz. May be I should have started from
Kripke immediately.
Have you hang on, in your toilet, the fundamentals two Kripke
principles?
*************************************************************************************************
*
*
* []p is true in alpha = For all beta such that (alpha R beta) we
have beta verifies p. *
*
*
* <>p is true in alpha = There is a beta verifying p such that
(alpha R beta) *
*
*
*************************************************************************************************
I must have a definition wrong somewhere.
Correct.
Do you see that (W, R) is reflexive entails that (W,R) is ideal?
If all worlds access to themselves, no world can be a cul-de-sac
world, as a cul-de-sac world don't access to any world, including
themselves.
Reflexive is alpha R alpha for all alpha, so no cul de sac is
possible.
Correct.
More precision later, notably for the transitive case.
Bruno
http://iridia.ulb.ac.be/~marchal/
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