On 06 Feb 2014, at 21:29, meekerdb wrote:
On 2/6/2014 12:14 PM, Bruno Marchal wrote:
In Kripke semantic all statements are relativized to the world you
are in. []A can be true in some world and false in another. The
meaning of "[]" is restricted, for each world, to the world they
can access (through the accessibility relation available in the
Kripke multiverse).
[]A still keep a meaning, but only in each world. So everything is
said when we define the new meaning of "[]" by the rule
[]A is true in alpha, by definition, means that A is true in all
world beta *accessible* from alpha.
And
<>A is true in alpha iff there is a world beta; where A is true,
accessible from alpha.
Suppose A is true in alpha,
OK. Nice.
but alpha is not accessible from alpha
OK.
and A is not true in any other world accessible from alpha.
OK.
Does it follow that <>A is not true in alpha?
Yes. That does follow.
How frustrating!
A is true, but not possible.
How could that makes sense?
Well, this does not make sense ... in the Leibnizian multiverse. For
sure.
I don't see the point allowing that worlds may not be accesible from
themselves? Does that have some application?
Yes.
First you prove to everybody that I can see in the future, as I
announced yesterday the discovery of a Kripke multiverse violating the
law []A -> A.
You just did.
Well, in alpha, to be sure, []A -> A is true (OK?), but []~A -> ~A is
falsified, as []~A is true (~A is true in all accessible world from
alpha), and ~A is false in alpha, as A is true is true in alpha, and
worlds obeys CPL).
That amounts to the same, as the laws do not depend on the valuation.
If []A -> A is a law, []~A -> ~A should follow.
Note that []~A -> ~A, is equivalent with (contraposition, double
negation): ~~A -> ~[]~A = A -> <>A
A -> <>A is the dual formulation of []A -> A.
As law, they are equivalent. But as formula in one world, they can
oppose to each other.
So you did find a Kripke multiverse violating the *law* []A -> A.
And you did find the culprit: those bizarre world which does not
access to themselves.
Does that have some application?
Yes.
1) An easy one, which plays some role in what I like to call the
simplest buddhist theory of life ever!
And that theory is a subtheory of G, and so will stay with us.
That theory models life by worlds accessibility.
To be alive at alpha means that <>t is true in alpha. It means that
there is, at least, one world accessible from alpha.
To die at alpha means that <>t is false in alpha. But t is true in
alpha, as t is true in all worlds, so the only way to have <>t false,
is that there are no accessible worlds from alpha, at all, including
itself.
That makes alpha into a cul-de-sac world.
So in Kripke semantics, ~<>t, or equivalently []f, characterizes the
cul-de-sac world.
Then the simplest buddhist theory of life ever is just the statement,
If you are alive, then you can die. It means that for all worlds alpha
where you are alive (<>t is true), you can access to a cul-de-sac world.
It means that everywhere, in all worlds we <>t -> <>[]f, or
equivalently <>t -> ~[]<>t.
2) If you interpret <>t by intelligent, and []f by stupid, you get
with the same multiverse, my general theory of intelligence and
stupidity.
3) if you interpret [] by provability (in PA, or in ZF), again, <>t ->
~[]<>t is a law. Read: if I am consistent, then I can't prove that I
am consistent.
It is easy to see that the law <>t -> ~[]<>t is a direct consequence
of the formula of Löb []([]A -> A) -> []A.
Just put t in place of A, and keep in mind that A -> f is just ~A, and
then contra-pose:
[]([]A -> A) -> []A
[]([]f -> f) -> []f
[](~[]f) -> []f
~[]f -> ~[](~[]f)
<>t -> ~[]<>t
The worlds in the Kripke mutiverse characterizing G are like that,
they don't access to themselves.
[]A-> A is not an arithmetical law from the 3p self-referential view
of the machine, but that is why the Theaetetus idea is applicable and
will give the non trivial S4Grz for the knower, or first person, fro
which []A -> A is indispensable.
Some might be astonished that []f is true in a cul-de-sac world. But
kripe semantics say that []f is true in alpha then f is true in all
accessible worlds from alpha.
This really means (for all beta): (alpha R beta) -> (beta satisfy
f).
But (alpha R beta) is always false, and (beta satisfy f) is always
false, so (alpha R beta) -> (beta satisfy f).
OK?
Bruno
Brent
--
You received this message because you are subscribed to the Google
Groups "Everything List" group.
To unsubscribe from this group and stop receiving emails from it,
send an email to [email protected].
To post to this group, send email to [email protected].
Visit this group at http://groups.google.com/group/everything-list.
For more options, visit https://groups.google.com/groups/opt_out.
http://iridia.ulb.ac.be/~marchal/
--
You received this message because you are subscribed to the Google Groups
"Everything List" group.
To unsubscribe from this group and stop receiving emails from it, send an email
to [email protected].
To post to this group, send email to [email protected].
Visit this group at http://groups.google.com/group/everything-list.
For more options, visit https://groups.google.com/groups/opt_out.
