On 2/7/2014 10:40 AM, Bruno Marchal wrote:
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?
Dunno. I'll have to think about it. One thing I find puzzling is that "accessible" seems
ill defined. I have an intuitive grasp of what "possible" and "necessary" mean. And I
know what "provable" means. But my intuitive idea of "accessible" say every world should
be accessible from itself. Logic is about formal relations of sentences so I understand
that "accessible" will have different applications, but what are some examples? Is
Robinson arithmetic accessible from Peano? Is ZFC accessible from arithmetic?
Brent
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