On 07 Feb 2014, at 23:21, meekerdb wrote:
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.
Normally, we will discuss this a lot.
One thing I find puzzling is that "accessible" seems ill defined.
Of course, it means just "binary relation", on some non empty set
called "multiverse" (here).
I have an intuitive grasp of what "possible" and "necessary" mean.
But that is only the alethic modalities. In PA the modal box [] will
represent "provability", and the "worlds" will be non self-accessible
because of Solovay theorem, and the fact that in the Kripke multiverse
in which the modal laws described what PA (or ZF) can prove about its
own provability (the modal logic G), all worlds will have to be non
accessible to themselves, and all worlds have to access cul-de-sac
world everywhere, making the "little buddhist theory" part of the
machine 3p-self-referential discourse (well defined by Gödel 1931, to
reassure John Clark).
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,
OK.
but what are some examples? Is Robinson arithmetic accessible from
Peano? Is ZFC accessible from arithmetic?
RA and ZFC are not worlds, but theories or machines (or relative
numbers). Later I can explain that we *can* (but are not obliged!)
interpret the worlds as maximal consistent set of beliefs, which are
some models of PA (or ZF).
But, this is demanding in math, and could be premature at this stage.
Let us be sure that any one (interested) see correctly the relation
between a (normal) modal logic and Kripke multiverse. WE are not yet
at the stage of giveing the completeness and soundness theorems.
I am busy today, but I will give more explicit definition soon.
Your intuition is normal. You will find may books on modal logic,
which considers that "[]A -> A" is a law. It is the Leibnizian laws
that the modal logicians will abandon with most hesitations ... But
the modal logic of provability of "rich" (Löbian) machine does not
give any choice in that matter. You will see. OK?
Bruno
Brent
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