On 13 Feb 2014, at 19:34, meekerdb wrote:
On 2/13/2014 1:10 AM, Bruno Marchal wrote:
What's the definition of G*?
G* is a quite peculiar modal logic. It has as axioms all the
theorem of G, + the axiom:
[]A -> A
But is NOT close for the necessitation rule (can you see why that
is impossible). This entails that G* has no Kripke semantics. But
it has some semantics in term of infinite sequence of G-multiverse.
By Solovay second theorem, G* axiomatizes what is true on the
machine. Not just what is provable by the machine.
G* minus G is not empty (it contains <>t, <><>t, <><><>t, ... for
example), and it axiomatizes the true but non provable modal
(provability) sentences.
It seems that the notation is inadequate since it depends on the
accesibility relation: For example if the accessibility relation
is T (for teleportation) then <T>M and <T>W may be false in Helsinki
Why.
Because teleportation isn't possible (so far as we know).
? Comp implies the possibility in principle of classical
teleportation, (UDA step 1). We don't need more.
Which brings up another point that bothers me: We are using [] as an
operator "necessary", and <> as "possible" as just symbols with a
defined syntax, but in application we must say what they mean. What
is necessary and what is possible are dependent on context; just as
above you casually assume that teleportation is possible - even
though you well know it isn't - just because you can write <T>.
This is similar to my complaint about arithmetical realism; it is a
sort of logical realism.
I use "[] and "<>" usually when I explain modal logic, through many
examples of different modal systems.
In the translation of UDA in arithmetic, all modalities are defined in
term of the provability predicate, that is the Gödel's Beweisbar.
What is necessary or possible depends on the worlds, yes, that is what
Kripke is all about.
All I explain is based on the fact that teleportation is possible
*theoretically*. Yes.
That it is hard to do in practice is not relevant. You could stop at
step 0, because the artificial brain is also impossible in practice
today. But it is not relevant.
We assume comp. They are both true, as H T M and H T W, if
teleportation is the accessibility relation.
while using F (for flying) would make <F>M and <F>W true.
OK, but it is the same with T.
No it's not. I can fly to Moscow.
By definition of the protocol in step three. If not you should have
made such remark at step 0, and just say no to the doctor. You just
say non-comp (even in theory).
The practicality of teleportation is not relevant for the theoretical
proof.
Bruno
so in the "eye of God", nothing changes.
But G, which represents the machine ability, does not prove that
equivalence, and this entails that []p and []p & <>t will obeys
different logics.
OK?
I'm not sure what you mean by "obey different logics"?
I meant different modal logics. It just means that they have
different theorems. They are different theories. For example G
proves []([]p ->p) -> []p, but Z and X does not prove that. Z
proves <><>A for all A, but G does not prove that. S4Grz proves []p
-> p, but G does not prove that. S4Grz proves []([]p ->p), but G
does not prove that, etc.
OK.
Brent
By incompleteness, despite G* proves the equivalence of []p, []p &
p, []p & <>t, are equivalent, as G cannot prove that equivalence,
they obeys different logic. They have different theorems. They are
different theories, and that's why we have 8 different hypostases.
That's how we got a theory of knowledge, a theory of observation,
etc, all based on the same arithmetyical truth. That corresponds to
the different "person points of view". You get the 1p view by the
"& p" constraints, and the matter by the "& p or & <>t"
constraints, and the non communicable parts, by the passage x to x*
for each logic x.
Bruno
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