On 2/14/2014 1:10 AM, Bruno Marchal wrote:
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.
I was merely using teleportation as an example to illustrate that "possible" is a relative
concept depending on the accessiblity relation. What does "possible in principle" mean?
Does it only mean "not self contradictory"? Does it mean consistent with our best
understanding of physics? Lawrence Krauss discusses the possibility in his book "The
Physics of Star Trek". He estimates that it would take more energy than available in the
Milky Way just to obtain the information to teleport a human being. Of course putting
that much mass/energy in the vicinity of the human being would create a black hole. So
what does "possible" really mean?
Brent
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
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