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). 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.
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.
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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