On 04 Oct 2011, at 01:00, Russell Standish wrote:
On Mon, Oct 03, 2011 at 05:31:21PM +0200, Bruno Marchal wrote:
The states are countable, but not the (3-)states + the neighborhhood
of (infinite) computations that you are mentioning yourselves.
Not sure if I see where is the problem. It seems that you have
answered it. The 1-OMs *are* set of histories, but with a particular
3-state, single out in the indexical way, and which will play the
role of the "Bp". The "& p" will force the logic of the
computational extensions to be different.
The way I was talking about it, there is a 1:1 correspondence between
the 3-states and the sets of histories making up the 1-OM. In that
case the cardinality of 1-OM is the same as that of the 3-states -
which you have already admitted is countable.
OK, I see your point. You are right on this, and I should have
perhapssaid "set of sets of histories". This is related to the
possible semantics of the first person logics (S4Grz1).
The 1-OMs are mutiplied by the computations going through it, making
it as great as the continuum of those computations going through, and
that can be understood intuitively by UDA-like reasoning. This come
from the rule Y = I. To have the measure on the 1-OMs, we have to
count the computations going through, not the states themselves. So an
1-OMs can be defined just by one computations (perhaps infinite), but
this does not entirely work, and that is why we have to take into
account the structure imposed by the logic to which the first person
obeys.
Perhaps I'm missing something? I don't quite get the "indexical" bit
for instance.
Examples of "indexical" are terms like "now, me, here, I, there".
Their meaning or referent depends on the locutor or of its current
(also an indexical) situation.
The 3-I of a machine, is an indexical, rather well handled by the
description of the machine as handled by the machine, like with
Gödel's beweisbar "B". The precise (arithmetical contento of "B"
varies from one machine to another, but if the machine verifies some
conditions (rich, ideally correct, etc.) it obeys the same modal
logics (G, G*).
The "1-I" is similarly well captured by the conjunction of the 3-I and
truth (Bp & p). This is a drastic change, because that "I" is no
describable by the machine, it is *not* arithmetical, and it changes
the logic of self-reference (which becomes a logic of evolving states).
Sorry for having been unclear, but I continue to doubt about the
relevance of the term OMs terms. It did mislead me in my answer too
you, and I should stick on the person-views and their modalities.
Ask for any clarification. Nothing is really simple here. I will add
some info in my reply to Brent.
Bruno
http://iridia.ulb.ac.be/~marchal/
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