On 15 Dec 2017, at 22:19, Brent Meeker wrote:
On 12/15/2017 9:24 AM, Bruno Marchal wrote:
that the statistics of the observable, in arithmetic from inside,
have to "interfere" to make Digital Mechanism making sense in
cognitive science, so MW-appearances is not bizarre at all: it has
to be like that. Eventually, the "negative amplitude of
probability" comes from the self-referential constraints (the logic
of []p & <>p on p sigma_1, for those who have studied a little bit).
Can you explicate this.
Usually, notions like necessity, certainty, probability 1, etc. are
assumed to obey []p -> p. This implies also []~p -> ~p, and thus p ->
<>p, and so, if we have []p -> p, we have [] -> <>p (in classical
normal modal logics).
Then provability, and even more "formal provability" was considered as
as *the* closer notion to knowledge we could hope for, and so it came
as a shock that no ("rich enough") theory can prove its own
consistency. This means for example that neither ZF nor PA can prove
~[]f, that is []f -> f, and so such machine cannot prove generally []p
->, and provability, for them, cannot works as a predicate for
knowledge, and is at most a (hopefully correct) belief.
Now, this makes also possible to retrieve a classical notion of
knowledge, by defining, for all arithmetical proposition p, the
knowledge of p by []p & true(p).
Unfortunately, we cannot define true(p) in arithmetic (Tarski), nor
can we define knowledge at all (Thomason, Scott-Montague). But for
eaxh arithmetical p, we can still mimic knowledge by []p & p, for each
p, and this lead to a way to associate canonically a knower to the
machine-prover. It obeys to a knowledge logic (with []p -> p becoming
trivial). That logic is captured soundly and completely by the logic
S4Grz (already described in many posts).
Similarly, the logic G of arithmetical self-reference cannot be a
logic of probability one, due to the fact that []p does not imply <>p
(which would again contradict incompleteness). It entails in the
Kripke semantics that each world can access to a cult-de-sac world in
which []p is always true, despite there is no worlds accessible to
verify such facts. We get a logic of probability by ensuring that "we
are not in a cul-de-sac world", which is the main default assumption
need in probability calculus. In that case, you can justify, for
example, that when you are duplicated in Washington and Moscow, the
probability of getting a cup of coffee is one, when the protocol
ensure the offering of coffee at both place: []p in that case means "p
is true in all accessible words, and there is at least one".
So, by incompleteness, [] & <>t provides a "probability one" notion,
not reducible to simple provability ([]p).
Then, by step 8, we are in arithmetic (in the model of arithmetic,
"model" in the logician's sense), and we translate computationalism by
restricting the accessible "p" to the leaves of the universal
dovetailing. By Gödel+Church-Turing-Kleene we can represent those
"leaves" by the semi-computable predicates: the sigma_1 sentences.
When we do this, we have to add the axiom "p->[]p" to G. This gives G1
(and G1*). It is enough thanks to a proof by Visser. For the logic of
the nuances brought by incompleteness, like []p & p, and []p & <>t, it
gives the logic S4Grz1 and the logic Z1*. Then, we can extract an
arithmetical interpretation of intuitionist logic from S4 (in a usual
well known way), and, a bit less well known, we can extract a minimal
quantum logic from B, and then from Z1* which is very close to B,
using a "reverse" Goldblatt transform (as Goldblatt showed how the
modal logic B (main axioms []p -> p, p -> []<>p, and NOT= []p -> []
[]p) is a modal version of minimal quantum logic.
Note that here "[] and "<>" are arithmetical predicate. We do not
assume more than Q, and use only internal interpretabilities of the
observer-machines.
This is explained in most of my papers, but the details are in the
long french text "Conscience et Mécanisme".
Bruno
Brent
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