Craig, Liz, Brent and/or anyone interested,
Again, it is just an attempt. Take it easy. I have to train myself,
and a bit yourself.
You might tell me if this helps, if only a little bit.
*** (it is also a second attempt to send this mail, as it looks my
emailer has some problem)
If you understand UDA step 1-7, normally you understand that physics
becomes:
1) a measure on computations,
2) when seen from some first person perspective.
Now, the machine which believe/assume elementary arithmetic (and
perhaps more as long as they remain arithmetically sound), will have
its rational believability notion acquires a non trivial modal logic,
known today as G (or GL, PrL, KW4, it has some story so get many names).
Actually the machines acquire a couple of logics: G and G*. G is the
logic of provability (believability) that the machine can believe, and
G* is the logic of believability (that the machine can believe or not,
and indeed G* extends properly G.
Technically:
Solovay's first theorem is that G proves A iff Peano Arithmetic proves
the natural translation of A in arithmetic, where the natural
transformation interpret the atomic p by arithmetical sentences,
preserves the boolean relations, and translate the modal box []A by
the arithmetical translation of provability in arithmetic, Gödel's
beweisbar predicate, applied to the transformation of A, that is
beweisbar(transformation of A)).
Solovay's second theorem is that G* proves A iff A is true (in the
standard model of Peano Arithmetic). G is decidable, and Solovay shows
that G* is representable in G, making G* decidable (at that
propositional level).
For example self-consistency, "I don't prove the false", with "I"
taken in the third person descriptive sense, the machine talks about
that machine, which happens to be itself. It represent more the code
you discuss with the doctor, *supposed* to be at the right
substitution level, than anything like a (conscious) first person
view, still less a "probability" on those views, yet.
To fix the idea I will use Peano Arithmetic, as generic ideally
correct machine, It is equivalent with Robinson arithmetic (which
seen as a machine, is already a Church-Post-Turing Universal Machine)
together with an infinity of induction axioms. For all arithmetical
formula F, PA believes
if F(0) and if for all n we have that (F(n) -> F(n+1)) then we have
the right to conclude that F(n) applies to all n.
Later, we might talk about Analysis, or second order arithmetic, which
admits a much more powerful induction axiom, so powerful that such a
theory is no more a well defined machine (more a set of possible
machines):
for all set S of natural numbers,
If 0 belongs to S, and if ((n belongs to S) implies (n+1 belongs to
S)) then we have the right conclude that all number belongs to S.
The number of subset of N is non enumerable, and this makes such a
belief set non well defined. Which sets are we talking about, what is
a set?
That is the Dedekind theory of the natural numbers, and it has many
interesting weakening in which you can develop the whole semantics of
Peano Arithmetic.
Aparte for Liz and Brent, but useful for Craig.
What we have seen:
Classical propositional logic. Both the proof theory and the
semantics, and the relation between.
Modal logic. Only the semantics. Not yet the proof theory.
And the crux of the matter remains: define the notion of believability
above in the arithmetical language (the thing Gödel basically did in
his 1931 paper).
Craig, we will define the notion of finite sequence (of symbols) in
arithmetic. I will only be able to sketch the basic idea. It is quite
standard, "well known" material in theoretical computer science and
mathematical logic, but I am aware this is not well known by the
general public. When done with all details, it is very long and
tiedous to follow, like programming in assembly language.
And physics?
UDA explains that physics is given by 1) a measure on computation 2)
"seen from inside".
By a theorem by Kleene, computation can be translated by true and
provable Sigma_1 sentences, and the Sigma_1 sentences will play the
role of the universal dovetailing. Such sentences enjoy, I mean
verify, the law
p <-> []p, in the G* minus G logic. That plays some key role.
That's the way comp is translated itself in arithmetic.
But what about the "seen from inside". G provides a rational believer,
but not a knower, which characterizes a first person view.
In modal logic, knowledge is axiomatized by a logic, called T, which
has as main axiom []A -> A. More introspective ability (and totally so
in some sense defined by Smullyan), are given by the formula, badly
named, 4, which is []A -> [][]A. The axiom K + T + 4 gives the theory
of knowledge S4.
The miracle here, is that the simplest and oldest definition of
knowledge, defining it by true belief, provides here a logic different
of G, and indeed a logic extending S4: S4Grz. It is the logic of"
beweisbar('p')-and-p, that is up to the transformation above, the
logic of []p & p.
Then, when restricting this to the sigma_1 sentences, the property
above entails already a quantum logic, like with the other nuances.
Instead of deriving physics from a measure on all computations, we
derive the logic of measure one, from the knowledge/probability
nuances given by Theaetetus on beweisbar. It provides modal logics
which describes an epistemical quantum logics (something which will
have to be explained too).
I might insist you buy the Mendelson book, I can explain if you have
problems. Smullyan's Forever undecided is a recreative introduction to
G.
In that theory, which is really the classical machine's own theory, it
is explained why truth, consciousness cannot be defined, why no one
can present comp as true, or talk like if he was sure of the
substitution level. That theory can be used also to make invalid many
argument against machine's intelligence or machine ability to manifest
a conscious experience.
Self-consciousness can be related to notion around self-consistency,
knowledge and truth.
The theatetus does not provide an *arithmetical* definition of
knowledge and from the first person view, you cannot name or describe
yourself in any third person way. In some precise sense, the 1-you is
not a machine. It is knowingly (by itself) different from all machines.
Apparently the raw universal consciousness might be related with the
fact that p <-> []p belongs to G* minus G (and also in the other
annuli), for the sigma_1 sentences. I am not sure, 'course. It would
be coherent with many account of NDE and brain perturbation reports.
Liz, there is no "r" in this month! Time to put the physicalist hat
down.
Bruno
http://iridia.ulb.ac.be/~marchal/
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