# Re: COMP, Quantum Logic and Gleason's Theorem

From: Bruno Marchal <marchal.domain.name.hidden>
Date: Tue, 27 Jan 2009 18:58:56 +0100

Günther,

AUDA is based on the self-reference logic of axiomatizable or
recursively enumerable theories, of machine. Those machines or
theories must be rich enough. In practice this means their theorems or
beliefs are close for induction.This is the work of Gödel and
followers, notably Löb, who found a nice generalzation of Gödel's
theorem and Solovay who proves the arithmetical completeness of the
logic he will call G and G'. Here is the key paper:

Solovay, R. M. (1976). Provability Interpretation of Modal Logic.
Israel Journal of Mathematics, 25:287-304.

I follow Boolos 1979 and Smullyan "Forever Undecided" in calling such
system G and G*. G has got many names K4W, PrL, GL.
There are four excellent books on this subject:

Boolos, G. (1979). The unprovability of consistency. Cambridge
University Press, London.

This is the oldest book. Probably the best for AUDA. And (very "lucky"
event) it has been reedited in paperback recently; I ordered it, and I
got it today (o frabjous day! Callooh! Callay! :). It contains a
chapter on the S4 intensional variant of G, and the theorem (in my
notation) that S4Grz = S4Grz*. The first person is the same from the
divine (true) view and the terrestrial (provable) view.
I have it now in three exemplars but two are wandering.

Boolos, G. (1993). The Logic of Provability. Cambridge University
Press, Cambridge.

This is the sequel, with the Russians' solutions to virtually all open
problems in Boolos 1979. The main problem was the question of the
axiomatizability of the first-order extension of G and G* (which I
note sometimes qG and qG*). And the answers, completely detailed in
Boolos' book, are as negative as they can possibly be. qG is PI_2
complete, and qG* is PI_1 complete *in* the Arithmetical Truth. The
divine intelligible of Peano Arithmetic is far more complex than Peano
Arithmetic's ONE, or God, in the arithmetical interpretation of
Plotinus.

Smoryński, P. (1985). Self-Reference and Modal Logic. Springer Verlag,
New York.

I have abandon this one sometimes ago, because of my eyes sight
defect, but with spectacles I have been able to distinguish tobacco
product from indices in formula, and by many tokens, it could be very
well suited for AUDA. The reason is that it develops the theory in
term of (computable) function instead of assertions, showing directly
the relation between computability and SIGMA_1 provability. Nice intro
from Hilbert's program to Gödel and Löb's theorem, and the Hilbert
Bernays versus Löb derivability conditions. It contains a chapter, a
bit too much blazed in the tone, on the algebraic approach to self-
reference, which indeed initiates originally the field in Italy
(Roberto Magari).
It contains also chapter on the Rosser intensional variants.

Smullyan, R. (1987). Forever Undecided. Knopf, New York.

This is a recreative introduction to the modal logic G. I was used
some times ago in this list to refer to that book by FU, and I don't
hesitate to use some of Smullyan's trick to ease the way toward self-
reference. It helps some, but can irritate others.
Note that Smullyan wrote *many* technical books around mathematical
self-reference, Gödel's theorems in many systems.

Modal logic is not so well known that such book can presuppose it, and
all those books introduce modal logic in a rather gentle way. But all
those books presuppose some familiarity with logic. Boolos Et Al. is
OK. It is difficult to choose among many good introduction to Logic.
By some aspect Epstein and Carnielly is very good too for our purpose.

Note that the original papers are readable (in this field). All this
for people who does not suffer from math anxiety which reminds me I
have to cure Kim soon or later. The seventh step requires some math.
AUDA requires to understand that those math are accessible to all
universal machine 'grasping" the induction principle, this is the work
of Gödel and Al.

I think the book by Rogers is also fundamental. Cutland's book is
nice, but it omits the study of the Arithmetical Hierarchy (SIGMA_0,
PI_0, SIGMA_1, PI_1, SIGMA_2, PI_2, ...).

AUDA without math = Plotinus (or Ibn Arabi or any serious and rational
mystic). Roughly speaking.

I will think about a layman explanation of AUDA without math, and
different from UDA.

Best regards,

Bruno

On 25 Jan 2009, at 18:45, Günther Greindl wrote:

>
> Hi Bruno,
>
>>> Goldblatt, Mathematics of Modality
>> Note that it is advanced stuff for people familiarized with
>> mathematical logic (it presupposes Mendelson's book, or Boolos &
>> Jeffrey).
>>
>> Two papers in that book are "part" of AUDA: the UDA explain to the
>> universal machine, and her opinion on the matter.
>
> I would like to add a "guide to AUDA" section on the resources page.
> Maybe you could specify the core references necessary for
> understanding
> the AUDA (if you like and have the time)?
>
> Here a first suggestion of what I am thinking of:
>
> Boolos Et Al. Computability and Logic. 2002. 4th Edition
>
> Chellas. Modal Logic. 1980.
>
> Goldblatt, Semantic Analysis of Orthologic and
> Arithmetical Necessity, Provability and Intuitionistic Logic
> to be found in Goldblatt, Mathematics of Modality. 1993.
>
>
>
> What do you think?
>
> Best Wishes,
> Günther
>
> >

http://iridia.ulb.ac.be/~marchal/

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Received on Tue Jan 27 2009 - 12:59:18 PST

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