Hi Bruno Marchal 

Could you explain a little about Bp & p duality ? Are they both
analytic, or does one of them us synthetic logic ?

Roger Clough, rclo...@verizon.net
Leibniz would say, "If there's no God, we'd have to invent him so everything 
could function."
----- Receiving the following content ----- 
From: Bruno Marchal 
Receiver: everything-list 
Time: 2012-08-23, 14:17:50
Subject: Re: Male Proof and female acceptance of proof

On 21 Aug 2012, at 21:42, Stephen P. King wrote:

On 8/21/2012 2:28 PM, Bruno Marchal wrote:

On 21 Aug 2012, at 12:12, Roger Clough wrote:

Hi Bruno and Stephen, 

This is the bicameral mind again. Right brain must accept left brain decisions 
for human safety.

Ought must rule over is (or else we'd all be nazis, Hume, for the safety of 
Passion must rule over reason (or else we'd all be nazis, Hume, for the safety 
of humanity)
Acceptace of proof dominates proof (common sense psychology)

Thus you can objectively, mathematically prove that 2+2=4, but you still have 
to subjectively accept that psychologically.
Woman always gets the last word.

No problem here. That fits nicely with the Bp versus Bp & p duality, which is 
just the difference between "rational belief" and "rational knowledge" (true 
rational belief).

It took time to realize that when we define the rational belief by formal 
proof, which makes sense in the ideal correct machine case, although knowledge 
and belief have the same content (the same arithmetical p are believed), still, 
they obey to different logics. This is a consequence of incompleteness. 
Rational beliefs obey to a modal logic known as G (or GL, Prl, K4W, etc.) and 
true rational belief obeys to a logic of knowledge (S4), indeed known as S4Grz. 

G is 

[](p -> q) -> ([]p -> []q)
[]p -> [][]p
[]([]p -> p) -> []p

with the rules A, A->B  /  B and A / []A

S4Grz is

[](p -> q) -> ([]p -> []q)
[]p -> [][]p
[]([](p -> []p) -> p) -> p

with the rules A, A->B  /  B and A / []A


Dear Bruno,

    It might help us immensely if you could tell us how to read these symbolic 
representations. Not all of us speak that language! There are English words for 
all of these symbols!


The only differences with elementary propositional logic are that we have one 
symbol more, the box "[]", and one more inference rule. 

It is a unary operator symbol, so if X is a formula, []X is a formula, like ~X.

The inference rule is that you can derive []p from p. Careful, this does not 
make p -> []p true in most modal logic.

I wrote often the box [] by using the letter B.

In the axiom above, it is better to not interpret the box, as this can confuse 
with the representation theorem which associate "meaning" mathematically.

I have often talked about Bp and Bp & p, with Bp having the arithmetical 
provability meaning (G?el 1931).
G above is the logic of G?el's beweisbar predicate. For example the second 
incompleteness theorem is given by Dt -> ~BDt, or <>t -> ~[]<>t, or 
consistent('t') -> NOT PROVABLE (CONSISTENT 't')), with for example t = "0=0", 
et 't' = G?el number of "0=0".

S4Grz above is the corresponding logic of the first person associated to the 
machine, given by beweisbar('p') & p, following Theatetus, and then Boolos, 
Goldblatt, Artemov. I have provided many explanations on this list, including 
an introduction to modal logic and the Kripke semantics, but you can also open 
some book in logic to help yourself.

G and S4Grz are the two machineries illustrating (and formalizing completely at 
the propositional modal) two important arithmetical hypostases discovered by 
the UM when looking inward. G is the logic of third person self-reference and 
S4Grz is the logic of the first person self-reference.

There are six other hypostases, or machine's points of view, three of them 
playing a role in the "creation of the collective persistent matter 
hallucination. Comp makes obligatory that persistence, and it can be tested, 
and it can be argued that the presence of p -> []<>p as a theorem in SGrz1 and 
Z1* and X1* confirms it in great part. Interactions can be defined in a manner 
similar to Girard, and then tested on those "material hypostases". I think that 
this is explained in the second part of the sane04 paper.
The "1" added to the system refers to the fact that we eventually limit the 
arithmetical translation of the sentence letters (p, q, r, ...) to the sigma_1 
sentences, which "models" the UD in arithmetic.

In particular Richard Ruquist's theory that fundamental physics is given by 
string theory becomes testable with respect to comp, as UDA shows that the 
physics is entirely retrievable from the S4Grz1, Z1* and/or X1*, and their 
first order modal extension. 

It is not as difficult as most paper your refer to, and it is only one paper, 
and you got the chance to ask any question to the author :)

You recently allude to a disagreement between us, but I (meta)disagree with 
such an idea: I use the scientific method, which means that you cannot disagree 
with me without showing a precise flaw at some step in the reasoning.

You seem to follow the seven first steps, so that in particular you grasp 
apparently that COMP + ROBUST-UNIVERSE entails the reversal physics/arithmetic, 
and the explanation why qualia and quanta separate. Are you sure you got this? 
Step 8 just eliminates the "ROBUST-UNIVERSE" assumption in step 7.

Then AUDA translates everything in UDA in terms of numbers and sequences of 
numbers, making the "body problem" into a problem of arithmetic. It is 
literally an infinite interview with the universal machine, made finite thanks 
to the modal logic above, and thanks to the Solovay arithmetical completeness 

You cannot both claim that there is a flaw, and at the same time invoke your 
dyslexia to justify you don't do the technical work to present it. 



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