On 09 May 2014, at 02:22, LizR wrote:
On 9 May 2014 05:07, Bruno Marchal <[email protected]> wrote:
On 08 May 2014, at 00:35, LizR wrote:
(By the way I think Max Tegmark does a good job of explaining what
this means in his book, even if he doesn't get to the reversal. He
says you need to assume a "capsule theory of memory" and talks about
observer moments a lot, which hammers home the point that any given
OM could be instantiated in a computer, in a Boltzmann brain, in
arithmetic etc.
In arithmetic?
Or Platonia, or wherever it is the UD lives.
OK. Note that the UD "lives" in a tiny part of arithmetic. What I call
the sigma_1 complete part. We will of course come back to this.
(to me the term OM is ambiguous, and people confuses easily first
person OM, which I think don't really exist or make sense, and third
person OM, which are "just" relative computational state).
OK, well this could get confusing then! I forget how Max Tegmark
defines an OM in the book. I think probably fairly informally. It
seems to me that if consciousness is somehow produced by computation
then it has to have states and steps between them, and an OM might
be a state. However I admit I don't know what an OM is, and some
might say the brain operates on a 1/10th of a second cycle or
similar, and hence an OM is quite a long time compared to the
possible underlying computational steps.
OK. "OM" is a term introduced by Nick Bostrom, and is (it seems to be)
an 1p-notion. But by the abandon of the consciousness-brain identity
thesis, that becomes a non trivial notion (which justifies their
handling by the modal logics).
I think if one doesn't get that point then the steps seem a lot
less intuitive. I just mention this FWIW. I haven't quite finished
his book yet but in places it seems like (Max's version of ) comp
for dummies, which is probably the right level for me)
Nice.
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.
OK, this is what you lost me! What are G and G* again? (Sorry!)
Well, we have seen some modal formula, and their Kripke semantic.
But logicians are not interested in only formula, but in theories,
and notably modal logicians study modal theories.
Classical modal logicians, as you can guess study classical modal
theories, and this means that they study modal theories (given by
some modal formulas) extending classical propositional logic.
If you remember, I gave you an axiomatization of classical
propositional logic. It was a finite set of classical propositional
formula, and inference rules, such that all classical tautologies
are provable from them. it was the following system.
It is the infinities of formula having the following shapes with A
for any formal formula (that is p, s, r, ... and their boolean
combinations):
(A -> (B -> A))
((A -> (B -> C)) -> ((A -> B) -> (A -> C)))
(((~B) -> (~A))) -> (((~B) -> A) -> B))
You can see, by using the truth table semantic that those are
tautologies.
The inference rule (the only means beyond substitution of A, B, C
for formal formula (like p, q, (p & (p V q)), (p -> (p -> p)), ..),
is the modus ponens rule.
From A and (A -> B) you can derive B.
I gave you a proof of (A -> A) as an example, and important theorem.
I hope you can see that the inference rules preserves
tautologicalness. If A is a tautology (true in all valuations of the
atomic formula, p, q, r, ...; has only "one" in its truth table),
and if (A -> B) is a tautology, then B is a tautology. OK?
And that theory can be shown to be complete. All tautologies are
provable.
The key notion here is "provable". "A is provable" means that there
is a sequence of formulas, which are either axioms, or are formula
derived from the modus ponens rule from preceding formula in the
sequence, ending to A.
Well, it is exactly the same for the modal theories, except that we
allow modal formulas.
Let me remind you that the following formula, called K, for Kripke,
is respected in all multiverses, where a multiverse is a set of
"worlds", together with a binary relation, called accessibility
relation. I leave the outer parentheses for reason or readibility.
K is [](A -> B) -> ([]A -> []B)
The proof is simple. If there is a world alpha with both [](A -> B)
and []A, but with []B false, contradicting K,
then in all beta such that (alpha R beta) you would have (A -> B)
and A, but ~B, which is impossible (the world obeys classical logic).
So, all theories having a Kripke semantics have K has axiom. All the
modal logical theories which will interest us (to derive phsyics and
theology from machine's self-reference) will extend that theory.
But note this: imagine that A is respected in a multiverse (or just
satisfied by one illuminated multiverse, its atomic formula are
valued on 1 or 0), then A is true in all worlds, but then []A has to
be true in all worlds too!
This means that the following rule of inference:
from A derive []A
will preserve the modal tautologicalness (satisfaction in
illuminated multiverse, respect by multiverse). And so is a sound
rule of inference.
So all modal theories (having Kripke semantics) have the two
following rules of inference:
Modus ponens: from A and (A -> B) you can derive B.
and
Necessitation rule: from A, you can derive []A.
Definition: I will say that a modal theory is *normal*, if it has
the axiom K, that is [](A -> B) -> ([]A -> []B), and if has the
modus ponens inference rule and the necessitation inference rule.
Let us give names to the formula so that we can describe a normal
theory in a simple way, without retyping complex formula:
K = [](A -> B) -> ([]A -> []B)
T = []A -> A
4 = []A -> [][]A
5 = <>A -> []<>A
C = <>A -> ~[]<>A
B = A -> []<>A
L = []([]A -> A) -> []A
Grz = []([](A -> []A) -> A) -> A
All OK so far.
Good.
Typical normal theories are KT, KT4 (known as S4), KT45 (known as
S5, the Leibnizian theory), KTB (the Brouwersche system), KD
(deontic logic).
What is D?
Oops. I knew I was forgetting some sentence. D is []A -> <>A. You have
already shown that it is chracterize by the ideal multiverse. Those
are without cul-de-sac worlds.
Now, I answer your question:
G is KL = KL4
So G is the theory with axiom
[](A -> B) -> ([]A -> []B)
[]([]A -> A) -> []A
(4 is derivable from that theory, I might show you this one day. The
Kripke semantics of G is slightly more complex, and two different
semantics are often used. In fact G characterize the finite
multiverse with an irreflexive transitive accessibility relations),
but also the class of multiverse without any infinite path a R b, b
R c, c R d, .... (those are variable for worlds and can be equal,
and so this entails irreflexivity).
Why is G so important?
By Solovay completeness theorem, G provides the logic of machine
self-reference. It characterizes the logic of what a machine, rich
enough to understand how a machine work, and classical logic, can
prove about its own provability ability.
G is the logic of arithmetically sound machine provability and its
dual, consistency.
Cf the "crux of the matter", the Gödel translation of "provable" in
arithmetic.
Let me sum up the sequel, and the representation theorems, in a
roughly non rigorous way, to avoid details and give the idea.
QL is for Quantum logic. It is the logic of quantum mechanics, with
the atomic formula being interpreted by rays in Hilbert spaces and,
arbitrary propositions by subpaces of an Hilbert space.
B proves R(A) iff QL proves A (where B is the normal theory KTB)
G proves R2(A) iff B proves A
Now I'm lost again. What is R, R2? (etc ... I see more of these
below but I will snip the rest to avoid my post having a long tail...)
They are the translation or representation of one modal logic into
another. They correspond to the Theaetetus' definition:
[]p ======> []p & p
or
[]p ======> []p & <>p & p
or the sigma_1 restriction, and then known theorem between modal logic
and intutionistic logic, and/or quantum logic(s).
More on this later.
I made first those translations, and then found the representation
theorems. I have not the time today, but in the next days I will give
you the precise representation links.
PS I wrote this thursday morning, but I can't send any mail.
Apparently this time my computer is not involved, looks like the
provider has some problem. 18h52 Hurray! I heard the mail sending. I
send this one, as I have to go.
I hope you get your mail sorted out soon (for both our sakes!)
Apparently we have to pray! Things seems OK, for now.
Bruno
http://iridia.ulb.ac.be/~marchal/
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