On 08 May 2014, at 00:35, LizR wrote:
On 6 May 2014 06:54, Bruno Marchal <[email protected]> wrote:
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.
OK...
*** (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.
Given the original assumptions, and assuming there are no
undiscovered errors in the steps - yes.
Fair enough (that is true for all reasoning, note). It is up to you to
convince you that there is no fatal error, and to improve in case you
find a non fatal error.
(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?
(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).
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
Typical normal theories are KT, KT4 (known as S4), KT45 (known as S5,
the Leibnizian theory), KTB (the Brouwersche system), KD (deontic
logic).
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
So G proves R2(R1(A) iff QL proves A.
By Solovay, the machine proves R3(R2(R1(A)) iff QL proves A. And
R3(R2(R1(A))) is the translation of the probability or uncertainty
measure defined in arithmetic.
Restricting it to sigma_1 proposition gives the logic of the
observable for machines, and by the representation theorem above it
obeys a quantum logic.
There is more to say, as Solovay theorem gives also G*, but that one
is NOT a normal modal logic.
Liz: only one exercise: print this, and put this in your diary, and
meditate on it. I know you have more ease with semantics than with
deduction. I will provide example next week (busy end of week). The
deductive part is really the "machine" functioning.
I use also the fact that computability (an absolute notion if you
agree with the Church-Turing notion) is a particular case of
provability (indeed sigma_1 provability).
It is interdisciplinary, so you need quantum mechanics, intutionistic
mathematics, provability theory, computability theory, cognitive
science, knowledge theory, etc. But by staying at the propositional
level, all this tunes into a not so big sets of modal logics and
representation theorems. The difficulty is only in knowing a bit of
logic, beyond the "easy" First indeterminacy notion.
Bon courage, and thanks for your effort. Don't feel discouraged by the
amount of work. It is only that: lot of works. If you get well the
UDA, that is already the main thing, or the main question. The math
part is obviously tricky, due to that interdisciplinary aspects and
the use of logics to manage the many needed bridges.
Bruno
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.
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