On 13 Feb 2014, at 04:03, meekerdb wrote:
On 2/12/2014 11:37 AM, Bruno Marchal wrote:
Liz, if Brent don't mind, my answer to Brent here contains a bit on
modal logic, directly related to the machine discourse (and this
will be justified later, as it is not obvious at all).
<snip>
which translates the UDA. the Gödel provability cannot be used
for the UD measure, due to the cul-de-sac worlds. That is why we
need []p & p, or []p & Dt, or []p & Dt & p.
Brent, do you see this?
Are you OK that in a cul-de-sac world we have []A for all A?
I understand that "W is a cul-de-sac world" means there is no world
accessible from W (including W itself), so "A is true in all worlds
accessible from W" is vacuously satisfied.
OK.
But then we also have []~A in W.
OK.
So []A in W doesn't say anything about the truth value of A in W.
That seems like a peculiar formulation.
?
This means only that modal logic is not truth-functional. We already
know that. If alpha R beta (and only beta), and if p is true in beta
and false in alpha, you have []p in alpha, and ~p in alpha. But you
could have p in alpha. []A truth value does not depend on the truth
value of A. We say that modal logic is not truth functional.
I repeat two arguments.
I recall first Kripke semantics:
All the worlds obeys CPL. And there is some fixed binary relation R
on that set of worlds (called "accessibility").
Then,
[]p is true in a world alpha if p is true in all worlds beta such
that alpha R beta
Or equivalently, (and dually):
<>p is true in a world alpha if it exists a world beta with p true
in beta and alpha R beta.
(re-verify that this entails well
<>p = ~[]~p
[]p = ~<>~p
~[]p = <>~p (jump law 1)
~<>p = []~p (jump law 2)
OK?)
Now consider some multiverse with zeta being a cul-de-sac world, like
{alpha, beta, gamma, zeta} with
alpha R beta, beta R gamma, gamma R zeta.
And nothing else. In that multiverse zeta is a cul-de-sac world.
OK?
Proposition. For any proposition A, []A is true in zeta.
Proof.
Imagine that []A is not true in Zeta. Zeta obeys CPL, so if []A is
not true, []A is false. OK? And if []A is false, then
~[]A is true, by classical logic. OK?
But if ~[]A is true, then <>~A is true, by the jump law 1 above. OK?
Then by Kripke semantics above, if <>~A is true in Zeta, it means
that there is a world accessible from Zeta, and in which ~A is true.
But that is impossible, given that Zeta is a culd-de-sac world.
Conclusion: []A cannot be false in Zeta.
But since A is any proposition it is also the case that []~A cannot
be false in Zeta. So while either A or ~A but not both are true in
Zeta, []A and []~A are both true.
Exact. That is why cul-de-sac world shoild be avoided: everything is
"necessary", and nothing is "possible" (when reading the box and
diamond with the alethic sense, which is some abuse, but can be useful
pedagogically).
Summary: []A is true, for any A, in any cul-de-sac world, of any
Kripke multiverse. This is a direct consequence of the jump law: as
[]A can only be false if <>~A is true, and all proposition
beginning by a diamond "<>" are false in a cul-de-sac world.
In particular []f is true in the cul-de-sac worlds. And in fact []f
is false in any non cul-de-sac world. So []f characterizes the cul-
de-sac worlds in Kripke semantics. OK?
definition: I will say that a world is transitory iff it is not
cul-de-sac world.
Now, the G modal logic has curious Kripke multiverse.
What's the definition of the G modal logic?
It is the logic obeyed by Gödel's beweisbar, when provable by the
machine.
By Solovay first theorem it is axiomatized by the axiom:
[]([]A -> A) -> []A)
With CPL (and thus the modus ponens rule), the K axioms [](A -> B) ->
([]A -> []B), and the necessitation rules A / []A.
[]A -> [][]A can be proved in that theory, like <>t -> ~[]<>t
("Gödel's incompleteness theorem").
On Kripke semantics is given by finite irreflexive and transitive
multiverse.
Another one is transitive multiverse without infinite path (a R b, b R
c, c R d, ...). In particular this implies irreflexivity, as a R a
entails a R a, a R a, a R a, etc.
No worlds can ever access to itself, but worse, all worlds access
to some cul-de-sac world. (cf the image "you die at each instant in
comp or in the little buddhist theory).
G proves <>t -> <>[]f. This says, in Kripke semantics, that if I am
in a transitory world, then I can access to a cul-de-sac world.
OK?
So let us come back in reality, and let us consider our common very
small multiverse {Helsinki, Washington, Moscou}, or {H, W, M} to be
shorter.
We are in the protocol of step 3. And suppose we are told that in M
and W, we will have a cup of coffee.
Then we would like to say that
"[](we-will have a cup-of-coffee)"
is true in Helsinki. Ou guardian angel G* told us that <>W and <>M
is true in Helsinki, so it looks like the probability one is well
captured by the modal box/ in all accessible world, I get a cup of
coffee.
But we can't listen to the guardian angel in that way, because <>W
and <>M, although true in H, are not provable by the little finite
creature, and we might be already in a cul-de-sac world, from the
machine's point of view. If we apply G, that is a possible case,
and so, to get a decent probability, we must assume explicitly some
world being accessible. That is the "act of faith" I often
mentionned. This is what we will do by defining probability 1, not by
[]p (in G)
but by []p & <>t
The probability of an "event" is one, if that event occurs in all
accessible worlds AND there is an accessible world.
G* proves []p <-> []p & <>t,
What's the definition of G*?
G* is a quite peculiar modal logic. It has as axioms all the theorem
of G, + the axiom:
[]A -> A
But is NOT close for the necessitation rule (can you see why that is
impossible). This entails that G* has no Kripke semantics. But it has
some semantics in term of infinite sequence of G-multiverse.
By Solovay second theorem, G* axiomatizes what is true on the machine.
Not just what is provable by the machine.
G* minus G is not empty (it contains <>t, <><>t, <><><>t, ... for
example), and it axiomatizes the true but non provable modal
(provability) sentences.
It seems that the notation is inadequate since it depends on the
accesibility relation: For example if the accessibility relation is
T (for teleportation) then <T>M and <T>W may be false in Helsinki
Why. We assume comp. They are both true, as H T M and H T W, if
teleportation is the accessibility relation.
while using F (for flying) would make <F>M and <F>W true.
OK, but it is the same with T.
so in the "eye of God", nothing changes.
But G, which represents the machine ability, does not prove that
equivalence, and this entails that []p and []p & <>t will obeys
different logics.
OK?
I'm not sure what you mean by "obey different logics"?
I meant different modal logics. It just means that they have different
theorems. They are different theories. For example G proves []([]p -
>p) -> []p, but Z and X does not prove that. Z proves <><>A for all
A, but G does not prove that. S4Grz proves []p -> p, but G does not
prove that. S4Grz proves []([]p ->p), but G does not prove that, etc.
By incompleteness, despite G* proves the equivalence of []p, []p & p,
[]p & <>t, are equivalent, as G cannot prove that equivalence, they
obeys different logic. They have different theorems. They are
different theories, and that's why we have 8 different hypostases.
That's how we got a theory of knowledge, a theory of observation, etc,
all based on the same arithmetyical truth. That corresponds to the
different "person points of view". You get the 1p view by the "& p"
constraints, and the matter by the "& p or & <>t" constraints, and the
non communicable parts, by the passage x to x* for each logic x.
Bruno
Brent
This allows him to identify specifically what makes some
computer program conscious: it's the ability to do induction and
diagnoalization and prove Goedel's theorems.
OK. But it is not a computable identification. We cannot
recognize, neither from code, nor from computational activity, is
an entity is Löbian or not.
I think you mean "we cannot *prove*". We can recognize
intelligent behavior and infer Lobian.
No we can't never be sure. We can in some case recognize that a
program computes the function factorial, but given
arbitrary programs and arbitrary computations, we can't necessarily
infer what is computed.
(But well, what you say can be true in some context; I can be OK).
We can just prove non constructively that such programs and
computations exists in a non computable distribution.
My problem with this is that I don't believe in arithmetical
realism in the sense required for this argument.
Then you have to find me two numbers a and b contradicting the
axioms of RA.
I think consciousness depends of consciousness *of* an external
world and thoughts just about Peano's arithmetic is not enough
to realize consciousness and the "ineffable=unprovable"
identification is gratuitous.
This lowers the level only, unless you add something non
computable in the local environment.
There are obvious physical and evolutionary reasons that qualia
would be ineffable. That's why I think step 8 is invalid
because it assumes dreams (of arithmetic?)
Once you accept comp, it is standard computer science to show
that *all* dreams are emulated in Arithmetic.
?? But the argument proposes emulating dreams by a physical (but
inert) computer - not Arithmetic.
It cannot be inert then. It might have inert part, fro some
computations, but that is in the course of the MGA reasoning.
You jump into another difficulties.
That arithmetic emulated all computations is part of standard
computer science.
In step 8, it is shown that IF comp is assumed, it makes no sense
to add anything more than arithmetic at the base level.
are possible independent of any external world - or looked at
another way, I think to make it work would
require that the 'inert' computation
simulate a whole world in which the consciousness would then
exist *relative* to that world.
I guess we will need to come back on step 8, soon or later. Not
sure what you mean by "inert computation"? re you alluding to the
"inert" device in Maudlin and MGA,
Yes.
OK. But I don't see the relation with the thread. You were
assessing Clark on step 3. You might have changed your mind and
jump on step 8, but I suggest we wait everyone grasp steps 1-7,
before looking at the more subtle step 8.
But no problem, we will come back at step 8.
Bruno
Brent
or to the static computations which exist in arithmetic. In that
case it is the usual argument against block-time or block-
universe, and this has been debunked repeatedly. Time and
activity are indexicals (indeed translated into *variants* of G*).
Bruno
Brent
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