On 31 May 2013, at 19:20, meekerdb wrote:
On 5/31/2013 8:46 AM, Bruno Marchal wrote:
On 30 May 2013, at 21:04, meekerdb wrote:
On 5/30/2013 2:24 AM, Bruno Marchal wrote:
On 29 May 2013, at 20:12, meekerdb wrote:
On 5/29/2013 12:38 AM, Bruno Marchal wrote:
I don't see the analogy. I don't think consciousness can be
negative, or even that it can be measured by one dimension.
"All-or-nothing" would be a function that is either 1 or 0.
The point is more that it is > 0, or 0.
If you can be conscious of red and green, then I'd say you are
more conscious than someone who is red/green colorblind
(albeit by a tiny amount).
That is about consciousness' content. Not on being or not
conscious.
In order to have beliefs about arithmetic requires that you be
conscious of numbers and have a language in which to express
axioms and propositions. I doubt that simpler animals have
this and so have different consciousness than humans.
Most plausibly. But this again is about the content, and the
character of consciousness, not the existence or not on some
consciousness.
You seem to regard consciousness as a kind of magic vessel which
exists even when it is empty. I think John Mikes is right when
he says it is a process. When a process isn't doing anything it
doesn't exist.
To be sure, I don't use this in the usual reasoning, but I have
to say that I am more and more open that there is something like
that, indeed.
But I agree that consciousness is related to a process, in part
(if not comp would be meaningless).
It just appears that such a process is very basic, that it is
emulated by (many) arithmetical relations, and that it is also
related to arithmetical truth (which is not emulable by any
machine, but machine are confronted to it).
Consciousness per se is not just a process: it is a first person
mental state relating some process with truth. What I say is that
such process can be kept very minimal.
I don't venture to say less consciousness because I think of
it as multi-dimensional and an animal may have some other
aspect of consciousness that we lack.
Sure. Bats have plausibly some richer qualia associated to
sound than humans. But what we discuss is that consciousness is
either present or not. Then it can take many different shapes,
and even intensity, up to the altered state of consciousness.
Cotard syndrom is also interesting. People having it believe
that they are dead, and some argue that they are not conscious,
but in fact what happen is that they lack the ability to put
any meaning on their consciousness.
"Put meaning on consciousness"? That makes no sense to me. They
are obviously conscious of some things. If they were
unconscious they couldn't respond.
There is a possibility that we can access a state where we are
conscious only of one thing, that we are conscious. It *is* part
of the unbelievable (G* minus G).
You mean unprovable? I get confused because it seems that you
sometimes use Bp to mean "proves p" and sometimes "believes p"
Hmm... you might read the Plotinus paper, or the second part of
sane04, or my old posts, or my recent post on Russell's FOAR.
I will tell you the whole thing.
1) I adopt Dennett' intentional stances. I will say that a machine
believes p if and only if the machine asserts p.
2) Being a bit tried listening to machine saying basically
anything, I limit myself to machine which believes in few things
(but not so few), that is, they believe in the classical
tautologies, and some arithmetical things like 0, successors, the
addition and multiplication laws.
(I think I so share those beliefs).
I assume that they are rational, so if they believe p and if they
believe p -> q, they can or will believe q.
In that case 'belief' can be shown to be defined in arithmetic by
Gödel's beweisbar Sigma_1 complete (Turing universal) predicate.
If the machine believes in enough induction axiom, she can proves
(believe) in its sigma_1 completeness, and she becomes Löbian,
meaning
that its mathematics of self-reference is governed by the logic G,
which has the Löb formula as its main axiom: B(Bp->p)->Bp. (Solovay
1976 first theorem)
From this you can see immediately that the machine cannot believe
that she is correct, that Bp -> p is always believable.
But this is where you seem to make a pun on "B". You start by
saying B means "proves" and then for a logic machine "proves" and
"asserts" and "believes" are all the same (extensionally) and so you
let "B" stand for both "proves" and "believes". But then you note
that the machine cannot prove she is correct and you substitute
"believe" for "prove" and conclude she cannot believe she is
correct. But logic is supposed to be a formalization of informal
reasoning.
The point is that the modal logic of *formal*, or *mechanical*
provability obeys G.
Then, and that's magic somehow, when you apply the Theaetetus idea,
you get a modal logic which formalize a non definable (by the machine)
informal notion of knowledge.
You informally reasoned to the conclusion that "proves" = "believes"
for the formal machine.
Not at all, I start from there. It happens that probability obeys an
axiomatic of belief. Even a special one which is based on the bizarre
Löbian placebo phenomenon.
But this is contrary to informal reasoning where "believes" means
"willing to act on" and is very different from "proves".
Noooo.... Of course, when we formalize, we do some simplification. But
I make this clear by using the intentional stance, and treating only
the case of ideally correct machine.
So I get the feeling that you have just incorrectly formalized the
informal reasoning and are playing a semantic trick to get "believe"
in place of "prove".
If you are willing to accept that you believe in arithmetic, and not
too believe much more, then all this works well. If you add new
beliefs, there are some caution to take, but basically as long as you
don't believe in a blattant arithmetical contradiction this will work.
It is just amazing that formal logic obeys the accepted law for
belief, and knowledge can only be something non formal, and non
directly formlizable (but still indirectly) assuming comp. I will come
back on this soon or probably later on Russell's list.
Bruno
I wonder if this is the crux of Russell's unease too?
Brent
Indeed she can show that this entails B(Bp -> p), by necessitation,
and then Bp, by Löb, and then p, and then she can proves all
sentences (with p = f, she is already inconsistent).
So Bp -> p is not always believable, despite being true for the
kind of machine I am considering, and thus, although Bp and Bp & p
are equivalent (we know the machine is correct), she cannot know
that.
So, thanks to incompleteness, or Löb, we can define a new abstract
modality []p = Bp & p, and this modality behaves like knowledge,
and gives the explanation why the machine cannot define it. She can
of course bet on comp, and define it in an abstract theory, like we
did, but the definition will refer, or be interpreted, by something
truly not definable in any third person way. Incompleteness shows,
at the least, the consistency of the definition of Theatetus for
knowledge, when applied to machine's believability, whatever the
axioms are as long as they are consistent with arithmetic or
computer science.
Incompleteness makes "provability" behaving not like a
"knowability", as most people thought, but like a "believability".
It makes the universal machine modest.
But it still seems absurd to me. It invites an infinite regress:
I am conscious of being conscious of being conscious of being...
Why?
Already Gödel's beweisbar is transitive: Bp -> BBp, and so if the
machine believes p, she can or will believe Bp, BBp, BBBp, BBBBp,
etc. The same occurs for the Theaetetical knowability described
above, but it does not occurs for observability, nor sensibility,
with the definitions provided.
There is no infinite regression, just an infinities of
consequences, something usual in arithmetic.
Bruno
Brent
It shows that consciousness seems independent of the ability to
interpret the consciousness content. Many pathological states
of consciousness exist, but none makes me feel like if
consciousness was not something (rich and variated) or nothing.
You refer to the content of consciousness, not consciousness
itself.
But you seem to contend that there can be consciousness without
content - which I find absurd.
There is always a content, but it looks like we can limit it to
one thing: "being conscious". This is coherent with Descartes
and mechanism. Consciousness is the fixed point of the doubt,
notably.
Bruno
http://iridia.ulb.ac.be/~marchal/
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