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. You informally reasoned to the conclusion that "proves" = "believes" for the formal machine. But this is contrary to informal reasoning where "believes" means "willing to act on" and is very different from "proves". 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".

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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