Hello Dan,

On 15 Feb 2013, at 05:31, freqflyer07281972 wrote:

Dear Bruno,

I would like to know what 'doxastic models of consciousness' means, as well as what means "S4Grz" - I know Craig was the one who originally used the term 'doxastic models' but you seemed to know right away what that meant, so I'd like to know from your perspective what it means;


Epistemic = knowledge

Doxastic = belief

Epistemic logic are modal (most of the time) logics where the modal box (here written with a B) represents a unary connector intended for the knower (Bp = "the agent know p", or "I know p").

Doxastic logic are modal (most of the time) logics where the modal box (here written with a B) represents a unary connector intended for the believer (Bp = "the agent believes p", or "I believe p").

The main difference between knowledge and belief is that knowledges are true, by definition, when beliefs can be false.

So among the axioms accepted for knowledge or epistemic logic, we have that Bp -> p (I know p entails p is true).
Contrariwise, modal doxastic logics will NOT have the axiom Bp -> p.

For the ideally self-referentially correct machine I consider, the belief B is modeled by provability. Before Gödel, most people (mathematicians and philosophers) would have thought that in this case we do have Bp -> p.

But as Gödel already remarked, the provability predicate, even in the correct case, cannot be modeled by a (normal) modal logic having Bp -> p. Indeed we would have Bf -> f, that is ~Bf, and that's consistency, which cannot be proven by the machine, despite it being true. That's why the logic of provability (belief) split into a true part and an believable, or provable part.

But that is also why the Theaetetus definition works non trivially when we define knowledge by Bp & p (that is I know p is I can justify it, and it is the case that p). Bp & p implies trivially p, and in the arithmetical setting we do get the classical modal logic of knowledge, known as S4. Indeed we get S4 + a new "axiom":

S4 is

Know p -> p   (main axiom for knowledge)
Know p -> Know Know p  (self-awareness, or introspective axiom)
Know (p -> q) -> (Know p -> Know q) (rational "omniscience", more used for "knowledgeable")

+ the logical inference rule (p/ know p). All this on the top of the classical propositional logic.

In the arithmetical context, we inherit the following axioms, named after a formula of Grzegorczyk, Grz):

Know (Know (p-> Know p) -> p) -> p.

It introduces a sort of antisymmetry on the Kripke accessibility relations, and avoid circular structure (in the finite world case, when used together with the other axioms). But there are other semantics too.

Note that the Bp of G represent an arithmetical sentence (beweisbar ('p'), with beweisbar defining provability in arithmetic, and 'p' being a representation in arithmetic of the sentence put for the proposition p). We have no choice in the modal logic, and Solovay provided the relevant completeness of G for the formal effective theories, which correspond to the rich ideally correct machines.

For Bp & p, we have no similar direct definition in arithmetic, but we can study them at the metalevel by modeling Bp & p for each individual instantiated sentences, so "I know 2+2 = 4" is, in arithmetic: beweisbar ('2+2=4') & 2 + 2 = 4.



moreover, I want to know S4Grz or be pointed towards an advanced level logic book so I can understand what that means.

S4Grz is quite well explained in Boolos 1979, and Boolos 1993. Together with the logics of self-reference G and G*. Known also as GL and GLS (Gödel, Löb, Solovay).

Boolos, G. (1979). The unprovability of consistency. Cambridge University Press, London.

Boolos, G. (1993). The Logic of Provability. Cambridge University Press, Cambridge.

A good book on Modal logic is the book by Chellas:

Chellas, B. F. (1980). Modal Logic, an introduction. Cambridge University Press, Cambridge.

To get matter from arithmetic, we need to add a consistency condition (so we get intelligible matter with Bp & Dt), and sensible matter with Bp & p & Dt. This gives quantum-like logic. It is an open, but well formulated problem to know if we get quantum computer from them, as we should, if we are machine, and if the classical theory of knowledge is correct, by the UD Argument.

This is explained (concisely, with reference) in the sane04 paper:

http://iridia.ulb.ac.be/~marchal/publications/SANE2004MARCHALAbstract.html



Finally, as a simple confirmation, I do assume that when you guys talk about Bp & p you mean the literal proposition "someone believes p & it is the case that it is p" --

OK.


if I don't get at least that, I should hang up my hat around here!

No worry :)

Best,

Bruno


On Wednesday, February 13, 2013 10:56:05 AM UTC-5, Bruno Marchal wrote:

On 12 Feb 2013, at 20:05, Craig Weinberg wrote:

When we talk about a Bp, relating to consciousness is that we are making an assumption about what a proposition is. In fact, if we look closely, a proposition can only be another level of B. p is really nothing but a group of sub-personal Beliefs (logarithmically nested as B^n)

?



which we are arbitrarily considered as a given condition...but there is no given condition in actual experience.

That's why we put Bp & p. To get the condition of 1p experience. It works as we get a non nameable, and non formalisable notion of knowledge. S4 and S4Grz do succeed in meta-formalizing a thoroughly non formalisable notion.




All experiences are contingent upon what the experiencer is capable of receiving or interacting with.

Any proposition that can be named relies on some pre-existing context (which is sensed or makes sense).

The problem with applying Doxastic models to consciousness is not only that it amputates the foundations of awareness,


It does not for the reason above. Note that even Bp & p can lead to falsity, in principle. Things get more complex when you add the non monotonic layers, that we need for natural languages and for the mundane type of belief or knowledge. Here, of course, with the goal of deriving the correct physical laws; it is simpler to consider the case of ideally correct machine, for which us, but not the machine itself can know the equivalence.

Bruno



but that the fact of the amputation will be hidden by the results. In Baudrillard's terms, this is a stage 3 simulacrum, (stage one = a true reflection, stage two = a perversion of the truth, stage three = a perversion which pretends not to be a perversion).

The third stage masks the absence of a profound reality, where the simulacrum pretends to be a faithful copy, but it is a copy with no original. Signs and images claim to represent something real, but no representation is taking place and arbitrary images are merely suggested as things which they have no relationship to. Baudrillard calls this the "order of sorcery", a regime of semantic algebra where all human meaning is conjured artificially to appear as a reference to the (increasingly) hermetic truth.

http://en.wikipedia.org/wiki/Simulacra_and_Simulation

This is made more important by the understanding that sense or awareness is the source of authenticity itself. This means that there can be no tolerance for any stage of simulation beyond 1. In my hypotheses, I am always trying to get at the 1 stage for that reason, because consciousness or experience, by definition, has no substitute.

Craig

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http://iridia.ulb.ac.be/~marchal/




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