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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