At 16:58 23/07/04 -0400, Jesse Mazer wrote:
>Bruno Marchal wrote:
>
>>All right. But modal logic are (traditionaly) extension of classical
>>logic, so that causal implication, or >natural language entailment, when
>>study mathematically are generally defined through modalities
>>+ >"material implication".
>>
>>So in a sense, you confuse yourself by premature anticipation.
>
>Well, I guess "in every possible world where X is true, Y is true also"
>can only be false if there's a possible world where the classical logical
>statement "X -> Y" is false (because in that possible world, X is true but
>Y is false). So perhaps the possible-world statement would be equivalent
>to the modal-logic statement "it is necessarily true that X->Y"--would
>this be an example of modal logics "extending" classical logic?
Exactly.
> In any case, in classical logic X -> Y can only be false if X is true in
> *our* world,
I would say: in our arithmetical platonia (see below).
> whereas the possible-world version of "if X then Y" does not require
> that X is true in our world, although it must be true in some possible world.
X could be false in all possible world. "if X then Y" will then be true in
all possible worlds.
>And like I said, I think the possible-world statement more accurately
>captures the meaning of the natural-language statement.
Yes but it is here that you anticipate, relatively to the goal which we
have ascribe to ourselves.
To sum up in a nutshell: the UD Argument shows comp transforms physics into
a "science of"
a measure on all (relative) computational histories. Those comp histories
are relatively
described in Arithmetical Platonia (the set of all true arithmetical
propositions). In particular physics must appears in the discourse of a
Self-Referentially Correct machine (SRC machine) when interviewed on the
geometry of their (maximal) consistent extensions.
p belongs to a consistent extension of a machine M when the machine does
not believe in -p (that is -B-p is true for the machine).
The SRC machine is supposed to be:
1) platonistic (the machine believes the classical tautologies)
2) arithmetical platonistic (the machine believes in the theorem of PA, say)
3) computationalist (by UDA the machine believs p -> Bp, with p atomic,
that is the atomic
"physical consistent state" are those generated by the DU. I will come back
on this
remember just that comp is translated by the modal formula p -> Bp
"1)" and "2)" makes the SRC machine a Loebian machine. It makes it, in FU's
terminology
a consistent, stable, normal, modest reasoner of type 4. That is, a
reasoner of type G.
So we get a modal logic, from which we can study the corresponding "multiverse"
(unlike those who want to capture directly "ordinary natural language
deduction" by an ad hoc choice of a modal logic)
"3)" forces us to add the comp axiom. I give the name "1" to the formula
"p->Bp" (1 is a shorthand for Sigma_1). We get a new theory which is just
G+1 (+and a weakening of the substitution rule: p can only represent an
atomic proposition [do you see why?]).
Now A. Visser, from Utrecht (in Holland) has proved the arithmetical
completeness for both (G+1) and its corresponding * logic (G+1)*. In honor
to Visser I call often V and V* the logic G+1 and (G+1)*. See the (big)
bible for a proof. Big bible = George Boolos 1993. (this is beyond FU
little bible).
If you identify a logic with its set of theorems you have the following
diamond where the edges represent inclusion:
V*
G* V (except that I'm too lazy for drawing the edges)
G
Going up in the north west direction is the non trivial godelian passage
from provability (believability) to truth. Going up in the north east
direction is the non trivial comp direction. Sometimes to fix the things I
say that G gives science and G* gives theology, V gives comp-science and V*
gives comp-theology. (But take this with some grain of salt).
Note that G* minus G (resp. V* minus V) gives all the unbelievable (comp)
but true propositions.
OK. At that stage we are not yet in a position to get physics. What is
missing? People on the list should be able to guess giving I insist on this
all the time. What is missing is the fundamental distinction between the
first and third person points of view, without which the UD Argument just
doesn't start. The four G, G*, V, V* gives only 3 person descriptions. G
for exemple gives a logic of self-referentially correct discourse, but the
machine talk about itself only from some description made (by construction)
at the right level. But the UDA shows physics appears through machine's
first point of view. Also G does not work for describing a probability
logic. Although the box []p in G describes p as true in all accessible
worlds, there exists necessarily (by Godel seconf theorem) accessible world
which are cul-de-sac worlds. We would like to have []p -> <>p, or in FU's
notation Bp -> -B-p. If the proba that p is one, we would like the proba
of -p being different of one!
Why not define Pp (meaning proba of p = 1) by Bp & -B-p ? We can: G* proves
Pp <-> Bp, but
the view for the machine is different: G* proves -B(Pp <-> Bp). So it makes
sense and it is the easiest way to cut out the cul-de-sac worlds. This is
only one of the "Theaetetus" attempts to define knowledge. Actually the
whole three main variants work in V*:
Pp = Bp & p
Pp = Bp & -B-p
Pp = Bp & -B-p & p
Work in which sense? In the sense that applying them on V* we get a modal
quasi-quantum logic.
That is a modal logic which describes a (sort of) quantum logic, that is a
logic where the probabilities are described by rays in Hilbert Space (and
variants). More on this latter probably.
I take this as a confirmation of the plausibility of comp. The second and
third Theaetetic variant
applied on V* minus V, describe the consistent (and true) measurable but
uncommunicable (un-sharable) truth, so that "qualia" are themselves
described by sort of quantum logics, although those should be more aptly
called qualia logics. It's the main advantage compared to traditional
empirical physics which methodologically put the 1-person under the carpet.
Bruno
http://iridia.ulb.ac.be/~marchal/
Received on Sat Jul 24 2004 - 11:06:24 PDT