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/



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