Le 19-nov.-05, à 22:56, Russell Standish a écrit :

On Sat, Nov 19, 2005 at 04:22:58PM +0100, Bruno Marchal wrote:
Now observation and knowledge are defined in the logics of
self-reference, i.e. by transformation of G and G*, and so are each
multiplied by two. Actually and amazingly for the knower (the first
person) G and G* give the same logic, like if the first person
conflates truth and provability. But for the notion of observation, G
and G* give again different logics, so that the observer can
distinguish communicable observations ("physical facts") and non
communicable observations (sensations, I would argue).

Are you now saying that your operators

   Pp = Bp & -B-p


   Op = Bp & p & -B-p

correspond to "to observe" (Op being "to validly observe" I
suppose)?. Previously, you would say that Pp is "to bet on p", and Op
"to correctly bet on p", which never really made sense to me. What's
the French word you would use for this - I may know it, or perhaps I
can figure the relevant English term from a dictionary.

Let me first explain in few words a plausible logician conception of a "multiverse". I borrow the term "multiverse" from FOR, but I think we should be neutral about what is really a universe, or a world, or a state, or an observer-moment: the only thing which matter is that we have many of them, and that they are related by a relation of accessibility. So a multiverse is just a set W (of elements called "worlds") and a binary relation R defined on it. Let us use the letter a, b, c, d, ... for the worlds. So aRb just means that the world b is accessible from the world a. You can travel from a to b. Note that I am not pretending that the "real multiverse" (perhaps the quantum one) is of that type, but it is good to begin with that familiar sort of Kripke multiverse, and then to correct it.

Now we assume that all the worlds obey classical logic: if p is true in world a, and if q is true in world a, then propositional formula like (p & q), (p -> q) etc. are true at a, and ~p is false at a, etc. In particular, all classical tautologies are true in all worlds of all multiverse independently of the assignment of truth value to the sentence letter p, q, r, etc.

The main idea of Kripke has consisted in saying that the modal formula Bp (also written []p) is true at world a, if p is true in all the worlds you can access from a. p is relatively necessary at a. For example, if the world are countries and if you have to pay taxes in all countries that you can access from where you are, then taxes are necessary (relatively to a).

That is, p is "necessary" at world a if p is true for all worlds b such that aRb. It is intuitively normal: a proposition is necessary for you if it is true in all world you can access.

Then a proposition is possible at world a if it is not necessary that ~a. So "possible p", written Dp, or <>p, can be seen as an abbreviation of ~B~p. Note that if Dp is true at a, it means there is an accessible world (where p is true) from a. In particular, given that the constant true t is true in all worlds, Dt really means I can access to some world (I am alive, if you want).

Now, there are relation between the structure of the multiverse, i.e. the nature of its accessibility relation, and the formula which are true in each world. It should be easy to guess that if the multiverse is reflexive (i.e. all worlds are accessible from themselves) then the formula Bp -> p is true in all the worlds, independently of the truth value of the sentence letters. Slightly less easy: the reverse is true: if Bp -> p is true in all worlds, independently of the assignment of true/false to the sentence letters, then the multiverse is reflexive. We say that the reflexive multiverse characterizes the formula Bp -> p. It means the formula remains invariant when we travel in that multiverse. It can be shown that the symmetrical multiverse, that is those where the accessibility relation is symmetric, characterizes the formula p -> BDp. The transitive multiverse characterizes Bp -> BBp. etc.

Of special interest in this thread are the dead-end world, or cul-de-sac observer-moment (we have use many name for them). A world a, in a multiverse W, is said to be a dead end or a cul-de-sac world if, when you are in a, there is no more world in which you can acceded. So, in such world no proposition are possible, so whatever proposition p is, ~Dp is always false. By classical logic B~p is always true. This is true whatever p is, in particular this is true for its negation ~p. So in a dead end world, all proposition are necessary and none is possible. Not a funny place!

Now, when B represents the Godel-Lob provability predicate, i.e. when B represents provability in or by a "sufficiently rich" formal system/machine, it can be shown that the "humble multiverse", that is those where all worlds have access to a dead end world, characterizes B. In that case Dp = ~B~p = "~p is not provable" = "p is consistent" (because if you cannot prove ~p, you will not get a contradiction by adding ~p as axiom, that is you will not prove ~p, that is p is consistent (with your formal theory or for your machine). So "humble multiverse" characterizes the formula Dp -> ~BDp, which is the second incompleteness theorem of Godel: consistent p -> not provable consistent p. So the humble multiverse characterize the machine's discourses.

This prevents us of defining "probability(p) = 1" in world a by Bp is true at a, because if a is a dead end then Bp is true (D~p is false) although the probabilities are senseless.

Now, as you know, I limit the interview of machine to the correct and consistent one. This is just a mathematical trick. For those machine Bp -> Dp is true, but not provable by the machine. So we can define "probability(p) = 1" by "Bp & Dp". It means that we define the probability one of a proposition p, by p is true in all accessible worlds and there is (at least one) accessible world. The incompleteness forces us to put explicitly the consistency as a requirement. It corresponds to the correct bets, or to the observation of laws (invariant truth of the multiverse).

Actually, adding that Dp to Bp is so much constraining that we loose the Kripke multiverse structure in the process, but we get instead a more interesting proximity relation, and ultimately we get (translating the comp hyp itself in the language of the machine) an orthogonality structure on the worlds of the multiverse, making it looking like the quantum multiverse inferred by the observing physicists.

Observation is implicitly defined here by measurement capable of selecting alternatives on which we are able to bet (or to gamble ?). The french word is "parier".

I will not explain the nuance between Bp & Dp and Bp & Dp & p. It just happen that they are not equivalent in the humble multiverse, that is they are not equivalent in the discourse of the correct machine, and this provide nuances. Arithmetical quantum logics appear for all first person nuances put on the provability predicate (with the comp hyp), giving three arithmetical interpretation of some quantum logic. Could explain this latter but I'm afraid it is a bit more technical.

Hoping this could help (to make logician's and physicist's talk closer perhaps). 'course, Kripke uses himself the term "frame" instead of multiverse, and "model" when each of the proposition letters (p, q, r, ...) are assigned to true or false (1 or 0, t or f) in each world.



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