On 21 Sep 2012, at 19:17, meekerdb wrote:

On 9/21/2012 1:22 AM, Bruno Marchal wrote:

On 20 Sep 2012, at 20:14, meekerdb wrote:

On 9/20/2012 10:31 AM, Bruno Marchal wrote:

On 20 Sep 2012, at 18:14, meekerdb wrote:

On 9/20/2012 2:05 AM, Bruno Marchal wrote:
A modal logic of probability is given by the behavior of the "probability one". In Kripke terms, P(x) = 1 in world alpha means that x is realized in all worlds accessible from alpha, and (key point) that we are not in a cul-de-sac world.

What does 'accessible' mean?


In modal logic semantic, it is a technical world for any element in set + a binary relation on it.

A mapping of the set onto itself?

?

A relation is not a map. A world can access more than one world.
For example {a, b} with the relation {(a, a), (a, b)}, or aRa, aRb.





When applied to probability, the idea is to interpret the worlds by the realization of some random experience, like throwing a coin would lead to two worlds accessible, one with head, the other with tail. In that modal (tail or head) is a certainty as (tail or head) is realized everywhere in the accessible worlds.

Then accessible means nomologically possible.

Accessible means only that some binary relation exists on a set. But in some concrete model of a multi-world or multi-situation context, nomological possibility is not excluded.

Then I don't understand what other kinds of possibility are allowed? I don't see how logical possibility could be considered an accessibility relation (at least not an interesting one) because it would allow Rxy where y was anything except not-x.





But in the worlds of the UD there is no nomological constraint, so there's no probability measure?

I am not sure why there is no nomological constraints in the UD. UD* is a highly structured entity. You might elaborate on this.

A nomological constraint is one of physics.

Why? Define perhaps "nomological".



But physics is derivative from part of the UD. The UD is structured only by arithmetic.

Why would this be not enough, given that physics will supervene on arithmetical relations (computations)?

Bruno







Generally speaking a different world is defined as not accessible. If you can go there, it's part of your same world.

Yes. OK. Sorry. Logician used the term world in a technical sense, and the worlds can be anything, depending of which modal logic is used, for what purpose, etc. Kripke semantic main used is in showing the independence of formula in different systems.

Bruno




Brent


This gives KD modal logics, with K: = [](p -> q)->([]p -> []q), and D: []p -> <>p. Of course with "[]" for Gödel's beweisbar we don't have that D is a theorem, so we ensure the D property by defining a new box, Bp = []p & <>t.

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