I clarify and progress a little bit. Then I jump a little bit.
(Sorry for quoting myself)

Ah! but Lee could have build an objection by saying that in Stathis' theory we die, or can die, at each "instant", or at each teleportation experiment. He told us this in its death thread.

The objection is that the idea that "Bp true in world w" represents "Probability(p) = 1 in world w", although quite obvious in "ideal multiverses" (multiiverse where all worlds or observer-moments are transient), is not obvious at all in "Stathis multiverse" where we die or can die at each observer-moment (that is, all transient world leads to (at least) a cul-de-sac world). This led us to the following problem "Bf is true or false in a dead-end". I will come back to it below.

Stathis was doing Kripke semantics, perhaps like Jourdain was doing prose. He suggests to define a state (world, OM, ..) as being "alive" when it is "transient":

The state/world/OM... x is "alive" when there is a y such that xRy

Note that x could be equal to y, in which case the world x reaches itself. Such a world/state/OM is of course transient, or alive.

and a state is "dead" when there is no such accessible world from x. x is terminal, or cul-de-sac, dead-end, etc.

Now in Stathis' theory, we die at each instant and this means that all transient states reach dead-end worlds!

Let me give you examples of a "Stathis's multiverse" ("near death" multiverses)

1) {w1, w2} with the accessibility relation w1Rw1, w1Rw2 (and nothing else). There is only one alive state (w1), and one dead state (w2). It follows stathis" rule that all alive states reach a dead-end. I suggest people (interested) do the drawing. Drawing does not survive the archive without coming up a little surreal!

2) {w, w1, w2, w3, w4, w5, w6, ...} with accessibility relation:
    w1 R w2,  w2 R w3, w3 R w4, etc., together with for all i wi R w.
Note that w is here a cul-de-sac world reached by all transient worlds wi

exercise: Draw 5 examples of finite "near death" multiverse, and 5 examples of infinite one (drawing "..." is permitted).

Now suppose x is alive. This means there is y such that xRy. But the proposition true, t, is true in all world, and thus it is true in y. This means Bf is false in x (by KRIPKE IMPORTANT LINE). It is just false that f is true in all accessible world from x, giving that in y t is true (and xRy). So in any world x which is alive, Bf is false. This means that ~Bf is true (worlds obeys classical logic). and giving that f equivalent with ~t, this means that ~B~t is true in the alive state.

Is that OK for everybody?  (among those interested)

What about ~B~t, or ~Bf,  in a dead-end state?
What about Bf in a dead-end state?

Is that question clear? Nobody wants to propose an answer? ( the answer is below).

This is a little bit tricky and I let you think (I must go now). It is important also for getting a "theory" (set of propositions through in all worlds in some multiverse, where a multiverse is just a set of worlds (OMs) with some specified accessibility relation among worlds (OMs).

Ouh la la!!!  Please read "true" instead of "through" !

The idea that a theory (set of propositions) should be true in all worlds is natural among physicists. SWE is supposed to be true "everywhere in the quantum multiverse". If a law changes it is arguably not a law. Many expected that my way to isolate physics would lead to "classical propositional calculus". The only laws would be the (classical) laws of logic! Well, my answer was that if that was the case (with comp) that would mean that physics is empty (no physical law at all). Only history and geography. A little like Smullyan describes the difference between math and physics in FU (Forever Undecided) page 47, when he says:

"Given any possible world, the set of all propositions that are true for that world, together with the set of all propositions which are false for that world, constitute the state of affairs holding for that world. A tautology, then is true, not only for this world, but for all possible worlds. The physical sciences are interested in the state of affairs that holds for the actual world, whereas pure mathematics and logic study all possible states of affairs".

Comp makes this statements false, even staying in classical logic (in Platonia). That's not obvious. Would Smullyan's statement be a consequence of comp, then, as I said, physics would be a branch of geography, and all physical laws would be contingent.

Bf ?

Let us come back to the question of the truth or the falsity of Bf in a dead-end observer moment w. (f = false, and a dead-end world/state/OM is an OM without reachable world, or without "successor" as Stathis said). Here w stands for an arbitrary cul-de-sac state.

Bf true in w would mean that for all y such that wRy, f is true in y.
This means that for all y :  if wRy then f is true in y.
But w is a cul-de-sac so wRy is always false, and "f is true in y" is also always false, so Bf true in w is equivalent with "for all y false -> false". But that is always true!!!

So Bf is true in any cul-de-sac world.

And this jeopardizes the reading of Bp as a probability statement. We have Bf true in w (cul-de-sac world), but to say that the probability of the falsity is one in a cul-de-sac would be ridiculous.

So the probability reading of a modal box, which I used in my explanation to Lee, is just irrelevant unless we make the "no cul-de-sac hypothesis".

But "Stathis' theory" says that there are cul-de-sac reachable from all transient worlds! So my answer to Lee is incompatible with my assessment to Stathis' theory on observer-moment!

 So what ?

And when you interview the Lobian machine, the first theory you get is Papaioannou's theory, which I recall says that if you are alive then you (can) die. (All transient OMs reach cul-de-sac OMs). [This will be explained, but also has been explained see links in my url]

Bf can be true only in cul-de-sac worlds (easy consequences of the KRIPKE IMPORTANT LINE in the preceding posts in this thread. Please verify by yourselves).
And Bf  is true in all cul-de-sac worlds (by the reasoning above).

So the "necessity" of the false, Bf, is the signature of the cul-de-sac world.

And so the negation of Bf, ~Bf, is the signature of the alive Oms, or transient worlds.

If all transient worlds leads to cul-de-sac worlds, in any world ~Bf -> ~B(~Bf), meaning that if you are in a transient state/OM/world you cannot bet you will reach a transient worlds.

Please verify that (~Bf -> ~B(~Bf)) is indeed true in all the worlds of any "Papaioannou's multiverse". Verify at least on the two examples above.

Please verify that, whatever the truth value of p is any world, (~Bp -> ~B(~Bp)) is also true in all "near death" multiverse.

let me jump a little bit:

Theorem: The following theory C, formalizes soundly and completely the "Papaioannou's tautologies (the proposition true in all worlds of all "mear death" multiverse):

Axiom 1  B(p->q) -> (Bp -> Bq)
Axiom 2  (~Bp  ->  ~B(~Bp))
Inference rule 1: if you can justify both p and (p -> q), then you can justify q
Inference rule 2: if you can justify p, then you can justify Bp

This is C.

We have the following "metatheorems":
Soundness: C proves only "near death" multiverse tautologies.
Completeness: all "near death" multiverse tautologies are provable in C

A cute theory, which alas proceeds from comp which motivates it (as illustrated by Stathis' thread), but which has just the problem that my "obvious" answers to Lee Corbin; which invokes naive probabilities, is just meaningless in Stathis' theory.

And the problem remains when you interview the lobian machine, it gives Stathis' theoy too!!under the form of the G and G* extensions of C, and this makes even worse the work to get a notion probability on the observer-moments.

Well, I have been stuck on that problem for *many* years ...

Look how lucky you are, I give you the solution in two words: Theaetetus (Plato).

I said I jump,




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