Le 14-déc.-05, à 01:34, Stathis Papaioannou a écrit :

In the multiverse, only other people end up in dead ends. Although from a third person perspective every entity in the multiverse could be said to exist only transiently because at every point of an entity's history we can say that there sprouts a dead end branch of zero extent, from a first person perspective, these branches cannot by definition ever be experienced.


All right.
Could I take this as a defence of the "Papaioannou multiverse" for some third person description: those where each world where you have a next state leads to a dead end? I call them "realist frames" in Conscience & Mechanism". Sometimes they are called "terminal frames" in the literature.

I know you have solved the "only if" part of following exercise:

(W, R) is reflexive     iff      (W,R) respects Bp -> p.

I will come back on the "if" part later.

Have you done this: showing that

(W,R) is a "Papaioannou multiverse" iff (W,R) respects Dt -> D(Bf).

Note that this question is a little bit academical. I have already explain how I will choose the modal logics. Actually I will not choose them, I will extract them from a conversation with the machine (and its "guardian angel"). This will leave no choice. It will happen that the formula Dt -> D(Bf) will appear in the discourse machine; indeed perhaps some of you know already that this is just the second incompleteness of Godel, once you interpret Bp by "the machine proves p", coded in some language the machine can use.

=============================
Exercises for those who begins the study of modal logics:
Does every one see that all the following formula are equivalent? :

Dt -> ~B(Dt)
Dt -> D(Bf)
BDt -> Bf
~Bf -> ~B(~Bf)


Those are equivalent (in all the modal logics we will meet), and the only things people should know to prove those equivalences are that:

1)
~Bp   is equivalent with D~p     (not necessary p = possible not p)
~Dp is equivalent with B~p      (not possible p = necessary not p)
Bp is equivalent with ~D~p
Dp is equivalent with ~B~p

From this you can deduce a nice memo: a not "~" can jump over boxes by transforming them into diamonds, and reciprocally:
For example:
~BBBBBBBBBBBBBBf is equivalent with DDDDDDDDDDDDDt

and 2)
the contraposition law:  (A -> B) is equivalent with (~B -> ~A).


I urge people who have difficulties NOT to hesitate to ask me question OUT of line. Too bad to miss the marvel of all marvels (G and G*) for reason of math-notation-anxiety!!!

Bruno






http://iridia.ulb.ac.be/~marchal/


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