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.
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
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
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:
~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:
~BBBBBBBBBBBBBBf is equivalent with DDDDDDDDDDDDDt
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!!!