OK, so ignoring Brent who I'm sure is way ahead of me...
The problem is to show that
(W, R) respects []A -> A if and only if R is reflexive,
Where reflexive means for all alpha, { alpha R alpha } (and *nothing more *is
implied!)
And []p means that p is true in all worlds reachable from the world being
considered.
(...I think. I just checked my diary and was told that "[]p means that p is
a law". Maybe that was the wrong page...)
OK, so anyway, before I get too confused let's consider world alpha which
is part of W. We know { alpha R alpha }. []p means p is true in all worlds
reachable from alpha (I think) which includes alpha itself, hence it means
that p has to be true in alpha, hence it means "[]p -> p". (Conversely, if
alpha wasn't reachable from itself, then p being true in all worlds
reachable from alpha *wouldn't* entail that p is true in alpha.)
QED, perhaps? Did I just prove something?
If so, I'still m not sure that proves "if any only if"...
Although... maybe it does. For []p to imply p in a world alpha, where []p
means "p is true in all worlds reachable from alpha", it can *only* imply p
is true if alpha is "reachable from alpha".
This applies to all worlds in (W, R) hence it must be reflexive.
I think.
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