Reflexive, Transitive and Symmetric applies only to the relation R that define
W1 ------> W2 --------> W3
W1 <-----> W2
And the Goedel-like formula
<>p --> -<>p
means: if p is true in at least one world accessed from w, then it is false that
in any world x accessed from w, that p is true in at least one world accessed from
|---->w121 p false
|---->w12: p true ---->|---->w122 p false
|---->w13: p false |---->w123 p false
w1--|---->w14: p false
|---->w15: p false |----->w161 p false
|---->w16 p false---->|----->w162 p false
|----->w163 p false
I have a little bit of trouble with reading -<>p from left to right....Is it - (
 ( <>p ) )?
Also does the frame of reference changes when you talk about  i.e., any world x?
In other words you start with world w but then, as you express  all (any)
world(s) x accessible from w do you have to change the frame to x when you pursue
the reasoning to <>?
If what I assumed is correct, then the Goedel-like formula makes sense.
Now reinterpreting  and <> to mean provable and consistent we get for
<>p --> -<>p
if p is consistent in at least one world accessed from w, then it is false that in
any world x accessed from w, that p is provable in at least one world accessed
Now going to your LASE
if p is true in w, then in any world x accessed from w, it is possible to access
one world where p is true.
This statement seems to be correct only when R is symmetric or if there is a
transitive loop back to w.
I assume the following:
A terminal world is defined as a world from which no other world is accessible.
A transitory world is one from which at least one other world is
An ideal world is one with no access to a terminal
A realist world is one with access to at least one terminal
world. w---->w1 --|
5. (W,R) respects p -> <>p iff (W,R) is ideal
if (if p is true in all worlds accessible from w, then p is true in at least one
world accessible from w.) is true then w does not lead to a terminal world.
In a terminal world nothing is possible, everything is false so p cannot be true.
The premise doesn't even hold. Since an ideal world does not lead to a terminal
world, then from an ideal world (W,R) respects p -> <>p is always true.
6. (W,R) respects <>p -> -<>p iff (W,R) is realist
if (if there is at least one world accessible from w where p is true, then it is
false that from any world x accessible from w, it is possible to access at least
one world in which p is true) is true, then w leads to at least one terminal
hmmm.. All this reasoning could be quickly shown with one drawing......When I read
this statement I suffer from a mental stack overflow. :-)