Reflexive, Transitive and Symmetric applies only to the relation R that define
accessibility. So:

Reflexive:

W ---->|
  <------|

Transitive:

W1 ------> W2 --------> W3


Symmetric

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
x:
                                            |---->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
from x:

Now going to your LASE

p-> []<>p

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.
---->w --|
A transitory world is one from which at least one other world is
accessible                   w---->w1
An ideal world is one with no access to a terminal
world                                              w---->w1
A realist world is one with access to at least one terminal
world.                                   w---->w1 --|


Your theorem

5. (W,R) respects []p -> <>p iff (W,R) is ideal
means
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
means
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
world.

hmmm.. All this reasoning could be quickly shown with one drawing......When I read
this statement I suffer from a mental stack overflow. :-)



George

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