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

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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