The frame (5.10) on page 323 is as follows
W = { X4, X5, X6 }
R = { (X4,X4), (x4,X5), (X5,X5), (X5,X6), (X6,X6) }
so each world as a relation to itself and apart form X6 not linking back to X4 the rest of the relations are transitive.
The books states that this frame does not satisfy []p -> [][]p
... here's my thinking ...
for X4, []p is true as p is true @ X4 and X5 (only worlds accessible via single step relation from X4)
for X4, [][]p is true as [] is true @ X4 and []p is true @ X5 (for the same reasons as X4, i.e. with itself and next world )
for X5, []p is true as already seen above
for X5, [][]p is true as same approach as X4 (i.e. []p is true @ X6 because of its reflexive relation)
for X6, []p is true because of the reflexive relation
but what about [][]p - my understanding is that []p must be true from all worlds accessible from X6 via single step relation, and in my view that relation is (X6,X6)
as pointed out the book says that this is not so i.e. []p -> [][]p is false, and the only what this can be false is if []p is true and [][]p is false - I can't get [][]p false
Any takers?
