So I'm trying to prove that, in a language with two diamonds <1>,<2>, the formula p -> <1> p is valid just in case for any x,y and R2(x,y) then R1(y,x). I have the "if" direction easily, that is to say, I've show that if the relations have this property then the formula is valid. But I'm stuck on the "only if". What I have so far is:
Suppose p -> <1>p is valid and w,w' are in W such that R2(w,w'). Let p = Av~A so that w ||- p and therefore w ||- <1>p so that w' ||- <1>p. So of course w' "accesses some world in which p is true" but how could you pin down that the world in question is w?
I wonder, are worlds distinguished by the sentences that are true in them? So could I form the sentence that is the conjunction of all true atomic sentences, or something like it, in or order to be able to infer at this point in the proof that the world which w' accesses wherein p is true, is the world w?