Classic model relation versus the Kripke model relation

[nomadreid]

Looking at an exposition of Kripke semantics, the relationship ||- in a Kripke model is obviously supposed to be similar to the model relation |= . A possible world (a node) looks suspiciously like a model for second-order formulas. But of course it cannot be this simple. What is the connection, if any, between ||- and |= ?

 

[nomadreid]

Given the lack of replies, I will assume that the only difference is one of context. Stick to one model, you have the classic model relation; looking at one of many models, you have the Kripke relation for possible worlds. I throw this up as a challenge: if it is wrong, please correct!