 It proves useful to enrich the above relational semantics by another binary relation \(\leq\) on the set of states \(W\). The relation \(\leq\) is assumed to be a partial order (i.e., it is reflexive, transitive and anti-symmetric), which allows one to think of it as a relation of possible expansion of information states. With such a reading it is natural to assume that the truth of atomic formulas \(p\) is persistent with respect to \(\leq\): if \(w \leq u\) and \({\cal M},w \models p\), then \({\cal M},u \models p.\) The conditions on \(\leq\) and \(C\) and the truth conditions for compound formulas should then be such that persistence (also called heredity) holds for arbitrary formulas, in particular for negated formulas \(\osim A\) if negation as impossibility is considered. A compatibility model is a structure \((W,C,\leq, v)\), where \((W,C,v)\) is a Kripke model, \(\leq\) is a partial order on \(W\), and the following condition is satisfied, which guarantees the heredity of negated formulas \(\osim A\): if \(wCu\), \(w' \leq w\), and \(u' \leq u\), then \(w'Cu'\). This condition is a constraint on the compatibility frame \((W,C,\leq)\) on which a model \((W,C,\leq,v)\) is based. The condition is not only useful (as will become clear), but also plausible, because it says that two information states, expansions of which are compatible, are themselves compatible.