Montague-Scott semantics

Kripke models as defined above cannot account for non-normal modal logics. To develop an adequate semantics for classical (and monotonic) modal logics we need a generalization of Kripke sematics, the so-called neighborhood semantics (also known as Montague semantics, or Montague-Scott semantics). A complete overview of the basic model theory of classical systems is found in [Che80].

Intuitively, $N(s)$ consists of the intensions of all formulae which are necessary at $s$, where the intension of a formulae is the set of all worlds where it is true. Thus, something is necessarily true at a world if and only if its intension is contained in the set of intensions of formulae considered necessary at that world.

Semantics for extensions of E, including the common monotonic and normal logics, can be obtained by restricting the class of neighborhood model through appropriate conditions ([Che80]). For example, EK is determined by the class of all neighborhood models satisfying the condition: for all $X,Y\subseteq S$ and $s\in S$, if $(S\setminus X)\cup Y \in N(s)$ and $X\in N(s)$ then $Y\in N(s)$.