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, consists of the intensions of all formulae which are necessary at , 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
and , if
and
then .