An intuitionistic Kripke model is a triple $\langle W,\leq, \Vdash \rangle$, where $\langle W,\leq \rangle$ is a preordered Kripke frame, and $\Vdash$ satisfies the following condition of hereditariness (or monotonicity):
if $P$ is a propositional variable, $w\leq u$, and $w\Vdash P$, then $u\Vdash P$.
Are there intermediate logics (excluding classical logic), including intermediate modal logics (i.e intermediate logics which contain modalities) for which there are no Kripke models in the above sense?
(If so,) what is the smallest such intermediate logic?
If there are no such intermediate logics, what is the proof of this claim?
I was thinking particularly of intuitionistic logics to which is adjoined some modality $\bigcirc$ which does not obey hereditariness. I.e, for which we have:
$P$ is a propositional variable, $w\leq u$, $w\Vdash \bigcirc P$ and $u\not\Vdash \bigcirc P$.
It has been observed below that classical logic can be given a Kripke model in the above sense. Does this entail that any intermediate logic can be given a Kripke model?