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 $u\not\Vdash \bigcirc P$.

Edit

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?

semanticsof propositional logic. It therefore makes no sense to ask whether there are logics which violate it. Even classical logic satisfies the monotonicity requirement, trivially so because there aren't any interesting Kripke models of classical logic. (But thereareKripke models of classical logic!) $\endgroup$ – Andrej Bauer Dec 17 '18 at 13:56