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 
$u\not\Vdash \bigcirc P$.




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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?



Superintuitionistic logics which are not hereditary/monotonic: impossible or possible?
Every propositional logic $L$ weaker than classical logic (i.e., any logic whose provable propositions are a subset of classically provable propositions) has a Kripke model. Just take $W = \{\star\}$, the frame with a single element, and the trivial preorder. The model is equivalent to the boolean algebra $\{\bot, \top\}$, therefore it validates all classically provable propositions and is a model of $L$.


I should also note that monotonicity is a property of Kripke models, not of logics. A Kripke model is just the Heyting algebra of the upper sets of a preorder, by design. Of course the upper sets are upward closed.


You mention modal logics, but for those we need to augment Kripke models with an accessibility relation, and in any case, monotonicity still holds by virtue of what a Kripke model is.


