First-order definability in modal logic

Journal of Symbolic Logic 40 (1):35-40 (1975)
It is shown that a formula of modal propositional logic has precisely the same models as a sentence of the first-order language of a single dyadic predicate iff its class of models is closed under ultraproducts. as a corollary, any modal formula definable by a set of first-order conditions is always definable by a single such condition. these results are then used to show that the formula (lmp 'validates' mlp) is not first-order definable
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DOI 10.2307/2272267
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Robert Goldblatt (1989). Varieties of Complex Algebras. Annals of Pure and Applied Logic 44 (3):173-242.

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