Describes first-order Kripke models for modal logic in which the domains of quantification of the possible worlds are anti-monotonic, that is, for any possible world $w$ and $w^{\prime }$ with domains $D_w$ and $D_{w^{\prime }}$, the following condition holds:

If $wR{w}^{\prime }$ then $D_{w^{\prime }} \subseteq D_w.$

The class of decreasing domain Kripke models for a normal modal logic corresponds to the validity of the Barcan formula

• • $\forall x\square \phi \left(x\right)\to \square \forall x\phi \left(x\right)$

Decreasing domains may be contrasted with the notion of an increasing domain; both are special cases of variable domain models.