In Kripke semantics for first-order modal logic and intuitionistic logic, the property holding of models that for any two possible worlds $w$ and $w^{\prime }$, the domains of quantification $D_w$ and $D_{w^{\prime }}$ are identical, i.e., $D_w=D_{w^{\prime }}$. A constant domain model contrasts with variable domain models, in which distinct possible worlds $w$ and $w^{\prime }$ may have distinct domains $D_w$ and $D_{w^{\prime }}$ where $D_w\ne D_{w^{\prime }}$. In the case of normal first-order modal logics, this semantic condition corresponds to accepting both the Barcan and converse Barcan formulae:

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

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

as axioms. With respect to first-order intuitionistic logic, constant domain models are characterized by the constant domain axiom:

• • $\forall x\left(\phi \left(x\right)\vee \psi \right)\to \left(\forall x\phi \left(x\right)\vee \psi \right)$ where $x$ is not free in $\psi$