Forcing and Interpolation in First-Order Hybrid Logic with rigid symbols

In this paper, we establish an analogue of Craig Interpolation Property for a many-sorted variant of first-order hybrid logic. We develop a forcing technique that dynamically adds new constants to the underlying signature in a way that preserves consistency, even in the presence of models with possibly empty domains. Using this forcing method, we derive general criteria that are sufficient for a signature square to satisfy both Robinson's consistency and Craig interpolation properties.