The language of space and spatial relations is a rich source of abstract semantic structure. We develop a probabilistic model that learns to understand utterances that describe spatial configurations of objects in a tabletop scene by seeking the meaning that best explains the sentence chosen. The inference problem is simplified by assuming that sentences express symbolic representations of (latent) semantic relations between referents and landmarks in space, and that given these symbolic representations, utterances and physical locations are conditionally independent. As such, the inference problem factors into a symbol-grounding component (linking propositions to physical locations) and a symbol-translation component (linking propositions to parse trees). We evaluate the model by eliciting production and comprehension data from human English speakers and find that our system recovers the referent of spatial utterances at a level of proficiency approaching human performance.