Humans can combine symbols to generate new meanings. Here, we studied the regional neural mechanisms that might make this possible. We asked participants to combine two discrete, symbolic features (a shape and a color) to make a novel spatial inference. Blood-oxygen-level-dependent (BOLD) data suggested that the hippocampus encoded elementary visual attributes in a high-dimensional, parallel format that permitted flexible individuation. In the ventromedial prefrontal cortex (vmPFC), posterior parietal cortex (PPC), and primary visual cortex (V1), neural patterns for novel stimuli (composites) could be predicted as a linear combination of signals for familiar stimuli (elements). In the vmPFC, this composition occurred in a high-dimensional format, but in PPC and V1, it took place in a low-dimensional, spatial frame of reference that was aligned with the response space. These data offer new insights into the neural circuits underlying compositional generalization.