For concreteness let us consider the topos of quivers. The inhabited graphs, i.e. those satisfying $\top\vdash (\exists x)\top$ are precisely the graphs $G$ having an edge since the formula on the right is interpreted as the image of the composite $G\overset{id}{\hookrightarrow} G\to 1$ but $1$ being the loop graph and in order to validate the sequent, its loop must be contained in the image requiring an edge in the source. This notion of being inhabited is stronger than being non-empty $\neq\empty$ but weaker as having a (global) element$1\to G$ which corresponds to graphs containing a loop. A uninhabited graph satisfying $(\exists x)\top\vdash \bot$ is necessarily empty whence edgeless graphs with a node are internally neither inhabited nor uninhabited.