Any model of the world a robot constructs on the basis of its sensor data is necessarily both incomplete, due to the robot’s limited window on the world, and uncertain, due to sensor and motor noise. This paper proposes a logic-based framework in which such models are constructed through an abductive process whereby sensor data is explained by hypothesising the existence, locations, and shapes of objects. Symbols appearing in the resulting explanations acquire meaning through the theory, and yet are grounded by the robot’s interaction with the world. The proposed framework draws on existing logic-based formalisms for representing action, continuous change, space, and shape. Noise is treated as a kind of non-determinism, and is dealt with by a consistency-based form of abduction.