  This paper gives a simple theory of inference to logically reason symbolic
knowledge fully from data over time. We take a Bayesian approach to model how
data causes symbolic knowledge. Probabilistic reasoning with symbolic knowledge
is modelled as a process of going the causality forwards and backwards. The
forward and backward processes correspond to an interpretation and inverse
interpretation of formal logic, respectively. The theory is applied to a
localisation problem to show a robot with broken or noisy sensors can
efficiently solve the problem in a fully data-driven fashion.
