  The tension between deduction and induction is perhaps the most fundamental
issue in areas such as philosophy, cognition and artificial intelligence (AI).
The deduction camp concerns itself with questions about the expressiveness of
formal languages for capturing knowledge about the world, together with proof
systems for reasoning from such knowledge bases. The learning camp attempts to
generalize from examples about partial descriptions about the world. In AI,
historically, these camps have loosely divided the development of the field,
but advances in cross-over areas such as statistical relational learning,
neuro-symbolic systems, and high-level control have illustrated that the
dichotomy is not very constructive, and perhaps even ill-formed. In this
article, we survey work that provides further evidence for the connections
between logic and learning. Our narrative is structured in terms of three
strands: logic versus learning, machine learning for logic, and logic for
machine learning, but naturally, there is considerable overlap. We place an
emphasis on the following "sore" point: there is a common misconception that
logic is for discrete properties, whereas probability theory and machine
learning, more generally, is for continuous properties. We report on results
that challenge this view on the limitations of logic, and expose the role that
logic can play for learning in infinite domains.
