  We consider the problem of answering queries about formulas of propositional
logic based on background knowledge partially represented explicitly as other
formulas, and partially represented as partially obscured examples
independently drawn from a fixed probability distribution, where the queries
are answered with respect to a weaker semantics than usual -- PAC-Semantics,
introduced by Valiant (2000) -- that is defined using the distribution of
examples. We describe a fairly general, efficient reduction to limited versions
of the decision problem for a proof system (e.g., bounded space treelike
resolution, bounded degree polynomial calculus, etc.) from corresponding
versions of the reasoning problem where some of the background knowledge is not
explicitly given as formulas, only learnable from the examples. Crucially, we
do not generate an explicit representation of the knowledge extracted from the
examples, and so the "learning" of the background knowledge is only done
implicitly. As a consequence, this approach can utilize formulas as background
knowledge that are not perfectly valid over the distribution---essentially the
analogue of agnostic learning here.
