  Explaining sophisticated machine-learning based systems is an important issue
at the foundations of AI. Recent efforts have shown various methods for
providing explanations. These approaches can be broadly divided into two
schools: those that provide a local and human interpreatable approximation of a
machine learning algorithm, and logical approaches that exactly characterise
one aspect of the decision. In this paper we focus upon the second school of
exact explanations with a rigorous logical foundation. There is an
epistemological problem with these exact methods. While they can furnish
complete explanations, such explanations may be too complex for humans to
understand or even to write down in human readable form. Interpretability
requires epistemically accessible explanations, explanations humans can grasp.
Yet what is a sufficiently complete epistemically accessible explanation still
needs clarification. We do this here in terms of counterfactuals, following
[Wachter et al., 2017]. With counterfactual explanations, many of the
assumptions needed to provide a complete explanation are left implicit. To do
so, counterfactual explanations exploit the properties of a particular data
point or sample, and as such are also local as well as partial explanations. We
explore how to move from local partial explanations to what we call complete
local explanations and then to global ones. But to preserve accessibility we
argue for the need for partiality. This partiality makes it possible to hide
explicit biases present in the algorithm that may be injurious or unfair.We
investigate how easy it is to uncover these biases in providing complete and
fair explanations by exploiting the structure of the set of counterfactuals
providing a complete local explanation.
