  Two types of explanations have been receiving increased attention in the
literature when analyzing the decisions made by classifiers. The first type
explains why a decision was made and is known as a sufficient reason for the
decision, also an abductive explanation or a PI-explanation. The second type
explains why some other decision was not made and is known as a necessary
reason for the decision, also a contrastive or counterfactual explanation.
These explanations were defined for classifiers with binary, discrete and, in
some cases, continuous features. We show that these explanations can be
significantly improved in the presence of non-binary features, leading to a new
class of explanations that relay more information about decisions and the
underlying classifiers. Necessary and sufficient reasons were also shown to be
the prime implicates and implicants of the complete reason for a decision,
which can be obtained using a quantification operator. We show that our
improved notions of necessary and sufficient reasons are also prime implicates
and implicants but for an improved notion of complete reason obtained by a new
quantification operator that we also define and study.
