  Miller recently proposed a definition of contrastive (counterfactual)
explanations based on the well-known Halpern-Pearl (HP) definitions of causes
and (non-contrastive) explanations. Crucially, the Miller definition was based
on the original HP definition of explanations, but this has since been modified
by Halpern; presumably because the original yields counterintuitive results in
many standard examples. More recently Borner has proposed a third definition,
observing that this modified HP definition may also yield counterintuitive
results. In this paper we show that the Miller definition inherits issues found
in the original HP definition. We address these issues by proposing two
improved variants based on the more robust modified HP and Borner definitions.
We analyse our new definitions and show that they retain the spirit of the
Miller definition where all three variants satisfy an alternative unified
definition that is modular with respect to an underlying definition of
non-contrastive explanations. To the best of our knowledge this paper also
provides the first explicit comparison between the original and modified HP
definitions.
