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

Modifications of the Miller definition of contrastive (counterfactual) explanations