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.

翻译：米勒近期基于著名的哈尔彭-珀尔因果定义与（非对比性）解释定义，提出了对比（反事实）解释的定义。关键在于，米勒定义基于原始的哈尔彭-珀尔解释定义，但该原始定义此后已被哈尔彭本人修改——推测原因在于原始定义在许多标准案例中会产生违反直觉的结果。博尔纳随后提出第三种定义，指出这种修改后的哈尔彭-珀尔定义亦可能产生反直觉结果。本文证明米勒定义继承了原始哈尔彭-珀尔定义所存在的问题。我们通过提出两个改进变体来解决这些问题，这两个变体分别基于更稳健的修改版哈尔彭-珀尔定义与博尔纳定义。本文分析新定义，并证明它们保留了米勒定义的精神——所有三个变体均满足一种替代性的统一定义，该定义基于底层非对比性解释定义而具有模块化特性。据我们所知，本文亦首次对原始与修改版哈尔彭-珀尔定义进行了明确比较。