I’ve been a bit disingenuous so far (or perhaps just dishonest), because the authors do not ever actually mention the paradox of the heap. I do think it is an unspoken foundation of their argument, but if that was all they had they wouldn’t have had something worth publishing. Their actual justification is more subtle: they define ‘person’ as an individual that is capable of attributing “value to [her] own existence” such that “being deprived of this existence represents a loss to her”, and then claim that neither fetuses nor newborns meet these criteria.
A rigorous and sensible definition to be sure, but I don’t think it’s capable of gaining the cultural traction that would be required to get people to accept infanticide. A simpler and broader concept is needed, and I think instinct does the work that such analytical wrangling about personhood never will.
For the record, they reach what is a compelling conclusion, if the premise is accepted as doing the work they claim it does: since “hardly can a newborn be said to have aims, as the future we imagine for it is merely a projection of our minds on its potential lives”, then “therefore, the rights and interests of the actual people involved [parents and the like, who do have aims] should represent the prevailing consideration in a decision about abortion and after-birth abortion.” (This is actually similar to the reasoning for infanticide given by pre-modern cultures, which I discuss here.)
So, I confess that I simplified the discussion to the paradox of the heap so that I could make my point without losing too many people in the process, and thus misrepresented the authors’ position somewhat. I figure though that anyone who cares will read the original, and anyone who doesn’t care is better served by making it to the end of my article than getting bogged down in this more philosophically esoteric and difficult-to-grasp concept.
But I think the authors’ concept does have some use. People who believe that personhood comes into existence gradually still need some guidelines on how to think about the person coming into existence. Being capable of having aims and of valuing ones own existence is a worthy consideration.