I love Sorites Paradoxes. :)

I'll take a shot at this, though. It's alive in the technical sense at the moment of conception, because the two cells that made it are alive. However, its personhood is called much further into question.

I'd propose two cut-offs for determining personhood. The first, and more reasonable one, is this: if the fetus can survive outside of the womb with the help of modern science, at such a time when its at least 50% likely that the fetus will be viable until typical infancy, then that's a person.

The second one is a bit more strict. When the fetus has developed to the point where its brain is at least 80% developed and it can breathe entirely on its own, then it's a person.

I'm inclined to accept the first one because we might as well utilize modern medicine, and because accepting the second definition would push dangerously into third-trimester abortion territory.

Edit: Empiricism. Right. Well, personhood is most commonly defined as the state or condition of being a person, and a person is distinct from a human in that it is a human with unique traits and behaviors. Since the brain is literally what makes us human, a fair percentage of its development should be a trait of personhood. Further, since being a person requires being alive (a human corpse is still a human but no longer a person to anyone other than the bereaved), the ability to at least partially sustain one's own life must be a trait of personhood. Machinery/modern medicine can be used as crutches for this purpose for obvious reasons (the most obvious being that if we didn't include their usage as acceptable, then people who were using wheelchairs or pacemakers or respirators wouldn't be considered people).

Thus, if a human fetus can survive outside of the womb, even with large amounts of medical assistance, I can consider it a person. The survival rate should be over 50% simply because of majority probability. If a fetus cannot currently survive to adequate capacity outside of the womb, it is not yet a person.