> At what point is a person a person? "at birth" seems arbitrary

All lines dividing up creation into named categories and entities are arbitrary.

That one is arbitrary is not an argument against it and in favor of a different (and equally arbitrary) one.

Can you explain what you mean? Are zygote, embryos, foetus and various categorisations of foetal development arbitrary? Or do you mean "arbitrary" specifically in terms of personhood (i.e. there might be a clearly distinguishable stage where the heart starts beating, but this bears no obvious relationship to personhood)

> That one is arbitrary is not an argument against it and in favor of a different (and equally arbitrary) one.

True, but that's not my aim. I am instead attacking the notion of personhood, which was a dependency of your own argument, not mine. If there are no meaningful indicators of personhood, the only way to avoid aborting a person is to set a limit at a point where a foetus is still likely to be a non-person, because detailed gauges of personhood are otherwise unavailable.

I think a better direction for this argument would be to burden you with the question; why aren't we allowed to terminate once a child is born? pre-empting the argument "but then there are no health risks / autonomy considerations" - does that mean all abortions are done for purely health-risk reasons?