This simple statement points to the real problem. When people think about Soul, they imagine it as something solid and material. For most people the Self and the Soul and synonymous.
The problem is that the Self isn't a thing, and neither is the Soul. A better term would be Selfing, a gerund. A gerund is a verb pressed into service as a noun. When people perceive their own perception, they are said to be self-aware. This process is Selfing. It's creating a thing out of an activity. Selfing is a process we all engage in moment-by-moment. This is easy to verify as doctors have demonstrated that there are times spent in deep anethesia when no selfing occurs. What happens to the soul when there is no brain activity?
The idea of the solid, substantial soul has been embraced by religion because it provides a mechanism to sustain Selfing after death. Apparently, God must create the conditions that mirror the brain in heaven in order to keep Selfing active. Or at least, that's what you need to believe if you want to believe in an eternal soul with endless Selfing. It seems like wishful thinking to me, but since no experiment can ever provide data about what happens after death, this is an area where science really can't make a statement.
The Terri Schiavo case revolved around the potential for Selfing. Medical experts concluded there was no selfing occurring, and the physical systems required for selfing to occur were so damaged that it would never occur again. The Terri Schiavo that existed as a person was never going to shine through the vegetative body lying in that hospital bed.
The danger here is whether or not we want to accept the judgment of doctors on whether or not selfing is occurring. Do we have so much faith in our current scientific understanding of consciousness that we feel we can say with certainty whether a living being is still experiencing subjective existence? How much selfing needs to be going on in order to be a person? At what point to you cross the threshold where you lose your rights as a living being? Those are the real issues raised by Terri Shiavo, and I don't know that our current level of understanding provides solid answers, at least not solid enough to make life or death decisions.