As Slughorn states, “Then, even if one’s own body is attacked or destroyed, one cannot die, for part of the soul remains earthbound and undamaged” (Book VI, Chapter 23, page 497). So if you can’t die because part of your soul is living somewhere else, how can a Horcrux be destroyed to begin with, since other Horcruxes still exist? It seems to me that, in order to truly destroy Voldemort’s soul, you have to destroy all seven pieces at once. That doesn’t work, since the diary and ring have been destroyed, but I theorize we’ll get this last piece of information in Book VII.