It is time to re-tell the famous story of Tibbles. Tibbles is a cat, who has just lost a tiny piece of hair Let T be all the parts of Tibbles - flesh and bones and internal organs and all - plus the now-detached piece of hair. Let T* be all the parts that now constitute Tibbles - namely all the parts that constitute T less the hair. Then (ignoring for now any problems about souls and animated bodies)
(1) Tibbles previously was identical with T.
(2) Tibbles now is identical with T*.
(3) T is not identical with T*
(4) Therefore (by substitutivity) Tibbles is not identical with T. Contradiction with (1) above.
Well, supposedly a contradiction. Tibbles is not now identical with T, but this does not contradict (1), which says that Tibbles was then identical with T. Is there any contradiction in being identical with some A in the past, and yet not being identical with A in the present? Does "A was identical with B" always entail "A is identical with B"? More later