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0. See: Rules for Users.
Yes. Because who doesn't like a good ship of Thesius analogy.
The you from 5 years ago was different from the you now but still considered you even though many cells have changed since then.
a man can never cross the same river twice since it is no longer the same river nor is he the same man
No but it wasn't the same axe you inherited ten seconds after you inherited it, either.
A broken handle means a trip to the hardware store. You explain the red stains away as bbq sauce.
The dwarfs of ankh-morpork say yes
Did he upgrade the head and handle? If so, it's still your grandpa's axe. Did you upgrade the axe? It might now be an axe of your own making. If "same" is extended not just to refer to the materials, but refers to fourth-dimensional quantum axe states, the axe stops being your grandpas axe the moment your grandpa isn't chopping with it, but becomes his axe any time he returns to use it.
But the axe has all the same memories and personality. It remembers the good and bad times. It tells you jokes and will help you move a refrigerator on the weekend
It was never really grampa’s axe in the first place—he was just licensing the technology.
It is our axe komrad