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In the TV show Only Fools and Horses a character called Trigger has a sweeping brush. He is proud that he has had the same brush for 17 years. However he reveals he's replaced both the head and handles many times over the years (hint: humour). In other words, though the object to him remains the same object, every part of it has been substituted many times.

It seems to me that there ought to be a word or phrase to describe this concept in philosophy, because it goes to the heart of the problem of identity of all objects both inanimate and animate.

What is that word or term?

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You're looking for the ship of Theseus:

The ship of Theseus, also known as Theseus's paradox, is a paradox that raises the question of whether an object which has had all its components replaced remains fundamentally the same object. The paradox is most notably recorded by Plutarch in Life of Theseus from the late 1st century. Plutarch asked whether a ship which was restored by replacing each and every one of its wooden parts, remained the same ship.

The Wikipedia page lists some variants, as well.

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As Labreur states, this problem is called The Ship of Theseus.

Spinoza, in his Ethics, provided a solution to this problem on the basis of his Descartian (amongst other influences) of his philosophy.

When a body or individual loses some of its parts which are replaced by others of the same nature, the body or individual will retain its nature as before, with no change in its form.

Bodies are not distinguished by difference of substance; what constitutes the form of the individual consists in the union of the bodies that are its parts (by The Definition); and this union is retained even if a continual change of constituent bodies occurs. So the individual will retain its nature, as before, through such a change.

  • Daniel C:It was a good question put in a witty way – Mozibur Ullah Dec 24 '13 at 19:23

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