The Ship of Theseus is a story used to illustrate how different systems for choosing which elements to use in defining the identity of a thing can disagree about whether or not that thing still exists under certain conditions. First it defines a thing's identity as the sum of its component parts (as a simple unstructured inventory), then shows how that contradicts with the usual intuition by gradually replacing those parts, showing that we actually tend to identify such things by higher patterns of their form and purpose.
According to Greek legend as reported by Plutarch,
“The ship wherein Theseus and the youth of Athens returned [from Crete] had thirty oars, and was preserved by the Athenians down even to the time of Demetrius Phalereus, for they took away the old planks as they decayed, putting in new and stronger timber in their place, insomuch that this ship became a standing example among the philosophers, for the logical question of things that grow; one side holding that the ship remained the same, and the other contending that it was not the same.”
Plutarch thus questions whether the ship would remain the same if it were entirely replaced, piece by piece. As a corollary, one can question what happens if the replaced parts were used to build a second ship. Which, if either, is the original Ship of Theseus?
The Greek philosopher Heraclitus is notable for his unusual view of identity. Arius Didymus quoted him as saying:
“Upon those who step into the same rivers, different and again different waters flow.”
Plutarch also informs us of Heraclitus' claim about stepping twice into the same river, citing that it cannot be done because "it scatters and again comes together, and approaches and recedes."
John Locke proposed a scenario regarding a favorite sock that develops a hole. He pondered whether the sock would still be the same after a patch was applied to the hole. If yes, then, would it still be the same sock after a second patch was applied? Indeed, would it still be the same sock many years later, even after all of the material of the original sock has been replaced with patches?
Grandfather's old axe
"Grandfather's old axe" is a colloquial expression of unknown origin describing something of which little original remains: "it's had three new heads and four new handles but it's still the same old axe." The phrase has also been used in banter as in: "This is George Washington's original axe...", while holding up an apparently new axe. This example is used explicitly to explain significant points of the plot in The Fifth Elephant by Terry Pratchett. A similar example was also seen in Only Fools and Horses, where Trigger (a central character) won an award for using the same broom to sweep the streets for twenty years, even though he’d replaced the head 17 times and the handle 14 times. The meaning of the expression and its relationship to murder are questioned in the opening of the novel John Dies at the End by David Wong.
One can think of many examples of objects which might fall prey to Theseus's paradox: buildings and automobiles for example can undergo complete replacement whilst still maintaining some aspect of their identity. Businesses, colleges and universities often change addresses and residences, thus completely "replacing" their old material structure for a new one, yet keeping the same purpose and often the same people that keep the organization functioning as it was. If two businesses merge, their identities merge (or one is consumed by the other). Similarly, the human body constantly creates new cells as old cells die. Average age of cells in an adult body may be less than 10 years.
If we relate identity to actions and phenomena, identity becomes even harder to grasp. Depending upon one's chosen perspective of what identifies or continues a hurricane, if a hurricane Evan collapses at a particular location and then one forms again at or near the same location, a person may be totally consistent to either choose to call the latter mentioned hurricane the same as the former (as in "Evan" was reinvigorated), or choose to call the latter a new hurricane "Frank" or "Georgia".
One could also see the bands Napalm Death, Zao and The Little River Band as contemporary examples of Theseus's paradox. Both band's current line-ups contain none of the founding members, yet they continue to use the same name.
A somewhat, more layman's example is in most Herbie films. In these films, when any part of the protagonist (a "living" car named Herbie) is replaced, no one thinks anything of it, however; if one were to replace all of Herbie's parts at once, which part would contain his "soul", and if a certain car part did contain his soul, would it be transferred if the part were attached to another car?